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Additional Topics A-1 Introduction to Power Functions . . . . . . . . . . . . . . . . . . . . . . . . AT2 A-2 Piecewise and Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . AT5 A-3 Technology Lab: Graphing to Solve Equations . . . . . . . . . . . . . .AT9 A-4 Patterns and Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AT10 A-5 Linear and Nonlinear Rates of Change . . . . . . . . . . . . . . . . . . AT14 A-6 Reasoning and Counterexamples . . . . . . . . . . . . . . . . . . . . . . AT18 ADDITIONAL TOPIC A-1 Objective Analyze the characteristics of power functions. Introduction to Power Functions A power function is a function that can be written in the form f (x) = ax n, where a and n are real numbers and a ≠ 0. This lesson will only address power functions for which n is a positive integer. Power Functions Vocabulary power function n=1 n=2 y = ax EXAMPLE 1 n=3 y = ax 2 y = ax n=4 3 y = ax n=5 4 y = ax 5 Graphing Power Functions Graph the power function y = x 5, and then describe the graph. Step 1 Make a table of values. x x5 = y -2 (-2) 5 = -32 -1 (-1) 5 = -1 0 05 = 0 1 15 = 1 2 2 5 = 32 The ordered pairs (−2, −32), (−1, −1), (0, 0), (1, 1) and (2, 32) lie on the graph. Step 2 Sketch the graph by plotting the ordered pairs from the table. 40 y 20 -4 -2 0 2 4x -20 -40 From left to right, the graph increases. It crosses the x-axis once, at the origin. 1. Graph the power function y = x 4, and then describe the graph. AT2 Additional Topics Substituting any real value of x into a power function results in a real number. Therefore, the domain of any power function is all real numbers. The range of a power function depends on the degree of the function. The degree of the power function f (x) = ax n is n. The table summarizes the possibilities for the range and the maximum and minimum values of a power function. Range, Maximum Value, and Minimum Value Odd Degree The range is all real numbers. There is no maximum or minimum value. EXAMPLE 2 Even Degree, a > 0 Even Degree , a < 0 The graph opens downward. The maximum value is 0. The range is y ≤ 0. The graph opens upward. The minimum value is 0. The range is y ≥ 0. Finding Maximum or Minimum Values Give the domain and range of each power function. Then give the minimum or maximum value, if any. A f (x) = -0.5x 7 The degree, 7, is odd. The domain is all real numbers. The range is all real numbers. There is no minimum or maximum value. B f (x) = -2x 4 The degree, 4, is even, and a < 0. The domain is all real numbers. The range is y ≤ 0. The maximum value is 0. Check The graph supports the answer. 1 x6 C f (x) = _ 20 The degree, 6, is even, and a > 0. The domain is all real numbers. The range is y ≥ 0. The minimum value is 0. Check The graph supports the answer. Give the domain and range of each power function. Then give the minimum or maximum value, if any. 1 x 10 2a. f (x) = -x 8 2b. f (x) = -2.5x 5 2c. f (x) = _ 4 A-1 Introduction to Power Functions AT3 A-1 Exercises Graph each power function, and then describe the graph. 1. y = x 3 2. y = x 6 3. y = x 7 Give the domain and range of each power function. Then give the minimum or maximum value, if any. 1 x 16 4. f(x) = -2x 11 5. f(x) = -_ 6. f(x) = 0.05x 8 8 7. Geometry A cube has sides of length x. a. Write a power function f that gives the area of each face of the cube. b. Write a power function g that gives the surface area of the cube. x 8. Geometry The volume V of a sphere of radius x is given by the 3 power function V(x) = __43 πx . Use a graphing calculator to determine the radius of a sphere that has a volume of 100 cm 3. Round your answer to the nearest tenth of a centimeter and explain your method. Determine whether each statement is always, sometimes, or never true. Explain. 9. A power function has two x-intercepts. 10. A power function has a minimum value of 1. 11. The graph of a power function that passes through the point (a, b) also passes through the point (−a, −b). 12. A power function that has a minimum value has an even degree. The end behavior of a function is a description of the function’s values as x increases or decreases. For example, you can describe the end behavior of f(x) = x 3 as follows: As x increases, f(x) increases. As x decreases, f(x) decreases. Describe the end behavior of each power function. 1 x7 13. f (x) = -4x 4 14. f (x) = _ 15. f (x) = -2x 9 3 16. Physics A certain metal sphere sinks in a lake according to the function d(t) = 2.1t 2 where t is the time in seconds and d(t) is the depth of the sphere in meters. Sketch a graph of the function. Then give a reasonable domain and range for the function when the sphere is dropped in a lake with a maximum depth of 100 meters. 17. Determine whether the power function f(x) = -3x 3 is increasing or decreasing on the interval -5 ≤ x ≤ -1. 18. Critical Thinking Is a direct variation a power function? Explain why or why not. 19. Graph f(x) = x 3, g(x) = x 3 + 1, and h(x) = x 3 - 2. Make a conjecture, in terms of a transformation, about the effect of b on the graph of f(x) = ax 3 + b. AT4 Additional Topics ADDITIONAL TOPIC A-2 Objective Recognize, identify, and graph piecewise and step functions. Vocabulary piecewise function step function greatest-integer function Piecewise and Step Functions A piecewise function is a function that is a combination of one or more functions. The rule for a piecewise function may be different for different parts of the function’s domain. 4 For example, the absolute value function y = ⎪x⎥ is a piecewise function and can be defined by the following rule: y y = |x| 2 x -4 0 -2 2 4 -2 ⎧-x if x < 0 y=⎨ ⎩ x if x ≥ 0 -4 To evaluate a piecewise function for a given input value, find the interval of the domain that contains the input value and apply the rule for that interval. EXAMPLE 1 Evaluating a Piecewise Function ⎧ x2 + 1 if x < 2 For f (x) = ⎨ , evaluate f (x) for x = −3 and x = 6. ⎩ -3x + 4 if x ≥ 2 2 f (-3) = (-3) + 1 = 10 Because -3 < 2 , use the rule for x < 2. f (6) = -3(6) + 4 = -14 Because 6 ≥ 2 , use the rule for x ≥ 2. ⎧3x - 3 if x < -1 1. For g(x) = ⎨ , evaluate g(x) for x = –4 and x = –1. ⎩ -x 2 if x ≥ -1 EXAMPLE 2 Graphing a Piecewise Function ⎧ -3 Graph f (x) = ⎨ ⎩ x2 + 1 if x < 0 if x ≥ 0 . This function is composed of two pieces. For x < 0, the function is the constant function f (x) = –3. Draw a horizontal ray to the left of (0, –3). Draw an open circle at (0, –3) to show that this point is not part of the graph. For x ≥ 0, the function is the quadratic function 2 f (x) = x + 1. Draw half of a parabola to the right of (0, 1). Draw a solid circle at (0, 1) to show that this point is part of the graph. ⎧ 2x 2. Graph g(x) = ⎨ ⎩ x -1 4 y 2 x -4 -2 0 2 4 -2 -4 if x < 1 . if x ≥ 1 A-2 Piecewise and Step Functions AT5 The graph and table at right give the price of admission to a theme park. The function is defined differently over different domain intervals (age groups), so the function is a piecewise function. Regular admission $25 Youth (under 15) $10 Seniors (55 and older) $20 Theme Park Admission Prices 30 Price ($) The function that describes the theme park admission prices is a step function. A step function is a piecewise function that is constant over each interval in its domain. Theme Park Admission Prices 20 10 0 3 20 30 40 Age (yr) 50 60 10 12 Parking Fees Step Functions The graph shows parking fees at a garage. Create a table and a verbal description to represent the graph. Step 1 Create a table. y 6 Fee ($) EXAMPLE 10 Use the endpoints of the segments of the graph to identify the intervals of the domain. 4 2 x 0 2 4 6 8 Hours Parking Fees Hours x Fee y ($) 0<x≤2 3 2<x≤4 4 4<x≤6 5 x>6 6 On the graph, an open circle means that a value is not included in the interval. A solid circle means that a value is included in the interval. Step 2 Write a verbal description. For any amount of time up to 2 hours, the parking fee is $3. For more than 2 hours and no more than 4 hours, the fee is $4. For more than 4 hours and no more than 6 hours, the fee is $5. For more than 6 hours, the fee is $6. Shipping Costs 6 Cost ($) 3. The graph shows the shipping costs for books that are ordered online. Create a table and a verbal description to represent the graph. y 4 2 x 0 AT6 Additional Topics 4 8 12 16 Weight (oz) 20 The greatest-integer function is a piecewise function, written f (x) = x, in which the number x is rounded down to the greatest integer that is less than or equal to x. For example, f (3.14) = 3.14 = 3. The graph of the greatestinteger function shows that it is a step function. 4 f(x) = x y 2 x -4 -2 0 2 4 -2 -4 EXAMPLE 4 Using the Greatest-Integer Function Write a function that gives the number of magazines that you can buy with x dollars if a magazine costs $3.75. Then use the function to find the number of magazines you can buy with $20. x The number of magazines you can buy with x dollars is ____ , rounded 3.75 down to the greatest integer that is less than or equal to this quotient. x . The function is f (x) = _ 3.75 To find how many magazines you can buy with $20, evaluate f (x) for x = 20. 20 = 5.__ f (20) = _ 3 = 5 3.75 You can buy 5 magazines with $20. 4. Write a function that gives the number of movie tickets that you can buy with x dollars if a movie ticket costs $10.95. Then use the function to find the number of movie tickets you can buy with $50. A-2 Exercises Evaluate each function for x = −2 and x = 3. ⎧ 2x + 1 if x < 0 ⎧-x 2 if x ≤ -1 1. f (x) = ⎨ 2. g (x) = ⎨ ⎩ 5x if x > -1 ⎩-x + 2 if x ≥ 0 ⎧ x 2 + 1 if x < -3 3. h(x) = ⎨ ⎩ 2x - 5 if x ≥ -3 Evaluate each function for x = −1 and x = 2. ⎧4x -1 if x < -1 5. f (x) = -x + 5 if -1 ≤ x < 2 ⎩ 3x if x ≥ 2 ⎨ ⎧ -5 4. f (x) = ⎨ if x < 3 ⎩ 3x + 1 if x ≥ 3 2 ⎧x 2 ⎨ 6. f (x) = 4x + 6 1 _ ⎩ 2x if x ≤ -2 if -2 ≤ x ≤ 1 if x ≥ 1 Graph each function. ⎧ x -1 if x < 0 7. f (x) = ⎨ if x ≥ 0 ⎩4 ⎧ -1 if x ≤ -1 8. f (x) = 2x if -1< x ≤ 2 ⎩ x - 4 if x > 2 ⎨ A-2 Piecewise and Step Functions AT7 Create a table and a verbal description to represent each graph. 9. 10. Admission Prices y 6 Weeks Price ($) 6 Weeks of Paid Vacation 4 2 y 4 2 x 0 4 8 12 Age (yr) x 0 16 2 4 6 8 Years of employment 11. Write a function that gives the number of energy bars that you can buy with x dollars if an energy bar costs $1.35. Then use the function to find the number of energy bars you can buy with $10. x 12. The function g (x) = ____ gives the number of gel pens you can buy with x dollars. 3.95 What is the price of one gel pen? How many gel pens can you buy with $15? Write a piecewise function for each graph. 13. 4 14. y 4 15. y -4 -2 2 4 x x x -4 -2 0 y 2 2 2 0 4 2 4 -4 -2 0 -2 -2 -2 -4 -4 -4 2 4 16. Entertainment The cost of admission to a state fair is $4 for children less than 12 years old and $8 for everyone 12 or older. a. Write a function that gives the cost of admission for a person who is x years old. b. Graph the function. 17. Pet Care The table shows the recommended amount of dog food based on a dog’s weight. a. Write a function that gives the amount of dog food in cups for a dog that weighs x pounds. b. Graph the function. Recommended Amount of Dog Food Weight of dog (lb) Amount of food (c) Less than 10 1 __ At least 10 and no more than 20 1 More than 20 1__12 2 18. Travel Carolyn leaves her house and drives for 3 hours at a constant rate of 60 mi/h. Then she stops for 1 hour to have lunch. After lunch, she continues to drive away from her house at a constant rate of 60 mi/h for another 2 hours. Graph a piecewise function that gives Carolyn’s distance in miles from her house after x hours. 19. Critical Thinking What are the domain and range of the piecewise function ⎧ x if x < 0 f(x) = ⎨ ? ⎩ x 2 if x ≥ 0 AT8 Additional Topics A-3 Graphing to Solve Nonlinear Equations You can use a graphing calculator to solve nonlinear equations, including quadratic and exponential equations. Activity Use a graph to solve x 2 – x – 5 = –3. Enter the left side of the equation as Y1 and the right side as Y2. Press . The x-values of the points where the graphs intersect (where Y1 = Y2) are the solutions of the equation. Notice that there is more than one solution. To find the coordinates of an intersection point: Y1 = x2 - x - 5 and select 5:intersect. Press Press to select Y1. Press again to select Y2. Use Press Y2 = -3 to move the cursor close to the intersection point. and . One point of intersection is (–1, –3), so one solution is x = –1. Repeat these steps to find the coordinates of the second intersection point. This point is (2, –3), so the second solution is x = 2. The solutions of x 2 – x – 5 = –3 are –1 and 2. Check x 2 - x - 5 = -3 x 2 - x - 5 = -3 (-1) 2 - (-1) - 5 -3 (2)2 - 2 - 5 -3 1 + 1 - 5 -3 4 - 2 - 5 -3 -3 -3 -3 -3 Use a graph to solve each equation. 1. x 2 + 6x + 9 = 1 () x 2. 1.5(2) x = 6 3. -x 2 + 3x - 4 = -4 () x 1 =4 2 _ 1 =6 4. 2 _ 5. 2 x 2 + 2x - 12 = -8 6. _ 2 3 3 7. Critical Thinking Explain how you could use a graphing calculator to show that the equation x 2 - 2x + 4 = 1 has no real solutions. 8. Critical Thinking Could you use the method described in the activity to solve an equation like x 2 + 4x = x + 4? Explain. A- 3 Technology Lab AT9 ADDITIONAL TOPIC A-4 Patterns and Recursion Objective Identify and extend patterns using recursion. In a recursive pattern or recursive sequence, each term is defined using one or more previous terms. For example, the sequence 1, 4, 7, 10, 13, ... can be defined recursively as follows: The first term is 1 and each term after the first is equal to the preceding term plus 3. Vocabulary recursive pattern You can use recursive techniques to identify patterns. The table summarizes the characteristics of four types of patterns. Using Recursive Techniques to Identify Patterns Type of Pattern EXAMPLE 1 Characteristics Linear First differences are constant. Quadratic Second differences are constant. Cubic Third differences are constant. Exponential Ratios between successive terms are constant. Identifying and Extending a Pattern Identify the type of pattern. Then find the next three numbers in the pattern. A 4, 6, 10, 16, 24, ... Find first, second, and, if necessary, third differences. 4 6 10 16 24 You may need to use trial and error when identifying a pattern. If first, second, and third differences are not constant, check for constant ratios. +2 +4 +6 +8 +2 +2 +2 Second differences are constant, so the pattern is quadratic. Extend the pattern by continuing the sequence of first and second differences. 4 6 10 16 24 34 46 60 +2 +4 +6 +8 + 10 + 12 + 14 +2 +2 +2 The next three numbers in the pattern are 34, 46, and 60. B __1 , __1 , 2, 8, 32 8 2 Find the ratio between successive terms. 1 1 __ __ 2 8 32 8 2 ×4 ×4 ×4 ×4 Ratios between terms are constant, so the pattern is exponential. Extend the pattern by continuing the sequence of ratios. 1 1 __ __ 2 8 32 128 512 2048 8 2 ×4 ×4 ×4 ×4 ×4 ×4 ×4 The next three numbers in the pattern are 128, 512, and 2048. AT10 Additional Topics Identify the type of pattern. Then find the next three numbers in the pattern. 1a. 56, 47, 38, 29, 20, ... 1b. 1, 8, 27, 64, 125, ... You can use a similar process to determine whether a function is linear, quadratic, cubic, or exponential. Note that before comparing y-values, you must first make sure there is a constant change in the corresponding x-values. Using Recursive Techniques to Identify Functions Type of Function EXAMPLE Characteristics (Given a Constant Change in x-values) Linear First differences of y-values are constant. Quadratic Second differences of y-values are constant. Cubic Third differences of y-values are constant. Exponential Ratios between successive y-values are constant. 2 Identifying a Function The ordered pairs {(–4, –4), (0, 0), (4, 4), (8, 32), (12, 108)} satisfy a function. Determine whether the function is linear, quadratic, cubic, or exponential. Then find three additional ordered pairs that satisfy the function. Make a table. Check for a constant change in the x-values. Then find first, second, and third differences of y-values. +4 +4 +4 +4 x -4 0 4 8 12 y -4 0 4 32 108 +4 +4 +0 + 28 + 24 + 24 + 76 + 48 + 24 There is a constant change in the x-values. Third differences are constant. The function is a cubic function. To find additional ordered pairs, extend the pattern by working backward from the constant third differences. +4 +4 +4 +4 +4 +4 +4 In Example 2, the constant third differences are 24. To extend the pattern, first find each second difference by adding 24 to the previous second difference. Then find each first difference by adding the second difference below to the previous first difference. x -4 0 4 8 12 16 20 24 y -4 0 4 32 108 256 500 864 +4 +4 +0 + 24 + 24 + 28 + 48 + 24 + 76 + 148 + 72 + 24 + 244 + 364 + 96 + 24 + 120 + 24 Three additional ordered pairs that satisfy this function are (16, 256), (20, 500), and (24, 864). A-4 Patterns and Recursion AT11 Several ordered pairs that satisfy a function are given. Determine whether the function is linear, quadratic, cubic, or exponential. Then find three additional ordered pairs that satisfy the function. 2a. {(0, 1), (1, 3), (2, 9), (3, 19), (4, 33)} ⎧ 1 1 , 5, _ 1 , 7, _ 1 , 9, _ 1 ⎫ 2b. ⎨ 1, _ , 3, _ ⎬ 54 6 18 162 ⎭ ⎩ 2 ( )( )( A-4 )( )( ) Exercises Identify the type of pattern. Then find the next three numbers in the pattern. 1. 25, 28, 31, 34, 37, ... 2. 20, 45, 80, 125, 180, ... 3. 128, 64, 32, 16, 8, ... 3 , 1, 1_ 1, _ 1 , 1_ 1 , ... 5. _ 4 2 2 4 4. 4, 32, 108, 256, 500, ... 6. 0.3, 0.03, 0.003, 0.0003, 0.00003, ... 7. 127, 66, 29, 10, 3, ... 8. 2, 8, 18, 32, 50, ... Several ordered pairs that satisfy a function are given. Determine whether the function is linear, quadratic, cubic, or exponential. Then find three additional ordered pairs that satisfy the function. 9. {(3, 1), (5, –3), (7, –7), (9, –11), (11, –15)} 10. {(–1, –2), (2, 7), (5, 124), (8, 511), (11, 1330)} ⎧ 1 1 , 4, _ 1 , 5, _ 1 , 6, _ 1 ⎫ 11. ⎨ 2, _ , 3, _ ⎬ 32 4 8 16 64 ⎭ ⎩ 12. {(–3, –7), (0, 2), (3, –7), (6, –34), (9, –79)} ( )( )( )( )( ) 13. {(0, 600), (10, 480), (20, 384), (30, 307.2), (40, 245.76)} 14. {(–8, 2), (–5, 7), (–2, 12), (1, 17), (4, 22)} 15. Entertainment The table shows the cost of using an online DVD rental service for different numbers of months. Online DVD Rentals a. Determine whether the function that models the data is linear, quadratic, cubic, or exponential. Explain. b. Graph the data in the table. c. What do you notice about your graph? Why does this make sense? d. Predict the cost of the service for 18 months. Months Cost ($) 3 50 6 92 9 134 12 176 15 218 16. A student claimed that the function shown in the table is a quadratic function. Do you agree or disagree? Explain. x 3 7 10 14 17 y 2 6 12 20 30 +4 +6 +2 AT12 Additional Topics +8 +2 + 10 +2 17. Business The table shows the annual sales for a small company. a. Determine whether the function that models the data is linear, quadratic, cubic, or exponential. Explain. b. Suppose sales continue to grow according to the pattern in the table. Predict the annual sales for 2011, 2012, and 2013. c. If the pattern continues, in what year will annual sales be $17,000 greater than the previous year’s sales? Annual Sales Year Sales ($) 2006 513,000 2007 516,000 2008 521,000 2009 528,000 2010 537,000 18. Critical Thinking Use the table for the following problems. a. b. c. d. x 0 1 y 3 6 2 3 4 Copy and complete the table so that the function is a linear function. Copy and complete the table so that the function is a quadratic function. Copy and complete the table so that the function is an exponential function. For which of these three types of functions is there more than one correct way to complete the table? Explain. Use the description to write the first five terms in each numerical pattern. 19. The first term is 8. Each following term is 11 less than the term before it. 20. The first term is 1000. Each following term is 40% of the term before it. 21. The first two terms are 1 and 2. Each following term is the sum of the two terms before it. Make a table for a function that has the given characteristics. Include at least five ordered pairs. 22. The function is linear. The first differences are -3. 23. The function is quadratic. The second differences are 6. 24. The function is cubic. The third differences are 1. A recursive formula for a sequence shows how to find the value of a term from one or more terms that come before it. For example, the recursive formula a n = a n-1 + 3 tells you that each term is equal to the preceding term plus 3. Given that a 1 = 5, you can use the formula to generate the sequence 5, 8, 11, 14, ... . Write the first four terms of each sequence. 25. a n = a n-1 + 2; a 1 = 12 26. a n = a n-1 - 7; a 1 =16 27. a n = 2a n-1; a 1 = 4 28. a n = 0.6a n-1; a 1 = 100 29. a n = 5a n-1 - 2; a 1 = 0 30. a n = (a n-1)2; a 1 = -2 31. A recursive function defines a function for whole numbers by referring to the value of the function at previous whole numbers. Consider the recursive function f (n) = f (n – 1) + 5 with f (0) = 1. a. According to the formula, f (1) = f (0) + 5. What is the value of f (1)? b. Use the formula to find f (2), f (3), f (4), and f (5). c. Graph f (n) by plotting points at x = 0, x = 1, x = 2, x = 3, x = 4, and x = 5. d. What do you notice about your graph? What does this tell you about f (n)? A-4 Patterns and Recursion AT13 ADDITIONAL TOPIC A-5 Objectives Identify linear and nonlinear rates of change. Compare rates of change. EXAMPLE Linear and Nonlinear Rates of Change Recall that a rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. change in dependent variable rate of change = ___ change in independent variable The table shows the price of one ounce of gold in 2005 and 2008. The year is the independent variable and the price is the dependent variable. The rate of change is 870-513 357 ________ = ___ = 119, or $119 per year. 2008-2005 3 1 Price of Gold Year Price ($/oz) 2005 513 2008 870 Identifying Constant and Variable Rates of Change Determine whether each function has a constant or variable rate of change. A {(0, 0), (1, 4), (3, 8), (6, 8), (8, 6)} Find the ratio of the amount of change in the dependent variable y to the corresponding amount of change in the independent variable x. +1 +2 +3 +2 x y 0 0 1 4 3 8 6 8 8 6 +4 +4 +0 The rates of change are 4 -2 __ = 4, __4 = 2, __0 = 0, and ___ = –1. 1 2 3 2 The function has a variable rate of change. -2 B {(0, 1), (1, 2), (4, 5), (6, 7), (7, 8)} Find the ratio of the amount of change in the dependent variable y to the corresponding amount of change in the independent variable x. x y +1 0 1 +1 The rates of change are +3 1 2 1 __ = 1, __3 = 1, __2 = 1, and __1 = 1. 4 5 +3 6 7 +2 The function has a constant rate of change. 7 8 +2 +1 1 3 2 1 +1 Determine whether each function has a constant or variable rate of change. 1a. {(–3, 10), (0, 7), (1, 6), (4, 3), (7, 0)} 1b. {(–2, –3), (2, 5), (3, 7), (5, 9), (8, 12)} AT14 Additional Topics The functions in Examples 1A and 1B are graphed below. Example 1A (variable rate of change) Example 1B (constant rate of change) 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Recall from Chapter 5 that a function is a linear function if and only if the function has a constant rate of change. The graph of such a function is a straight line and the rate of change is the slope of the line, as in Example 1B. A function with a variable rate of change, as in Example 1A, is a nonlinear function. Examples of nonlinear functions include quadratic functions and exponential functions. EXAMPLE 2 Identifying Linear and Nonlinear Functions Use rates of change to determine whether each function is linear or nonlinear. A x y +2 -2 0 +1 0 1 1 1.5 4 3 10 6 +3 +6 B +1 +4 + 0.5 +4 + 1.5 -2 +3 +4 x y -6 18 -2 - 16 2 2 2 +0 0 0 4 8 -2 +8 Find the rates of change. Find the rates of change. 0.5 = _ 1.5 = _ 3 =_ 1 _ 1 _ 1 1 _ _ 2 1 2 3 2 6 2 There is a constant rate of change, __12 , so this function is linear. 0 =0 _ -16 = -4 _ -2 = 1 _ 4 4 -2 The rates of change are not constant, so this function is nonlinear. 8 =_ 1 _ 4 2 Use rates of change to determine whether each function is linear or nonlinear. 2a. x y -2 1 _ 4 -1 1 _ 2 0 2b. x y -5 3 -1 3 1 3 1 3 3 3 8 7 3 4 16 When you are given a verbal description of a function, you can determine whether the function is linear or nonlinear by making a table of values and examining the rates of change. You can compare two functions by comparing their rates of change. A-5 Linear and Nonlinear Rates of Change AT15 EXAMPLE 3 Physical Science Application Two water tanks contain 512 gallons of water each. Tank A begins to drain, losing half of its volume of water every hour. Tank B begins to drain at the same time and loses 40 gallons of water every hour. Identify the function that gives the volume of water in each tank as linear or nonlinear. Which tank loses water more quickly between hour 4 and hour 5? Use the verbal descriptions to make a table for the volume of water in each tank. Time (h) 0 1 2 3 4 5 Water in Tank A (gal) 512 256 128 64 32 16 Time (h) 0 1 2 3 4 5 Water in Tank B (gal) 512 472 432 392 352 312 For tank A, the rates of change are –256, –128, –64, –32, and –16, so the rate of change is variable and the function is nonlinear. For tank B, the rates of change are all –40, so the rate of change is constant and the function is linear. Between hours 4 and 5, the volume of water in tank A decreases at a rate of 16 gallons per hour. The volume of water in tank B decreases at a rate of 40 gallons per hour. Tank B loses water more quickly. 3. Reka and Charlotte each invest $500. Each month, Charlotte’s investment grows by $25, while Reka’s investment grows by 5% of the previous month’s amount. Identify the function that gives the value of each investment as linear or nonlinear. Who is earning money more quickly between month 3 and month 4? A-5 Exercises Use rates of change to determine whether each function is linear or nonlinear. 1. 2. 3. 4. AT16 Additional Topics x 4 5 7 10 12 y –2 –1 1 4 6 x –2 3 4 6 8 y –4 6 8 14 20 x 0 3 9 12 18 y 14 12 8 6 2 x –8 –6 –4 –2 0 y –3 1 3 5 9 5. Hobbies Caitlin and Greg collect stamps. Each starts with a collection of 50 stamps. Caitlin adds 15 stamps to her collection each week. Greg adds 1 stamp to his collection the first week, 3 stamps the second week, 5 stamps the third week, and so on. Identify the function that gives the number of stamps in each collection as linear or nonlinear. Which collection is growing more quickly between week 5 and week 6? Determine whether each function has a constant or variable rate of change. 6. 4 7. y 4 8. y 2 2 2 -2 9. y = 2x 2 1x 12. y = _ 5 x 15. y = 3 √ 0 2 x x x -4 y 4 -4 4 0 -2 2 -4 4 0 -2 -2 -2 -2 -4 -4 -4 10. y + 1 = 3x 11. y = -7 13. y = 5 x x-3 16. y = _ 2x 14. y = x 2+ 1 2 4 17. x + y = 6.25 Determine whether each statement is sometimes, always, or never true. 18. A function whose graph is a straight line has a variable rate of change. 19. A quadratic function has a constant rate of change. 20. The rate of change of a linear function is negative. 21. The rate of change between two points on the graph of a nonlinear function is 0. 22. Critical Thinking The figure shows the graph of the x exponential function y = __12 . () 8 A 6 a. Find the rates of change between points A and B, between points B and C, and between points A and D. b. What do you notice about the rates of change you found in part a? Do you think this would be true for the rate of change between any two points on the graph? y 4 B C -4 -2 2 x D 0 2 4 c. How do your findings about the rates of change relate to the shape of the graph? Model Rocket a. Find the rates of change between points A and B and between points B and C. b. Which rate of change is greater? What does this tell you about the motion of the rocket? c. Find the rates of change between points C and D and between points D and E. d. What does the sign of the rates of change you found in part c tell you about the motion of the rocket? Explain. Height (ft) 23. A model rocket is launched from the ground. The graph shows the height of the rocket at various times. 90 80 70 60 50 40 30 20 10 0 C(2, 64) D(3, 48) B(1, 48) A(0, 0) E(4, 0) 1 2 3 4 5 Time (s) A-5 Linear and Nonlinear Rates of Change AT17 ADDITIONAL TOPIC A-6 Objectives Use inductive reasoning to make conjectures. Use deductive reasoning to prove conjectures, and find counterexamples to disprove conjectures. EXAMPLE Vocabulary inductive reasoning conjecture counterexample deductive reasoning Reasoning and Counterexamples Inductive reasoning is the process of concluding that a general rule or statement is true because specific cases are true. A statement based on inductive reasoning is called a conjecture. Row 1 Row 2 Row 3 When several numbers or geometric figures form a pattern and you assume the pattern will continue, you are using inductive reasoning. For example, you might use inductive reasoning to make the conjecture that the fourth row of the pattern at right will contain seven stars. 1 Using Inductive Reasoning Use inductive reasoning to make a conjecture about the value of the 100th term in the sequence 5, 9, 13, 17, 21, ... . Make a table. Examine the values and look for a pattern. Term 1st 2nd 3rd 4th 5th Value 5 9 13 17 21 4(1) + 1 4(2) + 1 4(3) + 1 4(4) + 1 4(5) + 1 Pattern Each value in the sequence is 4 times its position in the sequence, plus 1. The rule 4n + 1 can be used to find the nth term. A reasonable conjecture for the value of the 100th term is 4(100) + 1 = 401. 1. Use inductive reasoning to make a conjecture about the value of the 45th term in the sequence 1, 4, 9, 16, 25, ... . A counterexample is a case that proves that a conjecture or statement is false. One counterexample is enough to disprove a statement. Counterexamples Statement Counterexample No month has fewer than 30 days. February has fewer than 30 days, so the statement is false. All prime numbers are odd. The number 2 is a prime number, but it is not odd. This example shows that the statement is false. Every integer that is divisible by 2 is also divisible by 4. The integer 18 is divisible by 2, but it is not divisible by 4. This example shows that the statement is false. Recall that the Commutative and Associative Properties are true for addition and multiplication. The following example shows how you can use a counterexample to demonstrate that those properties are not true for other operations. AT18 Additional Topics EXAMPLE 2 Finding a Counterexample Find a counterexample to disprove the statement “The Associative Property is true for subtraction.” Find three real numbers a, b, and c such that a - (b - c) ≠ (a - b) - c. Try a = 10, b = 7, and c = 2. a - (b - c) (a - b) - c 10 - (7 - 2) (10 - 7) - 2 10 - 5 = 5 3-2=1 Since 10 - (7 - 2) ≠ (10 - 7) - 2, this is a counterexample that shows that the statement is false. 2. Find a counterexample to disprove the statement “The Commutative Property is true for division.” Inductive reasoning can be used to make conjectures, but it cannot be used to prove a statement. To prove a statement, you must use deductive reasoning. Deductive reasoning is the process of drawing conclusions from given facts, definitions, and properties. You may not realize it, but you use deductive reasoning every time you solve an equation. In fact, solving an equation can be thought of as a proof. EXAMPLE 3 Using Deductive Reasoning Use deductive reasoning to prove each statement. A If 3x + 4 = 19, then x = 5. Notice that each step of a proof is justified with a definition, a property, an operation, or a piece of given information. Statements Reasons 1. 3x + 4 = 19 Given 2. 3x + 4 - 4 = 19 - 4 Subtraction Property of Equality 3. 3x = 15 Subtraction 3x 15 4. __ = __ Division Property of Equality 5. x = 5 Division 3 3 B If 5(x – 3) = –20, then x = –1. Statements Reasons 1. 5(x - 3) = -20 Given 2. 5x - 5(3) = -20 Distributive Property 3. 5x - 15 = -20 Multiplication 4. 5x - 15 + 15 = -20 + 15 Addition Property of Equality 5. 5x = -5 Addition 5x -5 6. __ = ___ 5 5 Division Property of Equality 7. x = -1 Division Use deductive reasoning to prove each statement. 3a. If __2x -1 = 7, then x = 16. 3b. If 4(x + 2) = 8, then x = 0. A-6 Reasoning and Counterexamples AT19 A-6 Exercises Use inductive reasoning to make a conjecture about the value of the 40th term in each sequence. 1. 7, 8, 9, 10, 11, ... 2. 4, 7, 10, 13, 16, ... 3. 5, 10, 15, 20, 25, 30, ... 4. 0, 3, 6, 9, 12, ... 5. 3, 5, 3, 5, 3, ... 6. 1, 2, 3, 1, 2, 3, 1, ... Use inductive reasoning to make a conjecture about the next item in each pattern. 3, _ 15 , ... 7, _ 1, _ 7. d, e, f, d, e, f, ... 8. _ 9. 0.1, 0.02, 0.003, ... 2 4 8 16 , , , ... 10. 66, 57, 48, 39, ... 11. −1, 5, −3, 5, −5, 5, ... 12. Find a counterexample to disprove each statement. 13. The Associative Property is true for division. 14. The Commutative Property is true for subtraction. Use deductive reasoning to fill in each missing term. 15. The Red-Q company makes only red clothes. This shirt is made by Red-Q. Therefore, this shirt is ? . 16. Lance goes to the library every went to the library today. ? . Today is Wednesday. Therefore, Lance Tell whether each statement is true or false. If the statement is false, give a counterexample to disprove the statement. 17. If n is an even number, then 3n is also an even number. 18. The sum of two odd numbers is also an odd number. 19. If a number is divisible by 6, then it is divisible by 12. 20. The product of two odd numbers is an odd number. 21. Every number that is a multiple of 4 is a multiple of 2. 22. If m is a multiple of 3 and n is a multiple of 4, then m + n is a multiple of 7. 23. Fill in the missing statement or reason to complete the proof of the following statement: If –2(x + 6) = –18, then x = 3. Statements Reasons 1. -2(x + 6) = -18 Given 2. (-2 )x + (-2)6 = -18 Distributive Property ? 3. a. Multiplication 4. -2x - 12 + 12 = -18 + 12 b. 5. -2x = -6 Addition -2x -6 6. ____ = ___ c. 7. x = 3 Division -2 -2 ? ? Use deductive reasoning to prove each statement. AT20 24. If x – 16 = 41, then x = 57. 25. If 3x – 17 = 13, then x = 10. 26. If 7(x – 2) = 21, then x = 5. 27. If 3(x + 1) = –9, then x = –4. Additional Topics 28. Geometry A student drew the set of figures shown below. Then the student made the following conjecture: Given three points in a plane, it is possible to draw exactly three distinct lines through the points. Do you agree or disagree? Explain. 29. Geometry An exterior angle of a triangle is Remote 1 the angle formed by extending one side of the Exterior angle triangle. An exterior angle of a triangle has two interior angles remote interior angles. In the figure, ∠4 is an 2 3 4 exterior angle. Its remote interior angles are ∠1 and ∠2 . a. Draw a triangle and an exterior angle. Use a protractor to measure the exterior angle and its remote interior angles. What do you notice about the sum of the measures of the remote interior angles? b. Repeat the process in part a several more times. Make a conjecture. c. What type of reasoning did you use to make your conjecture? Explain. Determine whether inductive or deductive reasoning was used in each situation. Explain. 30. Christine visited San Diego four times. Each time it was raining. Christine concludes, “San Diego is a very rainy city.” 31. Stephanie knows that her cousin’s new pet is a lizard. She also knows that every lizard is a reptile and that every reptile has scales. Stephanie concludes that her cousin’s new pet has scales. 32. According to Aaron’s textbook, the sum of the angle measures in any triangle is 180°. Aaron finds that two angles of a triangle measure 50° and 60°. Aaron concludes that the third angle must measure 70°. 33. Looking at the sequence 3, 9, 27, 81, ..., David concludes that the next number must be 243 because each term is 3 times the previous term. A conditional statement is a statement that can be written in the form “If p, then q.” The contrapositive of a conditional statement is the statement “If not q, then not p.” The contrapositive of a true conditional statement is also true. Use the contrapositive to make a true conjecture based on each conditional statement. 34. If today is Tuesday, then tomorrow is Wednesday. 35. If Sam lives in Detroit, then Sam lives in Michigan. 36. If n is an even number, then n + 1 is an odd number. 37. If 8x + 3 = 19, then x = 2. 38. Logic The mockingbird, the cardinal, the goldfinch, and the grouse are the state birds of Illinois, New Jersey, Pennsylvania, and Tennessee, but not necessarily in that order. Use deductive reasoning and the following clues to match each bird with its state. • The cardinal is not the state bird of Tennessee. • The state bird of Pennsylvania begins with the same letter as the state bird of New Jersey. • The state whose capital is Trenton has the goldfinch as its state bird. A-6 Reasoning and Counterexamples AT21 Student Handbook Extra Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S4 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S6 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S8 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S10 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S12 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S14 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S16 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S18 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S20 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S22 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S24 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S26 Application Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S28 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S28 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S29 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S30 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S31 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S32 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S33 Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S34 Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S35 Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S36 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .S37 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S38 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S39 S2 Student Handbook Problem-Solving Handbook . . . . . . . . . . . . . . . . . . . . . . . PS2 Draw a Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS2 Make a Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS3 Guess and Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS4 Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS5 Find a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS6 Make a Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS7 Solve a Simpler Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS8 Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS9 Use a Venn Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS10 Make an Organized List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS11 Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SA2 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G1 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IN1 Symbols and Formulas . . . . . . . . . . . . . . inside back cover S3 Extra Practice Chapter 1 Lesson 1-1 Skills Practice Give two ways to write each algebraic expression in words. 1. x + 8 2. 6(y) 3. g - 4 12 4. _ h Evaluate each expression for a = 4, b = 2, and c = 5. a 5. b + c 6. _ 7. c - a 8. ab b Write an algebraic expression for each verbal expression. Then evaluate the algebraic expression for the given values of y. Verbal 9. Lesson 1-2 Lesson 1-3 Algebraic y=9 y=6 y reduced by 4 10. the quotient of y and 3 11. 5 more than y 12. the sum of y and 2 Add or subtract using a number line. 13. -7 - 9 14. -2.2 + 4.3 1 - 2_ 1 15. -5 _ 2 2 Subtract. 17. 12 - 47 18. 1.3 - 9.2 Compare. Write <, >, or =. 20. -5 - (-8) -4 - 9 21. ⎪-6 - (-2)⎥ 1 - 4_ 2 19. 1_ 3 3 7-4 Evaluate the expression g - (-7) for each value of g. 2 23. g = 121 24. g = 1.25 25. g = - _ 5 Find the value of each expression. 27. -24 ÷ (-8) 28. 5(-9) ( ) 6 2 ÷ -_ 30. _ 7 7 16. 3.4 - 6.5 ( ) 22. -2 - 5 7 - 14 1 26. g = -8 _ 3 29. -5.2 ÷ (-1.3) 9 ÷0 32. _ 10 4 31. 0 ÷ - _ 5 Evaluate each expression for x = -8, y = 6, and z = -4. y 33. xy 34. yz 35. _ z z 36. _ x Let a represent a positive number, b represent a negative number, and z represent zero. Tell whether each expression is positive, negative, zero, or undefined. a ab 37. ab 38. -bz 39. - _ 40. _ z b Lesson 1-4 Write each expression as repeated multiplication. Then simplify the expression. 41. 3 3 42. -2 4 43. (-5)3 44. (-1) 5 Write each expression using a base and an exponent. 45. 5 · 5 · 5 · 5 · 5 46. 4 · 4 · 4 47. 2 · 2 · 2 · 2 Write the exponent that makes each equation true. 48. 2 ■ = 16 S4 Extra Practice 49. 4 ■ = 256 50. (-3)■ = 81 51. -5 ■ = -125 Chapter 1 Lesson 1-5 Skills Practice Find each root. 52. - √ 64 53. √ 144 Compare. Write <, >, or =. 55. √ 118 11 56. 6 √ 35 54. 57. 14 8 _ √ 125 3 √ 196 58. √ 50 Write all classifications that apply to each real number. 59. -44 60. √ 49 61. 15.982 Lesson 1-6 1 62. _ 9 Evaluate each expression for the given value of the variable. 63. 22 - 3g + 5 for g = 4 64. 12 - 30 ÷ h for h = 6 65. Simplify each expression. 66. 4 + 12 ÷ ⎪3 - 9⎥ 67. -36 - √ 4 + 15 ÷ 3 Translate each word phrase into an algebraic expression. 69. the quotient of 8 and the difference of a and 5 7 (11j + j) + 6 for j = 3 √ 5 - √ 12(3) 68. __ -4 + √ 2(8) 70. the sum of -9 and the square root of the product of 7 and c Lesson 1-7 Simplify each expression. 71. -5 + 38 + 5 + 62 1 - 42 + 7 _ 2 1 · 4 · 25 72. 2 _ 73. _ 5 3 3 Write each product using the Distributive Property. Then simplify. 74. 12(108) 75. 7(89) 76. 11(33) Simplify each expression by combining like terms. 77. 7a - 3a 78. -2b - 12b 79. 4c + 5c 2 - c Simplify each expression. Justify each step with an operation or property. 80. 6(p - 2) + 3p 81. 8q - 3 + 5q(2 + q) 82. -4 + 3r - 7(2s - r) Lesson 1-8 Graph each point. 83. A(2, 3) 84. B(-4, 1) 85. C (0, -2) 86. D (-4, -1) Name the quadrant in which each point lies. 87. J 88. K 89. L 90. M 91. N 92. P * Ó { Þ { Ý Ó ä Ó Ó { { Generate ordered pairs for each function for x = -2, -1, 0, 1, and 2. Graph the ordered pairs and describe the pattern. 93. y = x - 3 94. y = -2x 95. y = -x 2 96. y = ⎪3x⎥ Write an equation for each rule. Use the given values for x to generate ordered pairs. Graph the ordered pairs and describe the pattern. 97. y is equal to the sum of one-third of x and -2; x = -6, -3, 0, 3, and 6. 98. y is equal to 4 less than x squared; x = -2, -1, 0, 1, and 2. Extra Practice S5 Chapter 2 Lesson 2-1 Skills Practice Solve each equation. Check your answer. 1. x - 9 = 5 2. 4 = y - 12 3 =7 3. a + _ 5 4. 7.3 = b + 3.4 5. -6 + j = 5 6. -1.7 = -6.1 + k Write an equation to represent each relationship. Then solve the equation. 7. A number decreased by 7 is equal to 10. 8. The sum of 6 and a number is -3. Lesson 2-2 Solve each equation. Check your answer. n = 15 9. _ 5 k 10. -6 = _ 4 r =5 11. _ 2.6 12. 3b = 27 13. 56 = -7d 14. -3.6 = -2f 1z=3 15. _ 4 4g 16. 12 = _ 5 1 a = -5 17. _ 3 Write an equation to represent each relationship. Then solve the equation. 18. A number multiplied by 4 is -20. 19. The quotient of a number and 5 is 7. Lesson 2-3 Solve each equation. Check your answer. 20. 2k + 7 = 15 21. 11 - 5m = -4 2 b + 6 = 10 23. _ 5 f 24. _ - 4 = 2 3 22. 23 = 9 - 2d 25. 6n + 4 = 22 Write an equation to represent each relationship. Solve each equation. 26. The difference of 11 and 4 times a number equals 3. 27. Thirteen less than 5 times a number is equal to 7. Lesson 2-4 Solve each equation. Check your answer. 28. 5b - 3 = 4b + 1 29. 3g + 7 = 11g - 17 30. -8 + 4y = y - 6 + 3y - 2 31. 7 + 3d - 5 = -1 + 2d - 12 + d Write an equation to represent each relationship. Then solve the equation. 32. Three more than one-half a number is the same as 17 minus three times the number. 33. Two times the difference of a number and 4 is the same as 5 less than the number. Lesson 2-5 Lesson 2-6 Solve each equation for the indicated variable. 5 - c = d - 7 for c 34. q - 3r = 2 for r 35. _ 6 y _ 36. 2x + 3 = 5 for y 37. 2fgh - 3g = 10 for h 4 Solve each equation. Check your answer. 38. ⎪a⎥ = 13 39. ⎪x⎥ - 16 = 3 f 41. ⎪7s ⎪ - 6 = 8 42. _ + 1 = 15 2 ⎪ 44. 500 = 25 ⎪z ⎪+ 200 S6 Extra Practice ⎥ 45. ⎪7j + 14⎥ - 5 = 16 40. ⎪g + 5⎥ = 11 43. ⎪p - 5⎥ - 12 = -9 ⎪p - 2⎥ - 15 46. __ = -1 5 Chapter 2 Lesson 2-7 Skills Practice 47. A long-distance runner ran 9000 meters in 30 minutes. Find the unit rate in meters per minute. 48. A hummingbird flapped its wings 770 times in 14 seconds. Find the unit rate in flaps per second. 49. A car traveled 210 miles in 3 hours. Find the unit rate in miles per hour. 50. A printer printed 60 pages in 5 minutes. Find the unit rate in pages per minute. Lesson 2-8 Solve each proportion. h =_ 5 51. _ 4 6 5 2 _ 52. _ m=5 r =_ 10 53. _ 7 3 x 2=2 _ 54. _ 3 8 5 =_ 3 55. _ x - 3 10 b-2 =_ 7 56. _ 4 12 Find the value of x in each diagram. 57. ABCD ∼ EFGH 58. JKL ∼ MNO ÝÊvÌ nÊ £äÊvÌ £äÊvÌ {ÊvÌ ÈÊ ÝÊ £{Ê " Lesson 59. Find 25% of 60. 60. Find 40% of 95. 2-9 61. What percent of 75 is 15? 62. What percent of 60 is 33? 63. 91 is what percent of 65? 64. 35% of what number is 24.5? Write each decimal or fraction as a percent. 9 4 65. _ 66. 0.55 67. _ 5 6 Write each percent as a decimal and as a fraction. 69. 32% 70. 24% 71. 37.5% 68. 0.0375 72. 118.75% Write each list in order from least to greatest. 6 , 0.19 5 , 9.2, 117%, _ 9 , 8.8% 2 , 0.28, 1.9%, _ 74. _ 73. _ 17 25 11 3 Lesson 75. Estimate the tax on a $2438.00 clarinet when the sales tax is 7.9%. 2-10 76. Estimate the tip on a $21.32 check using a tip rate of 20%. Lesson Find each percent change. Tell whether it is a percent increase or decrease. 77. 10 to 25 78. 40 to 2 79. 800 to 160 2-11 80. 4 to 14 81. 8 to 30 82. 60 to 36 83. Find the result when 45 is increased by 40%. 84. Find the result when 120 is decreased by 70%. Extra Practice S7 Chapter 3 Lesson 3-1 Skills Practice Describe the solutions of each inequality in words. 1. 3 + v < -2 2. 15 ≤ k + 4 3. -3 + n > 6 4. 1 - 4x ≥ -2 Graph each inequality. 5. f ≥ 2 6. m < -1 8. (-1 - 1)2 ≤ p 2 7. √4 + 32 > c Write the inequality shown by each graph. 9. 10. ä £ Ó Î { x È ä Ó { È n £ä £Ó Î Ó £ ä £ 11. 12. 13. Î Ó £ ä £ Ó Î -6 -4 -2 0 2 4 6 Î { x È 14. Ó Î ä £ Ó Write each inequality with the variable on the left. Graph the solutions. 15. 14 > b 16. 9 ≤ g 17. -2 < x 18. -4 ≥ k Lesson 3-2 Solve each inequality and graph the solutions. 19. 8 ≥ d - 4 20. -5 < 10 + w 21. a + 4 ≤ 7 22. 9 + j > 2 Write an inequality to represent each statement. Solve the inequality and graph the solutions. 23. Five more than a number v is less than or equal to 9. 24. A number t decreased by 2 is at least 7. 25. Three less than a number r is less than -1. 26. A number k increased by 1 is at most -2. Use the inequality 4 + z ≤ 11 to fill in the missing numbers. 27. z ≤ 28. z ≤4 29. z - 3 ≤ Lesson Solve each inequality and graph the solutions. 3-3 30. 24 > 4b 31. 27g ≤ 81 34. 4p < -2 3s > 3 35. _ 8 -2e ≥ 4 39. _ 5 38. -3k ≤ -12 h 42. 9 > _ -2 43. 49 > -7m x <3 32. _ 5 3d 36. 0 ≥ _ 7 33. 10y ≥ 2 40. 8 < -12y 41. -3.5 > 14c 44. 60 ≤ -12c 1 q < -6 45. - _ 3 a ≥_ 3 37. _ 4 8 Write an inequality for each statement. Solve the inequality and graph the solutions. 1 and a number is not more than 6. 46. The product of _ 2 47. The quotient of r and -5 is greater than 3. 48. The product of -11 and a number is greater than -33. 49. The quotient of w and -4 is less than or equal to -6. S8 Extra Practice Chapter 3 Lesson 3-4 Skills Practice Solve each inequality and graph the solutions. 50. 3t - 2 < 5 51. -6 < 5b - 4 2f + 3 52. 4 < _ 2 3 2 4 1 _ _ _ _ 53. 10 ≤ 3(4 - r) 54. + h < 55. (10k - 2) > 1 5 3 4 3 3 8q - 2 2 < -3 3 2 2 56. -n - 3 < -2 57. 37 - 4d ≤ √ 3 +4 58. - _ ) ( 4 Use the inequality -6 - 2w ≥ 10 to fill in the missing numbers. 59. w ≤ 60. w - 3 ≤ 61. +w≤1 Write an inequality for each statement. Solve the inequality and graph the solutions. 62. Twelve is less than or equal to the product of 6 and the difference of 5 and a number. 63. The difference of one-third a number and 8 is more than -4. 64. One-fourth of the sum of 2x and 4 is more than 5. Lesson Solve each inequality and graph the solutions. 3-5 65. 4v - 2 ≤ 3v 66. 2(7 - s) > 4(s + 2) 5 ≥_ 1u-_ 1u 67. _ 3 2 6 69. 4(k + 2) ≥ 4k + 5 70. 2(5 - b) ≤ 3 - 2b Solve each inequality. 68. 3 + 3c < 6 + 3c Write an inequality to represent each relationship. Solve your inequality. 71. The difference of three times a number and 5 is more than the number times 4. 72. One less than a number is greater than the product of 3 and the difference of 5 and the number. Lesson 3-6 Solve each compound inequality and graph the solutions. 73. 6 < 3 + x < 8 74. -1 ≤ b + 4 ≤ 3 75. k + 5 ≤ -3 OR k + 5 ≥ 1 76. r - 3 > 2 OR r + 1 < 4 Write the compound inequality shown by each graph. 77. 78. Î Ó £ ä £ Ó È { Ó Î ä Ó { È Write and graph a compound inequality for the numbers described. 79. all real numbers less than 2 and greater than or equal to -1 80. all real numbers between -3 and 1 Lesson Solve each inequality and graph the solutions. 3-7 81. ⎪n + 5⎥ ≤ 26 82. ⎪x⎥ + 6 < 13 83. 4⎪k⎥ ≤ 12 84. ⎪c - 8⎥ > 18 85. 6⎪p⎥ ≥ 48 86. ⎪3 + t⎥ - 1 ≥ 5 88. 2⎪w⎥ + 5 < 3 89. ⎪s⎥ + 12 > 8 Solve each inequality. 87. ⎪a⎥ -2 ≤ -5 Write and solve an absolute-value inequality for each expression. Graph the solutions on a number line. 90. All numbers whose absolute value is greater than 14. 91. All numbers whose absolute value multiplied by 3 is less than 27. Extra Practice S9 Chapter 4 Lesson 4-1 Skills Practice Choose the graph that best represents each situation. 1. A person blows up a balloon with a steady airstream. 2. A person blows up a balloon and then lets it deflate. 3. A person blows up a balloon slowly at first and then uses more and more air. À>« Ê 6Õi /i Lesson 4-2 À>« Ê 6Õi 6Õi À>« Ê /i /i Express each relation as a table, as a graph, and as a mapping diagram. ⎧ ⎫ ⎫ ⎧ 4. ⎨(0, 2), (-1, 3), (-2, 5)⎬ 5. ⎨(2, 8), (4, 6), (6, 4), (8, 2)⎬ ⎩ ⎭ ⎩ ⎭ Give the domain and range of each relation. Tell whether the relation is a function. Explain. ⎫ ⎧ ⎫ ⎧ 7. ⎨(5, 4), (0, 2), (5, -3), (0, 1)⎬ 6. ⎨(3, 4), (-1, 2), (2, -3), (5, 0)⎬ ⎭ ⎩ ⎭ ⎩ y 8. 9. x 2 0 1 2 -1 y 1 0 -1 -2 -3 8 6 4 2 0 Lesson 4-3 2 4 6 8 x Determine a relationship between the x- and y-variables. Write an equation. ⎧ ⎫ 10. ⎨(1, 3), (2, 6), (3, 9), (4, 12)⎬ 11. x 1 2 3 4 ⎩ ⎭ y 1 4 9 16 Identify the independent and dependent variables. Write an equation in function notation for each situation. 12. A science tutor charges students $15 per hour. 13. A circus charges a $10 entry fee and $1.50 for each pony ride. 14. For f (a) = 6 - 4a, find f (a) when a = 2 and when a = -3. 2 d + 3, find g (d) when d = 10 and when d = -5. 15. For g (d) = _ 5 16. For h (w) = 2 - w 2, find h (w) when w = -1 and when w = -2. 17. Complete the table for f (t ) = 7 + 3t. t f(t) S10 Extra Practice 0 1 2 3 18. Complete the table for h(s) = 2s + s 3 - 6. s h(s) -1 0 1 2 Chapter 4 Lesson 4-4 Skills Practice Graph each function for the given domain. ⎧ ⎫ ⎧ ⎫ 19. 2x - y = 2; D: ⎨-2, -1, 0, 1⎬ 20. f(x) = x 2 - 1; D: ⎨-3, -1, 0, 2⎬ ⎩ ⎭ ⎩ ⎭ Graph each function. 21. f(x) = 4 - 2x 22. y + 3 = 2x 23. y = -5 + x 2 5 - 2x to find the value of y when x = _ 1. 24. Use a graph of the function y = _ 2 2 Check your answer. 25. Find the value of x so that (x, 4) satisfies y = -x + 8. 26. Find the value of y so that (-3, y) satisfies y = 15 - 2x 2. Lesson For each function, determine whether the given points are on the graph. x + 4; -3, 3 and 3, 5 27. y = _ 28. y = x 2 - 1; (-2, 3) and (2, 5) ) ( ( ) 3 Describe the correlation illustrated by each scatter plot. 4-5 29. 30. Þ 31. Þ Ý Þ Ý Ý Identify the correlation you would expect to see between each pair of data sets. Explain. 32. the number of chess pieces captured and the number of pieces still on the board 33. a person’s height and the color of the person’s eyes Choose the scatter plot that best represents the described relationship. Explain. À>« Ê 34. the number of students in a class and the À>« Ê Þ Þ grades on a test 35. the number of students in a class and the number of empty desks Ý Ý Lesson 4-6 Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 36. -10, -7, -4, -1, … 37. 8, 5, 1, -4, … 38. 1, -2, 3, -4, … 39. -19, -9, 1, 11, … Find the indicated term of each arithmetic sequence. 40. 15th term: -5, -1, 3, 7, … 41. 20th term: a 1 = 2; d = -5 42. 12th term: 8, 16, 24, 32, … 43. 21st term: 5.2, 5.17, 5.14, 5.11, … Find the common difference for each arithmetic sequence. 7, _ 10 , … 1 , 1, _ 44. 0, 7, 14, 21, … 45. 132, 121, 110, 99, … 46. _ 4 4 4 47. 1.4, 2.2, 3, 3.8, … 48. -7, -2, 3, 8, … 49. 7.28, 7.21, 7.14, 7.07, … Find the next four terms in each arithmetic sequence. 50. -3, -6, -9, -12, … 51. 2, 9, 16, 23, … 5, … 1, _ 1 , 1, _ 52. - _ 53. -4.3, -3.2, -2.1, -1, … 3 3 3 Extra Practice S11 Chapter 5 Lesson 5-1 Skills Practice Identify whether each graph represents a function. Explain. If the graph does represent a function, is the function linear? 1. { 2. Þ 3. Þ { Þ È Ó Ó Ý È { ä Ó { Ý { Ó Ó ä Ó { Ó Ó { Ý { { Ó ä Tell whether the given ordered pairs satisfy a linear function. Explain. 4. 5. x 2 5 8 x -4 -2 0 2 4 y 7 6 5 4 y 3 12 8 7 Ó { 11 14 3 3 Lesson Tell whether each equation is linear. If so, write the equation in standard form and give the values of A, B, and C. x = 4 - 2y 6. y = 8 - 3x 7. _ 8. -3 + xy = 2 9. 4x = -3 - 3y 3 Find the x- and y-intercepts. 5-2 10. -4x = 2y - 1 11. x - y = 3 12. 2x - 3y = 12 13. 2.5x + 2.5y = 5 Use intercepts to graph the line described by each equation. 14. 15 = -3x - 5y 15. 4y = 2x + 8 16. y = 6 - 3x Lesson Find the slope of each line. 5-3 18. { 19. Þ Þ n Ó 17. -2y = x + 2 { Ý { Lesson 5-4 Ó ä Ó { Ý n { Ó { { n 5-5 { n Find the slope of the line that contains each pair of points. 20. (-1, 2) and (-4, 8) 21. (2, 6) and (0, 1) 22. (-2, 3) and (4, 0) Find the slope of the line described by each equation. 23. 2y = 42 - 6x 24. 3x + 4y = 12 Lesson ä 25. 3x = 15 + 5y Find the coordinates of the midpoint of each segment. −− 26. AB with endpoints A(-3, 4) and B(1, 5) −− 27. CD with endpoints C(9, -8) and D(9, -2) Find the distance, to the nearest hundredth, between each pair of points. 28. A(0, 6) and B(4, 8) 29. J(-1, 7) and K(-4, 5) 30. S(-3, -9) and T(2, 3) S12 Extra Practice Chapter 5 Lesson 5-6 Skills Practice Tell whether each equation represents a direct variation. If so, identify the constant of variation. 31. x - 2y = 0 32. x - y = 3 33. 3y = 2x 34. The value of y varies directly with x, and y = 2 when x = -3. Find y when x = 6. 35. The value of y varies directly with x, and y = -3 when x = 9. Find y when x = 12. Lesson Write the equation that describes each line in slope-intercept form. 5-7 36. slope = 2, y-intercept = -2 38. slope = -2, (5, 4) is on the line. 40. y (-3, 2) 2 -2 0 (3, 0) x -2 5-8 Lesson 5-9 0 -2 2 -2 Lesson 37. slope = 0.25, y-intercept = 4 1 , (-8, 0) is on the line. 39. slope = _ 3 41. y x (2, -1) (-2, -5) Write each equation in slope-intercept form. Then graph the line described by the equation. 1 x=2 42. 2y = x - 3 43. -3x - 2y = 1 44. 2y - _ 2 Write an equation in point-slope form for the line with the given slope that contains the given point. 1 ; (2, 4) 45. slope = 2; (0, 3) 46. slope = -1; (1, -1) 47. slope = _ 2 Write the equation that describes each line in slope-intercept form. 48. slope = 3, (-2, -5) is on the line. 49. (-1, 1) and (1, -2) are on the line. 50. (3, 1) and (2, -3) are on the line. 51. x-intercept = 4, y-intercept = -5 Write an equation in slope-intercept form for the line that is parallel to the given line and that passes through the given point. 52. y = -2x + 3; (1, 4) 53. y = x - 5; (2, -4) 54. y = 3x; (-1, 5) Write an equation in slope-intercept form for the line that is perpendicular to the given line and that passes through the given point. 55. y = x + 1; (3, -2) Lesson 5-10 56. y = -4x - 1; (-1, 0) 57. y = 4x + 5; (2, -1) Graph f (x) and g (x). Then describe the transformation(s) from the graph of f (x) to the graph of g (x). 1 58. f (x) = x, g(x) = x + 2 59. f (x) = x, g (x) = x - _ 2 60. f (x) = 6x + 1, g(x) = 2 x + 1 61. f (x) = 3x - 1, g (x) = 9x - 1 1x 62. f (x) = x, g(x) = 2x - 1 63. f (x) = x + 1, g (x) = - _ 2 Extra Practice S13 Chapter 6 Lesson 6-1 Skills Practice Tell whether the ordered pair is a solution of the given system. ⎧ 2x - 3y = -7 ⎧4x + 3y = -2 ⎧ -2x - 3y = 1 1. (1, 3); ⎨ 2. (-2, 2); ⎨ 3. (4, -3); ⎨ ⎩ -5x + 3y = 4 ⎩ -2x - 2y = 2 ⎩ x + 2y = -2 Use the given graph to find the solution of each system. ⎧ 1 _ y = 2 x - 1 ⎧y = x + 1 4. ⎨ 5. ⎨ 1x+3 ⎩ y = -x + 1 y = - _ 2 ⎩ { Þ Þ { Ó Ó Ý { Ó ä Ó {Ý { Ó ä Ó Ó { { Ó { Solve each system by graphing. Check your answer. ⎧y = x + 1 6. ⎨ ⎩ y = -2x - 2 Lesson 6-2 ⎧3x + y = -8 7. ⎨ 1 ⎩ 3y = _ x - 5 2 Solve each system by substitution. ⎧y = 12 - 3x ⎧2x + y = -6 9. ⎨ 10. ⎨ ⎩ y = 2x - 3 ⎩ -5x + y = 1 ⎧2x + 3y = 2 12. ⎨ 1 ⎩ - _ x + 2y = -6 2 ⎧3x - 2y = -3 13. ⎨ ⎩ y = 7 - 4x ⎧x = 2 - 2y 8. ⎨ ⎩ -1 = -2x - 3y ⎧y = 11 - 3x 11. ⎨ ⎩ -2x + y = 1 ⎧4y - 2x = -2 14. ⎨ ⎩ x + 3y = -4 Two angles whose measures have a sum of 90° are called complementary angles. For Exercises 15–17, x and y represent the measures of complementary angles. Use this information and the equation given in each exercise to find the measure of each angle. 15. y = 9x - 10 16. y - 4x = 15 17. y = 2x + 15 Lesson 6-3 S14 Solve each system by elimination. ⎧x - 3y = -1 ⎧-3x - y = 1 18. ⎨ 19. ⎨ ⎩ -x + 2y = -2 ⎩ 5x + y = -5 ⎧-x - 3y = -1 20. ⎨ ⎩ 3x + 3y = 9 ⎧3x - 2y = 2 21. ⎨ ⎩ 3x + y = 8 ⎧5x - 2y = -15 22. ⎨ ⎩ 2x - 2y = -12 ⎧-4x - 2y = -4 23. ⎨ ⎩ -4x + 3y = -24 ⎧-3x - 3y = 3 24. ⎨ ⎩ 2x + y = -4 ⎧4x - 3y = -1 25. ⎨ ⎩ 2x - 2y = -4 ⎧3x + 6y = 0 26. ⎨ ⎩ 7x + 4y = 20 Extra Practice Chapter 6 Lesson 6-4 Skills Practice Solve each system of linear equations. ⎧y = 2x + 4 ⎧-y = 3 - 5x 27. ⎨ 28. ⎨ ⎩ -2x + y = 6 ⎩ y - 5x = 6 ⎧y + 2 = 3x 29. ⎨ ⎩ 3x - y = -1 ⎧y - 1 = -3x 31. ⎨ ⎩ 12x + 4y = 4 ⎧2y = 6 - 6x 30. ⎨ ⎩ 3y + 9x = 9 ⎧4x - 2y = 4 32. ⎨ ⎩ 3y = 6 (x - 1) Classify each system. Give the number of solutions. ⎧2y = 2 (4x - 3) 33. ⎨ ⎩ y - 1 = 4x ⎧3y + 6x = 9 34. ⎨ ⎩ 2(y - 3) = -4x ⎧3x - 13 = 2y 35. ⎨ ⎩ -3y = 2x Lesson Tell whether the ordered pair is a solution of the given inequality. 6-5 36. (3, 6); y > 2x + 4 37. (-2, -8); y ≤ 3x - 2 38. (-3, 3); y ≥ -2x + 5 Graph the solutions of each linear inequality. 39. y > 2x 40. y ≤ -3x + 2 41. y ≥ 2x - 1 42. -y < -x + 4 43. y ≥ -2x + 4 44. y > -x - 3 1 x + 1_ 1 45. y < _ 2 2 46. y ≤ 4x - (-1) Write an inequality to represent each graph. 47. n 48. Þ Þ n { { Ý n Lesson 6-6 { ä { Ý n n { ä { { n n { n Tell whether the ordered pair is a solution of the given system. ⎧y > 3x - 3 ⎧y > -3x - 2 ⎧y > 2x 49. (2, 5); ⎨ 50. (3, 9); ⎨ 51. (2, 3); ⎨ ⎩y ≥ x + 1 ⎩ y < 2x + 3 ⎩y ≤ x - 3 Graph each system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. ⎧x + 4y < 2 52. ⎨ ⎩ 2y > 3x + 8 ⎧y ≤ 6 - 2x 53. ⎨ ⎩ x - 2y < -2 ⎧2x - 2 > -3y 54. ⎨ ⎩ -x + 3y ≥ -10 Graph each system of linear inequalities. Describe the solutions. ⎧y > 2x + 1 55. ⎨ ⎩ y < 2x - 2 ⎧y < 3x - 1 56. ⎨ ⎩ y > 3x - 4 ⎧y ≥ -x + 2 57. ⎨ ⎩ y ≥ -x + 5 ⎧y ≥ 2x - 3 58. ⎨ ⎩ y ≥ 2x + 3 ⎧y > -4x - 2 59. ⎨ ⎩ y ≤ -4x - 5 ⎧y ≥ -2x + 1 60. ⎨ ⎩ y < -2x + 6 Extra Practice S15 Chapter 7 Skills Practice Lesson Simplify. 7-1 1. 3 -4 2. 5 -3 3. -4 0 4. -2 -5 6. (-2)-4 7. 1-7 8. (-4)-3 9. (-5)0 5. 6 -3 10. (-1)-5 Evaluate each expression for the given value(s) of the variable(s). 11. x -4 for x = 2 12. (c + 3)-3 for c = -6 13. 3j -7k -1 for j = -2 and k = 3 14. (2n - 2)-4 for n = 3 Simplify. 15. b 4g -5 k -3 16. _ r5 17. 5s -3c 0 z -4 18. _ 5t -2 f2 19. _ 3a -4 -3t 4 20. _ q -5 a 0k -4 21. _ p2 22. 3f -1y -5 25. 10 6 26. 10 -8 Lesson Find the value of each power of 10. 7-2 23. 10 -7 24. 10 9 Write each number as a power of 10. 27. 10,000,000 28. 0.00001 29. 10,000,000,000,000 Find the value of each expression. 30. 72.19 × 10 -2 31. 0.096 × 10 -7 32. 7384.5 × 10 6 Write each number in scientific notation. 33. 3,605,000 34. 0.0063 35. 100,500,000 38. (k 4) Lesson Simplify. 7-3 36. 3 4 · 3 2 37. r 7 · r 0 39. (b 4) 40. (c 3d 2) · (c d 2) 4 41. (-3q 3) -2 3 3 -2 Find the missing exponent in each expression. a 43. (a 3b ■) = _ b6 3 42. a ■a 6 = a 9 Lesson Simplify. 7-4 3 11 45. _ 38 44 · 53 46. _ 2 3 · 43 · 53 b 44. (a 4b -2) · a 3 = _ a5 9 4 ■ 6h 4 47. _ 12h 3 r 6s 5 48. _ r 5s 6 Simplify each quotient and write the answer in scientific notation. 49. (4 × 10 7) ÷ (1.6 × 10 5) Simplify. () 2 52. _ 3 S16 Extra Practice 4 50. (10 × 10 4) ÷ (2 × 10 7) ( ) x 2y 2 53. _ y3 2 () 4 54. _ 5 -3 51. (2.5 × 10 8) ÷ (5 × 10 3) 55. ( ) 2xy 2 _ 3(xy)2 -3 Chapter 7 Skills Practice Lesson Simplify each expression. 7-5 1 _ 1 _ 1 _ 56. 27 3 57. 256 4 58. 169 2 59. 0 5 60. 4 2 61. 49 2 62. 36 2 63. 16 4 1 _ 3 _ 3 _ 3 _ 5 _ Simplify. All variables represent nonnegative numbers. 1 _ 64. √ x2 y6 Lesson 7-6 65. √ a 9 b 15 3 Find the degree of each monomial. 68. 4 7 69. x 3 y Find the degree of each polynomial. 72. a 2 b + b - 2 2 73. 5x 4 y 2 - y 5 z 2 g ) ( √ 1 _ 7 3 (m 8) 2 66. _ √ m4 67. r 6 st 2 70. _ 2 71. 9 0 74. 3g 4 h + h 2 + 4j 6 75. 4nm 7 - m 6 p3 + p 5 60 √ t 14 Write each polynomial in standard form. Then give the leading coefficient. 1 t3 + t - _ 1 t5 + 4 76. 4r - 5r 3 + 2r 2 77. -3b 2 + 7b 6 + 4 - b 78. _ 2 3 Classify each polynomial according to its degree and number of terms. 80. -4x 2 + x 6 - 4 + x 3 81. x 3 - 7 2 79. 3x 2 + 4x - 5 Lesson 7-7 Lesson 7-7 Add or subtract. 82. 4y 3 - 2y + 3y 3 83. 9k 2 + 5 - 10k 2 - 6 84. 7 - 3n 2 + 4 + 2n 2 85. (9x 6 - 5x 2 + 3) + (6 x 2 - 5) 86. (2y 5 - 5y 2) + (3y 5 - y 3 + 2y 2) 87. (r 3 + 2r + 1) - (2r 3 - 4) 88. (10s 2 + 5) - (5s 2 + 3s - 2) 89. (2s 7 - 6s 3 + 2) - (3s 7 + 2) Multiply. 90. (3a 7)(2a 4) 91. (-3xy 3)(2x 2z)(yz 4) 92. (4k 3m)(-2k 2m 2) 93. 3jk 2(2j 2 + k) 94. 4q 3r 2 (2qr 2 + 3q) 95. 3xy 2(2x 2y - 3y) 96. (x - 3)(x + 1) 97. (x - 2)(x - 3) 98. (x 2 + 2xy)(3x 2y - 2) 99. (x 2 - 3x)(2xy - 3y) 102. (x + 3)(2x 4 - 3x 2 - 5) 100. (x - 2)(x 2 + 3x - 4) 101. (2x - 1)(-2x 2 - 3x + 4) 103. (3a + b)(2a 2 + ab - 2b 2) 104. (a 2 - b)(3a 2 - 2ab + 3b 2) Lesson Multiply. 7-8 105. (x + 3) 2 106. (3 + 2x) 2 107. (4x + 2y)2 108. (3x - 2)2 109. (5 - 2x) 2 110. (3x - 5y)2 111. (3 + x)(3 - x) 112. (x - 5)(x + 5) 113. (2x + 1)(2x - 1) 114. (x 2 + 4)(x 2 - 4) 115. (2 + 3x 3)(2 - 3x 3) 116. (4x 3 - 3y)(4x 3 + 3y) Extra Practice S17 Chapter 8 Lesson 8-1 Skills Practice Write the prime factorization of each number. 1. 24 2. 78 3. 88 4. 63 5. 128 6. 102 7. 71 8. 125 Find the GCF of each pair of numbers. 9. 18 and 66 10. 24 and 104 11. 30 and 75 12. 24 and 120 13. 36 and 99 14. 42 and 72 Find the GCF of each pair of monomials. 15. 4a 3 and 9a 4 16. 6q 2 and 15q 5 17. 6x 2 and 14y 3 18. 4z 2 and 10z 5 19. 5g 3 and 9g 20. 12x 2 and 21y 2 Lesson Factor each polynomial. Check your answer. 8-2 21. 6b 2 - 15b 3 22. 11t 4 - 9t 3 23. 10v 3 - 25v 24. 12r + 16r 3 25. 17a 4 - 35a 2 26. 9f + 18f 5 + 12f 2 27. 3(a + 3) + 4a(a + 3) 28. 5(k - 4) - 2k (k - 4) 29. 5(c - 3) + 4c 2(c - 3) 30. 3(t - 4) + t (t - 4) 31. 5(2r - 1) - s(2r - 1) 32. 7(3d + 4) - 2e(3d + 4) Factor each expression. Factor each polynomial by grouping. Check your answer. 33. x 3 + 3x 2 - 2x - 6 34. 2m 3 - 3m 2 + 8m - 12 35. 3k 3 - k 2 + 15k - 5 36. 15r 3 + 25r 2 - 6r - 10 37. 12n 3 - 6n 2 - 10n + 5 38. 4z 3 - 3z 2 + 4z - 3 39. 2k 2 - 3k + 12 - 8k 40. 3p 2 - 2p + 8 - 12p 41. 10d 2 - 6d + 9 - 15d 42. 6a 3 - 4a 2 + 10 - 15a 43. 12s 3 - 2s 2 + 3 - 18s 44. 4c 3 - 3c 2 + 15 - 20c Lesson Factor each trinomial. Check your answer. 8-3 45. x 2 + 15x + 36 46. x 2 + 13x + 40 47. x 2 + 10x + 16 48. x 2 - 9x + 18 49. x 2 - 11x + 28 50. x 2 - 13x + 42 51. x 2 + 4x - 21 52. x 2 - 5x - 36 53. x 2 - 7x - 30 54. Factor c 2 - 2c - 48. Show that the original polynomial and the factored form describe the same sequence of values for c = 0, 1, 2, 3, and 4. Copy and complete the table. S18 x 2 + bx + c Sign of c Binomial factors Sign of Numbers in Binomials x 2 + 9x + 20 Positive (x + 4)(x + 5) Both positive 55. x - x - 20 ? ? ? 56. x - 2x - 8 ? ? ? 57. x - 6x + 8 ? ? ? Extra Practice 2 2 2 Chapter 8 Skills Practice Lesson Factor each trinomial. Check your answer. 8-4 58. 2x 2 + 13x + 15 59. 3x 2 + 14x + 16 60. 8x 2 - 16x + 6 61. 6x 2 + 11x + 4 62. 3x 2 - 11x + 6 63. 10x 2 - 31x + 15 64. 6x 2 - 5x - 4 65. 8x 2 - 14x - 15 66. 4x 2 - 11x + 6 67. 12x 2 - 13x + 3 68. 6x 2 - 7x - 10 69. 6x 2 + 7x - 3 70. 2x 2 + 5x - 12 71. 6x 2 - 5x - 6 72. 8x 2 + 10x - 3 73. 10x 2 - 11x - 6 74. 4x 2 - x - 5 75. 6x 2 - 7x - 20 76. -2x 2 + 11x - 5 77. -6x 2 - x + 12 78. -8x 2 - 10x - 3 79. -4x 2 + 16x - 15 80. -10x 2 + 21x + 10 81. -3x 2 + 13x - 14 Lesson 8-5 Determine whether each trinomial is a perfect square. If so, factor. If not, explain why. 82. x 2 - 8x + 16 83. 4x 2 - 4x + 1 84. x 2 - 8x + 9 85. 9x 2 - 14x + 4 86. 4x 2 + 12x + 9 87. x 2 + 8x - 16 88. 9x 2 - 42x + 49 89. 4x 2 + 18x + 25 90. 16x 2 - 24x + 9 Determine whether each trinomial is the difference of two squares. If so, factor. If not, explain why. 91. 4 - 16x 4 92. -t 2 - 35 93. c 2 - 25 94. g 5 - 9 95. v 4 - 64 96. x 2 - 120 97. x 2 - 36 98. 9m 2 - 15 99. 25c 2 - 16 Find the missing term in each perfect-square trinomial. 100. 4x 2 - 20x + 101. 9x 2 + 103. 9b 2 - 104. + 25 +1 + 28a + 49 102. - 56x + 49 105. 4a 2 + 4a + Lesson Tell whether each expression is completely factored. If not, factor. 8-6 106. 5(16x 2 + 4) 107. 3r (4x - 9) 108. (9d - 6)(2d - 7) 109. (5 - h)(6 - 5h) 110. 12y 2 - 2y - 24 111. 3f (2f 2 + 5fg + 2g 2) Factor each polynomial completely. Check your answer. 112. 12b 3 - 48b 113. 24w 4 - 20w 3 - 16w 2 114. 18k 3 - 32k 115. 4a 3 + 12a 2 - a 2b - 3ab 116. 3x 3y - 6x 2y 2 + 3xy 3 117. 36p 2q - 64q 3 118. 32a 4 - 8a 2 119. m 3 + 5m 2n + 6mn 2 120. 4x 2 - 3x 2 - 16x + 48x 121. 18d 2 + 3d - 6 122. 2r 2 - 9r - 18 123. 8y 2 + 4y - 4 124. 81 - 36u 2 125. 8x 4 + 12x 2 - 20 126. 10j 3 + 15j 2 - 70j 127. 27z 3 - 18z 2 + 3z 128. 4b 2 + 2b - 72 129. 3f 2 - 3g 2 Extra Practice S19 Chapter 9 Lesson 9-1 Skills Practice Tell whether each function is quadratic. Explain. 1. y + 4x 2 = 2x - 3 2. 4x - y = 3 4. 5. x -6 -4 -2 0 2 y -5 -6 -4 2 11 3. 3x 2 - 4 = y + x x 0 1 2 3 4 y -5 -5 -3 1 7 Tell whether the graph of each quadratic function opens upward or downward. Then use a table of values to graph each function. 2 x2 6. y = -3x 2 7. y = _ 8. y = x 2 + 2 9. y = -4x 2 + 2x 3 Identify the vertex of each parabola. Then find the domain and range. 10. 11. y 12. y 2 2 x x -2 -4 -2 y 6 2 2 8 -2 4 -4 2 -2 x 2 Lesson 9-2 4 6 Find the zeros of each quadratic function and the axis of symmetry of each parabola from the graph. 13. 8 14. y 2 15. y 2 y x x 6 -2 0 2 4 -2 0 4 -2 -2 2 -4 -4 2 4 x -4 -2 0 Find the vertex. 16. y = 3x 2 - 6x + 2 Lesson 9-3 2 4 17. y = -2x 2 + 8x - 3 18. y = x 2 + 2x - 4 Graph each quadratic function. 19. y = x 2 - 4x + 1 20. y = -x 2 - x + 4 21. y = 3x 2 - 3x + 1 22. y - 2 = 2x 2 24. y - 4 = x 2 + 2x 23. y + 3x 2 = 3x - 1 Lesson Order the functions from narrowest to widest. 9-4 25. f (x) = 2x 2, g(x) = -4x 2, h(x) = -x 2 1 x 2, h(x) = -2x 2 26. f (x) = 3x 2, g(x) = _ 2 1 x2 27. f (x) = 4x 2, g(x) = x 2, h(x) = - _ 28. f (x) = 2x 2, g(x) = 5x 2, h(x) = -3x 2 4 Compare the graph of each function with the graph of f (x) = x 2. 1 x2 29. g(x) = 2x 2 - 2 30. g(x) = - _ 31. g(x) = -3x 2 + 1 2 S20 8 Extra Practice Chapter 9 Lesson 9-5 Lesson 9-6 Skills Practice Solve each quadratic equation by graphing the related function. 32. x 2 - x - 2 = 0 33. x 2 - 2x + 8 = 0 34. 2x 2 + 4x - 6 = 0 35. 2x 2 + 9x = -4 36. 2x 2 + 3 = 0 37. 2x 2 - 2x - 12 = 0 38. 3x 2 = -3x + 6 39. x 2 = 4 40. 2x 2 + 6x - 20 = 0 41. -3x 2 - 2 = 0 42. x 2 = -2x + 8 43. x 2 - 2x = 15 Use the Zero Product Property to solve each equation. Check your answer. 44. (x + 3)(x - 2) = 0 45. (x - 4)(x + 2) = 0 46. (x)(x - 4) = 0 47. (2x + 6)(x - 2) = 0 48. (3x - 1)(x + 3) = 0 49. (x)(2x - 4) = 0 Solve each quadratic equation by factoring. Check your answer. 50. x 2 + 5x + 6 = 0 51. x 2 - 3x - 4 = 0 52. x 2 + x - 12 = 0 Lesson 9-7 Lesson 9-8 Lesson 9-9 53. x 2 + x - 6 = 0 54. x 2 - 6x + 5 = 0 55. x 2 + 4x - 12 = 0 56. x 2 = 6x - 9 57. 2x 2 + 4x = 6 58. x 2 + 2x = -1 59. 3x 2 = 3x + 6 60. x 2 = x + 12 61. 4x 2 + 8x + 4 = 0 Solve using square roots. Check your answer. 62. x 2 = 169 63. x 2 = 121 64. x 2 = 289 65. x 2 = -64 66. x 2 = 81 67. x 2 = -441 68. 4x 2 - 196 = 0 69. 0 = 3x 2 - 48 70. 24x 2 + 96 = 0 71. 10x 2 - 75 = 15 72. 0 = 4x 2 + 144 73. 5x 2 - 105 = 20 Solve. Round to the nearest hundredth. 74. 4x 2 = 160 75. 0 = 3x 2 - 66 76. 250 - 5x 2 = 0 77. 0 = 9x 2 - 72 79. 6x 2 = 78 78. 48 - 2x 2 = 42 Complete the square to form a perfect-square trinomial. 80. x 2 - 8x + 81. x 2 + x + 82. x 2 + 10x + 83. x 2 - 5x + 85. x 2 - 7x + 84. x 2 + 6x + Solve by completing the square. 86. x 2 + 6x = 91 87. x 2 + 10x = -16 88. x 2 - 4x = 12 89. x 2 - 8x = -12 90. x 2 - 12x = -35 91. -x 2 - 6x = 5 92. -x 2 - 4x + 77 = 0 93. -x 2 = 10x + 9 94. -x 2 + 63 = -2x Solve using the quadratic formula. 95. x 2 + 3x - 4 = 0 96. x 2 - 2x - 8 = 0 98. x 2 - x - 10 = 0 99. 2x 2 - x - 4 = 0 97. x 2 + 2x - 3 = 0 100. 2x 2 + 3x - 3 = 0 Find the number of real solutions of each equation using the discriminant. 101. x 2 + 4x + 1 = 0 102. 2x 2 - 3x + 2 = 0 103. x 2 - 5x + 2 = 0 104. 2x 2 - 4x + 2 = 0 105. x 2 + 2x - 5 = 0 106. 2x 2 - 2x - 3 = 0 Extra Practice S21 Use the circle graph for Exercises 5–7. 5. Which candidate received the fewest votes? äÈ äx Óä 6Ì}ÊÀÊ-ÌÕ`iÌ`ÞÊ*ÀiÃ`iÌ 7. A total of 400 students voted in the election. How many votes did Velez receive? 10-2 ä{ 9i>À 6. Which two candidates received approximately the same number of votes? Lesson Óä Óä ä£ 4. Estimate the amount by which the population decreased from 2005 to 2006. äÎ 3. During which one-year period did the population increase by the greatest amount? Óä £x £ä x ä Óä 2. Estimate the population in 2005. *«Õ>ÌÊvÊ`Ûi äÓ 10-1 Use the line graph for Exercises 1–4. 1. In what year was the population the greatest? Óä Lesson Skills Practice Óä Chapter 10 The daily high temperatures in degrees Celsius during a two-week period in Madison, Wisconsin, are given at right. 8. Use the data to make a stem-and-leaf plot. 9. Use the data to make a frequency table with intervals. 10. Use the frequency table from Exercise 9 to make a histogram for the data. >Àià £ä¯ 6iiâ În¯ >Và Óx¯ 9>} Óǯ High Temperatures (oC) 22 25 28 33 29 24 19 19 18 25 32 30 32 25 11. Use the data to make a cumulative frequency table. Lesson 10-3 Find the mean, median, mode, and range of each data set. 12. 42, 45, 48, 45 13. 66, 68, 68, 62, 61, 68, 65, 60 Identify the outlier in each data set, and determine how the outlier affects the mean, median, mode, and range of the data. 14. 4, 8, 15, 8, 71, 7, 6 15. 36, 7, 50, 40, 38, 48, 40 Use the data to make a box-and-whisker plot. 16. 7, 8, 10, 2, 5, 1, 10, 8, 5, 5 17. 54, 64, 50, 48, 53, 55, 57 Lesson 10-4 18. The graph shows the ages of people who listen to a radio program. a. Explain why the graph is misleading. }iÃÊvÊ,>`Ê*À}À>ÊÃÌiiÀà b. What might someone believe because of the graph? c. Who might want to use this graph? Explain. ÓxÊÌÊÎÈ 19. A researcher surveys people at the Elmwood library about the number of hours they spend reading each day. Explain why the following statement is misleading: “People in Elmwood read for an average of 1.5 hours per day.” S22 Extra Practice Îä¯ 1`iÀÊ£n £x¯ £nÊÌÊÓ{ £x¯ Chapter 10 Lesson 10-5 Skills Practice 20. Identify the sample space and the outcome shown for the spinner at right. Write impossible, unlikely, as likely as not, likely, or certain to describe each event. 21. Two people sitting next to each other on a bus have the same birthday. 22. Dylan rolls a number greater than 1 on a standard number cube. An experiment consists of randomly choosing a fruit snack from a box. Use the results in the table to find the experimental probability of each event. 23. choosing a blueberry fruit snack Cherry 8 Peach 6 24. choosing a cherry fruit snack Blueberry 6 Outcome Frequency 25. not choosing a cherry fruit snack Lesson 10-6 Find the theoretical probability of each outcome. 26. rolling an even number on a number cube 27. tossing two coins and both landing tails up 28. randomly choosing a prime number from a bag that contains ten slips of paper numbered 1 through 10 29. The probability of choosing a green marble from a bag is __37 . What is the probability of not choosing a green marble? 30. The odds against winning a game are 8 : 3. What is the probability of winning the game? Lesson 10-7 Tell whether each set of events is independent or dependent. Explain your answer. 31. You pick a bottle from a basket containing chilled drinks, and then your friend chooses a bottle. 32. You roll a 6 on a number cube and you toss a coin that lands heads up. 33. A number cube is rolled three times. What is the probability of rolling three numbers greater than 4? 34. An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 blue marbles, 2 orange marbles, and 5 yellow marbles. What is the probability of selecting a blue marble and then a yellow marble? 35. Madeleine has 3 nickels and 5 quarters in her pocket. She randomly chooses one coin and does not replace it. Then she randomly chooses another coin. What is the probability that she chooses two quarters? Lesson 10-8 For Exercises 36 and 37, tell whether each situation involves combinations or permutations. Then give the number of possible outcomes. 36. How many different ways can three photographs be arranged in a row on a wall? 37. How many different smoothies can be made by blending two of the following fruits: oranges, bananas, strawberries, and peaches? 38. There are 6 entrants in a livestock competition at a county fair. How many different ways can the first-place, second-place, and third-place ribbons be awarded? 39. An amusement park has 7 roller coasters. How many different ways can Jacinto choose 4 different roller coasters to ride? Extra Practice S23 Chapter 11 Lesson 11-1 Skills Practice Find the next three terms in each geometric sequence. 1. 1, 5, 25, 125 … 2. 736, 368, 184, 92, … 3. -2, 6, -18, 54, … 1 1 1, _ 1 , 1, 3, … _ _ 4. 8, 2, , , … 5. 7, -14, 28, -56, … 6. _ 2 8 9 3 7. The first term of a geometric sequence is 2, and the common ratio is 3. What is the 8th term of the sequence? 8. What is the 8th term of the geometric sequence 600, 300, 150, 75, …? Lesson 11-2 Tell whether each set of ordered pairs satisfies an exponential function. Explain your answer. ⎧ ⎧ 1 , 0, 2 , 1, 8 , 2, 32 ⎫⎬ 1 , 0, 0 , 1, _ 1 , 2, 4 ⎫⎬ 9. ⎨ -1, _ 10. ⎨ -1, - _ ) ( ) ( ) ( ( ) ( ) 2 2 2 ⎩ ⎭ ⎩ ⎭ ( ( ) ( )( ) ⎧ 1 , 2, _ 1 ⎫⎬ 11. ⎨(-1, 4), (0, 1), 1, _ 4 16 ⎭ ⎩ ⎧ ⎫ 12. ⎨(0, 0), (1, 3), (2, 12), (3, 27)⎬ ⎩ ⎭ Graph each exponential function. x 1 (4)x 13. y = 3(2) 14. y = _ 2 x x 1 1 _ 16. y = - (2) 17. y = 5 _ 2 2 () Lesson 11-3 ( ) ) 15. y = -3 x 18. y = -2(0.25) x Write an exponential growth function to model each situation. Then find the value of the function after the given amount of time. 19. The rent for an apartment is $6600 per year and increasing at a rate of 4% per year; 5 years. 20. A museum has 1200 members and the number of members is increasing at a rate of 2% per year; 8 years. Write a compound interest function to model each situation. Then find the balance after the given number of years. 21. $4000 invested at a rate of 4% compounded quarterly; 3 years 22. $5200 invested at a rate of 2.5% compounded annually; 6 years Write an exponential decay function to model each situation. Then find the value of the function after the given amount of time. 23. The cost of a stereo system is $800 and is decreasing at a rate of 6% per year; 5 years. 24. The population of a town is 14,000 and is decreasing at a rate of 2% per year; 10 years. Lesson 11-4 S24 Graph each data set. Which kind of model best describes the data? ⎧ ⎫ 25. ⎨(0, 3), (1, 0), (2, -1), (3, 0), (4, 3)⎬ ⎭ ⎧⎩ ⎫ 26. ⎨(-4, -4), (-3, -3.5), (-2, -3), (-1, -2.5), (0, -2), (1, -1.5)⎬ ⎩⎧ ⎭ ⎫ 27. ⎨(0, 4), (1, 2), (2, 1), (3, 0.5), (4, 0.25)⎬ ⎩ ⎭ Look for a pattern in each data set to determine which kind of model best describes the data. ⎧ ⎫ 28. ⎨(-1, -5), (0, -5), (1, -3), (2, 1), (3, 7)⎬ ⎧⎩ ⎫ ⎭ 29. ⎨(0, 0.25), (1, 0.5), (2, 1), (3, 2), (4, 4)⎬ ⎩⎧ ⎭ ⎫ 30. ⎨(-2, 11), (-1, 8), (0, 5), (1, 2), (2, -1)⎬ ⎩ ⎭ Extra Practice Chapter 11 Lesson 11-5 Skills Practice Find the domain of each square-root function. 31. y = √ x+1 32. y = √ x-2+4 x 34. y = √ 3x - 6 35. y = 1 + _ 3 Graph each square-root function. 37. y = √ x+2 38. y = √ x-3 √ 40. y = - √ x Lesson Simplify each expression. 11-6 43. 46. 128 √_ 2 3 _ √ 48 33. y = √ 4+x 36. y = √ 4x - 1 39. y = √ 3x + 1 41. y = 2 √ x+1 42. y = 3 √ x-2 2 44. √7 + 24 2 45. 47. y 2 + 4y + 4 √ (4 - x)2 √ 48. √ 52 - 42 Simplify. All variables represent nonnegative numbers. 72 49. √ 50. 75x 4y 3 √ 51. 64 √_ x 53. 16a _ √ 25b 54. 52. Lesson 11-7 Lesson 11-8 6 4 2 11 _ √ 81 18x _ √ 49x 4 3 Add or subtract. 55. 5 √ 7 + 3 √7 56. 6 √ 2 + √ 2 57. 5 √ 3 - 2 √3 - 9 √ 58. √ 5 + 7 √5 5 59. 2 √ y + 4 √ y - 3 √ y - 3 √ 60. 5 √ 3 + 4 √2 3 Simplify each expression. 61. √ 75 + √ 27 62. √ 45 - √ 20 63. 2 √ 12 + √ 18 64. 3 √ 27x + √ 48x 65. 5 √ 20y - 2 √ 80y 66. √ 28a + 2 √ 63a - √ 175a 67. √ 50y - 2 √ 18y + 3 √ 8y 68. √ 12x - √ 27x - √ 5x 69. 5 √ 180s - 6 √ 80s Multiply. Write each product in simplest form. 70. √ 5 √ 10 71. √ 6 √ 12 73. (2 √ 7) 2 76. 2 √ 5 ( √ 20 + 3) 79. (3 + √ 5 )(8 - √ 5) 74. √ 6x √ 15x 77. √ 2x (3 + √ 8x ) 80. (4 + √ 2) 2 72. (3 √ 3) 2 ) 75. √ 3 (2 + √27 78. (4 + √ 3 )(1 - √ 3) 81. (5 - √ 3) 2 Simplify each quotient. √ 5 82. _ √ 3 √ 5 7 85. _ √ 50 Lesson 11-9 2 √ 7 83. _ √ 5 √ 12a 86. _ √ 32 Solve each equation. Check your answer. 88. √x 89. √ 3x = 9 = 11 √ 3 84. _ √ 20 √ 200x 87. _ √ 28 90. √ -2x = 10 91. 5 = √ -4x 92. 94. √ 3x + 1 = 4 95. √ 2x + 5 = 3 96. √ x-4+1=7 97. √ 6 - 3x - 2 = 4 98. √ 6 - x - 5 = -3 99. 4 √ x = 20 √ x + 5 = 12 93. √ x -4=1 Extra Practice S25 Chapter 12 Lesson 12-1 Skills Practice Tell whether each relationship is an inverse variation. Explain. 1. x y 4 2. x y 8 2 8 16 16 32 3. x y 6 -1 24 3 4 2 -12 32 6 2 4 -6 64 12 1 8 -3 4. 3xy = 10 5. y - x = 6 6. 6xy = -1 7. Write and graph the inverse variation in which y = 4 when x = 3. 1 when x = 6. 8. Write and graph the inverse variation in which y = _ 2 9. Let x 1 = 6, y 1 = 8, and x 2 = 12. Let y vary inversely as x. Find y 2. 10. Let x 1 = -4, y 1 = -2, and y 2 = 16. Let y vary inversely as x. Find x 2. Lesson 12-2 Identify the excluded values for each rational function. 16 11. y = _ x 3 13. y = - _ x+5 20 14. y = _ x + 20 8 16. y = _ x+5 7 17. y = _ -6 3x - 2 3 18. y = _ +4 2x - 2 1 19. y = _ x+3 1 20. y = _ x-2 1 +4 21. y = _ x 3 22. y = _ x-2 1 +2 23. y = _ x-3 1 -6 24. y = _ x-5 1 +5 25. y = _ x+2 1 +1 26. y = _ x+5 1 12. y = _ x-1 Identify the asymptotes. 2 15. y = _ x-4 Graph each function. Lesson Find the excluded value(s) of each rational expression. 12-3 3 27. _ 7x -2 28. _ x2 - x 6 29. __ x 2 + x - 12 p+1 30. __ 2 p + 4p - 5 Simplify each rational expression, if possible. Identify excluded values. 4m 2 31. _ 12m 7x 5 32. _ 28x 4x 2 - 8x 33. _ x-2 2y 34. _ y-1 5x 3 + 20x 2 35. _ x+4 a+1 36. _ a-2 3y 3 + 3y 37. _ y2 + 1 x 3 + 4x 38. _ x2 + 4 Simplify each rational expression, if possible. S26 b+2 39. __ b 2 + 5b + 6 x-3 40. __ x 2 - 6x + 9 y 2 - 4y - 5 41. __ y 2 - 2y - 3 (m + 2)2 42. __ m 2 - 6m - 16 x2 - 9 43. __ 2 x + x - 12 2-m 44. _ 3m 2 - 6m x-4 45. _ 12x 2 - 3x 3 6 - 3x 46. __ x 2 - 6x + 8 Extra Practice Chapter 12 Skills Practice Lesson Multiply. Simplify your answer. 12-4 ab 4a 3 · _ 47. _ 3 b 6a 2 x-2 ·_ 2x - 10 50. _ x-5 3 1 (x 2 - 2x - 8) 53. _ 2x + 4 4r 3 + 8r _ · 2r 56. _ r3 3r + 6 4b 2 + 4 b 2 - 1 59. _ · _ b-1 8b 2 + 8 6x 3y _ 8x 2yz 2 48. _ · 4y 4 3xz 5 x-3 ·_ 8 49. _ 2 4x - 12 2 3 9b 2 b ·_ _ 51. a 6a 3c 12b 5c 2 3x (x 2 + x - 30) 54. _ 4x - 20 3x + 6 3x · _ 52. _ 2x + 4 9 2y 55. _ (y 2 + 10y + 25) 3y + 15 x2 + x x-3 57. _ ·_ 2 x - x - 6 6x 2 + 6x pq + 2q 3pq + 3 60. _ · _ pq + 1 pq 2 + 2q 2 2 2 - 3a - 10 · a - 2a - 3 __ __ 58. a 2 a-5 a -a-6 x 2 + 4x + 3 63. __ ÷ (x 2 - 1) 3x 3 + 9x 2 p 2 - 2p p-1 __ ÷ 64. __ p 2 + 4p - 5 p 2 + 3p - 10 x2 + 1 1 - 3x 66. _ + _ x-1 x-1 2x 2x 2 + __ 67. __ 2 x - 2x - 3 x 2 - 2x - 3 r 2 + 3r + 2 2r + 6 61. _ · _ 4r + 4 r 2 - 2r - 8 Divide. Simplify your answer. 3x 2y 3 6y 4 _ 62. _ ÷ x 2z 5 x 3z 2 Lesson Add. Simplify your answer. 12-5 5x 3x + _ 65. _ 4x 3 4x 3 Subtract. Simplify your answer. 5 -_ 2 68. _ 6y 4 6y 4 Lesson 12-6 5a 2 + 1 15a + 1 m 2 + 2m m + 12 69. _ -_ 70. _ -_ 2 2 a -a-6 a -a-6 m2 - 9 m2 - 9 Find the LCM of the given expressions. 71. 8x 5y 8, 6x 4y 9 72. x 2 - 4, x 2 + 7x + 10 73. d 2 - 2d - 3, d 2 + d - 12 Add or subtract. Simplify your answer. 3 5 5 -_ 1 75. _ +_ 74. _ y 2 4y 2 x2 - x - 6 x + 2 3x - _ x 76. _ x-2 2-x Divide. 77. (12y 5 - 16y 2 + 4y) ÷ 4y 2 78. (6m 4 - 18m + 3) ÷ 6m 2 79. (16x 4 + 20x 3 - 4x) ÷ -4x 3 2 - 4b - 5 __ 80. b b+1 2x 2 + 9x + 4 81. __ x+4 Divide using long division. 83. (a 2 - 5a - 6) ÷ (a + 1) 84. (2x 2 + 10x + 8) ÷ (x + 4) 85. (3y 2 - 11y + 10) ÷ (y - 2) a 2 - 13a - 5 __ 82. 6 3a + 1 86. (3x 2 - 2x - 7) ÷ (x - 2) 87. (2x 2 + 2x - 9) ÷ (x + 3) 88. (5x 3 + 2x 2 - 4) ÷ (x - 2) Lesson 12-7 Solve. Check your answer. 5 =_ 10 4 4 =_ 89. _ 90. _ t x+1 x-1 t+9 8 3 4 =_ 1 92. _ 93. _ =_ y a-2 a+1 2y + 4 x 3 3 1 1 2 96. _ + _ = _ 95. _ + _ = - _2 x 2 2m 2 2 m Solve. Identify any extraneous solutions. x+2 3 =_ x-5 2=_ 98. _ 99. _ x x+4 x+4 x2 - 4 8 6 _ 91. _ m = m+1 5 6 94. _ =_ 4w - 2 5w - 2 3 _ 10 97. 1 - _ a = a2 4x - 7 = _ 16 100. _ x-4 x-4 Extra Practice S27 Extra Practice Chapter 1 Applications Practice Biology Use the following information for Exercises 1 and 2. In general, skin cells in the human body contain 46 chromosomes. (Lesson 1-1) 1. Write an expression for the number of chromosomes in c skin cells. 2. Find the number of chromosomes in 8, 15, and 50 skin cells. 3. On a winter day in Fairbanks, Alaska, the temperature dropped from 12 °F to -16 °F. How many degrees did the temperature drop? (Lesson 1-2) 4. Geography The elevation of the Dead Sea in Jordan is -411 meters. The greatest elevation on Earth is Mt. Everest, at 8850 meters. What is the difference in elevation between these two locations? (Lesson 1-2) 5. Jeremy is raising money for his school by selling magazine subscriptions. Each subscription costs $16.75. During the first week, he sells 12 subscriptions. How much money does he raise? (Lesson 1-3) 6. As a service charge, Nadine’s checking account is adjusted by -$3 each month. What is the total amount of the adjustment over the course of one year? (Lesson 1-3) 7. To go from one figure to the next in the sequence of figures, each square is split into four smaller squares. How many squares will be in Figure 5? (Lesson 1-4) 10. An art museum exhibits a square painting that has an area of 75 square feet. Find its side length to the nearest tenth. (Lesson 1-5) 11. Travel The base of the Washington Monument in Washington, D.C., is a square with an area of 336 yards. Find the length of one side of the monument’s base to the nearest tenth. (Lesson 1-5) 12. The toll to cross a bridge is $2 for cars, $5 for trucks, and $10 for buses. The total amount of money collected can be found using the expression 2C + 5T + 10B. Use the table to find the total amount of money collected between 10 A.M. and 11 A.M. (Lesson 1-6) Bridge Tolls, 10 A.M. to 11 A.M. Type of Vehicle Car C Truck T Bus B Number 104 20 3 13. The expression __59 (F - 32) converts a temperature F in degrees Fahrenheit to a temperature in degrees Celsius. Convert 77 °F to degrees Celsius. (Lesson 1-6) Use the following information for Exercises 14 and 15. An airplane has 12 rows of seats in first class and 35 rows of seats in coach. Each row has the same number of seats. (Lesson 1-7) 14. The total number of seats in the plane is 12x + 35x, where x is the number of seats in a row. Simplify the expression. }ÕÀiÊä S28 }ÕÀiÊ£ }ÕÀiÊÓ 15. Find the total number of seats given that the plane has 6 seats per row. 8. When you fold a sheet of paper in half and then open it, the crease creates 2 regions. Folding the paper in half 2 times creates 4 regions. How many regions do you create when you fold a sheet of paper in half 5 times? (Lesson 1-4) Use the following information for Exercises 16 and 17. A sales representative earns $680 per week plus a $40 commission for each sale. (Lesson 1-8) 9. Dan began his stamp collection with just 5 stamps in the first year. Every year thereafter, his collection grew 5 times as large as the year before. How many stamps were in Dan’s collection after 4 years? (Lesson 1-4) 17. Write ordered pairs for the representative’s weekly earnings for 5, 8, and 10 sales. Extra Practice 16. Write a rule for the sales representative’s weekly earnings. Chapter 2 Applications Practice 1. Economics In 2004, the average price of an ounce of gold was $47 more than the average price in 2003. The 2004 price was $410. Write and solve an equation to find the average price of an ounce of gold in 2003. (Lesson 2-1) 2. During a renovation, 36 seats were removed from a theater. The theater now seats 580 people. Write and solve an equation to find the number of seats in the theater before the renovation. (Lesson 2-1) 3. A case of juice drinks contains 12 bottles and costs $18. Write and solve an equation to find the cost of each drink. (Lesson 2-2) 4. Astronomy Objects weigh about 3 times as much on Earth as they do on Mars. A rock weighs 42 lb on Mars. Write and solve an equation to find the rock’s weight on Earth. (Lesson 2-2) 9. Charles is hanging a poster on his wall. He wants the top of the poster to be 84 inches from the floor but would be happy for it to be 3 inches higher or lower. Write and solve an absolute-value equation to find the maximum and minimum acceptable heights. (Lesson 2-6) 10. The ratio of students to adults on a school trip is 9 : 2. There are 6 adults on the trip. How many students are there? (Lesson 2-7) 11. A cheetah can reach speeds of up to 103 feet per second. Use dimensional analysis to convert the cheetah’s speed to miles per hour. Round to the nearest tenth. (Lesson 2-7) 12. Write and solve a proportion to find the height of the flagpole. (Lesson 2-8) 5. The county fair’s admission fee is $8 and each ride costs $2.50. Sonia spent a total of $25.50. How many rides did she go on? (Lesson 2-3) 6. At the beginning of a block party, the temperature was 84°. During the party, the temperature dropped 3° every hour. At the end of the party, the temperature was 66°. How long was the party? (Lesson 2-3) 7. Consumer Economics A health insurance policy costs $700 per year, plus $15 for each visit to the doctor’s office. A different plan costs $560 per year, but each office visit is $50. Find the number of office visits for which the two plans have the same total cost. (Lesson 2-4) 8. Geometry The formula A = __12 bh gives the area A of a triangle with base b and height h. (Lesson 2-5) a. Solve A = 1 __ bh 2 for h. b. Find the height of a triangle with an area of 30 square feet and a base of 6 feet. ¶ x°{ÊvÌ n°£ÊvÌ ÓÇÊvÌ 13. Paul has 8 jazz CDs. The jazz CDs are 5% of his collection. How many CDs does Paul have? (Lesson 2-9) 14. Miguel earns an annual salary of $38,000 plus a 3.5% commission on sales. His sales for one year were $90,000. Find his total salary for the year. (Lesson 2-10) 15. How long would it take $3600 to earn simple interest of $450 at an annual interest rate of 5%? (Lesson 2-10) 16. A store sells swimsuits at a 30% discount. What is the final price of a swimsuit that originally sold for $28? (Lesson 2-11) 17. Mei sells jam at a farmer’s market for $4.20 per jar. Each jar costs Mei $3 to make. What is the markup as a percent? (Lesson 2-11) Extra Practice S29 Chapter 3 Applications Practice 1. At a food-processing factory, each box of cereal must weigh at least 15 ounces. Define a variable and write an inequality for the acceptable weights of the cereal boxes. Graph the solutions. (Lesson 3-1) 8. The admission fee at an amusement park is $12, and each ride costs $3.50. The park also offers an all-day pass with unlimited rides for $33. For what numbers of rides is it cheaper to buy the all-day pass? (Lesson 3-4) 2. In order to qualify for a discounted entry fee at a museum, a visitor must be less than 13 years old. Define a variable and write an inequality for the ages that qualify for the discounted entry fee. Graph the solutions. (Lesson 3-1) 9. The table shows the cost of Internet access at two different cafes. For how many hours of access is the cost at Cyber Station less than the cost at Web World? (Lesson 3-5) 3. A restaurant can seat no more than 102 customers at one time. There are already 96 customers in the restaurant. Write and solve an inequality to find out how many additional customers could be seated in the restaurant. (Lesson 3-2) 4. Meteorology A hurricane is a tropical storm with a wind speed of at least 74 mi/h. A meteorologist is tracking a storm whose current wind speed is 63 mi/h. Write and solve an inequality to find out how much greater the wind speed must be in order for this storm to be considered a hurricane. (Lesson 3-2) Hobbies Use the following information for Exercises 5–7. When setting up an aquarium, it is recommended that you have no more than one inch of fish per gallon of water. For example, in a 30-gallon tank, the total length of the fish should be at most 30 inches. (Lesson 3-3) Freshwater Fish Name Length (in.) Red tail catfish 3.5 Blue gourami 1.5 5. Write an inequality to show the possible numbers of blue gourami you can put in a 10-gallon aquarium. 6. Find the possible numbers of blue gourami you can put in a 10-gallon aquarium. 7. Find the possible numbers of red tail catfish you can put in a 20-gallon aquarium. S30 Extra Practice Internet Access Cafe Cost Cyber Station $12 one-time membership fee $1.50 per hour Web World No membership fee $2.25 per hour 10. Larissa is considering two summer jobs. A job at the mall pays $400 per week plus $15 for every hour of overtime. A job at the movie theater pays $360 per week plus $20 for every hour of overtime. How many hours of overtime would Larissa have to work in order for the job at the movie theater to pay a higher salary than the job at the mall? (Lesson 3-5) 11. Health For maximum safety, it is recommended that food be stored at a temperature between 34 °F and 40 °F inclusive. Write a compound inequality to show the temperatures that are within the recommended range. Graph the solutions. (Lesson 3-6) 12. Physics Color is determined by the wavelength of light. Wavelengths are measured in nanometers (nm). Our eyes see the color green when light has a wavelength between 492 nm and 577 nm inclusive. Write a compound inequality to show the wavelengths that produce green light. Graph the solutions. (Lesson 3-6) 13. Allison ran a mile in 8 minutes. She wants to run a second mile within 0.75 minute of her time for the first mile. Write and solve an absolute-value inequality to find the range of acceptable times for the second mile. (Lesson 3-7) Chapter 4 Applications Practice 1. Donnell drove on the highway at a constant speed and then slowed down as she approached her exit. Sketch a graph to show the speed of Donnell’s car over time. Tell whether the graph is continuous or discrete. (Lesson 4-1) 2. Lori is buying mineral water for a party. The bottles are available in six-packs. Sketch a graph showing the number of bottles Lori will have if she buys 1, 2, 3, 4, or 5 six-packs. Tell whether the graph is continuous or discrete. (Lesson 4-1) 3. Health To exercise effectively, it is important to know your maximum heart rate. You can calculate your maximum heart rate in beats per minute by subtracting your age from 220. (Lesson 4-2) a. Express the age x and the maximum heart rate y as a relation in table form by showing the maximum heart rate for people who are 20, 30, 35, and 40 years old. 7. The function y = 3.5x describes the number of miles y that the average turtle can walk in x hours. Graph the function. Use the graph to estimate how many miles a turtle can walk in 4.5 hours. (Lesson 4-4) 8. Earth Science The Kangerdlugssuaq glacier in Greenland is flowing into the sea at the rate of 1.6 meters per hour. The function y = 1.6x describes the number of meters y that flow into the sea in x hours. Graph the function. Use the graph to estimate the number of meters that flow into the sea in 8 hours. (Lesson 4-4) 9. The scatter plot shows a relationship between the number of lemonades sold in a day and the day’s high temperature. Based on this relationship, predict the number of lemonades that will be sold on a day when the high temperature is 96 °F. (Lesson 4-5) i>`iÊ->ià b. Is this relation a function? Explain. Season Statistics Wins Home Runs 95 185 93 133 80 140 93 167 5. Michael uses 5.5 cups of flour for each loaf of bread that he bakes. He plans to bake a maximum of 4 loaves. Write a function to describe the number of cups of flour used. Find a reasonable domain and range for the function. (Lesson 4-3) 6. A gym offers the following special rate. New members pay a $425 initiation fee and then pay $90 per year for 1, 2, or 3 years. Write a function to describe the situation. Find a reasonable domain and range for the function. (Lesson 4-3) Õ«ÃÊÃ` 4. Sports The table shows the number of games won by four baseball teams and the number of home runs each team hit. Is this relation a function? Explain. (Lesson 4-2) nä Èä {ä Óä ä Óä {ä Èä nä } ÊÌi«iÀ>ÌÕÀiÊc® 10. In month 1 the Elmwood Public Library had 85 Spanish books in its collection. Each month, the librarian plans to order 8 new Spanish books. How many Spanish books will the library have in month 15? (Lesson 4-6) 11. Nikki purchases a card that she can use to ride the bus in her town. Each time she rides the bus $1.50 is deducted from the value of the card. After her first ride, there is $43.50 left on the card. How much money will be left on the card after Nikki has taken 12 bus rides? (Lesson 4-6) Extra Practice S31 Chapter 5 Applications Practice 1. Jennifer is having prints made of her photographs. Each print costs $1.50. The function f (x) = 1.50x gives the total cost of the x prints. Graph this function and give its domain and range. (Lesson 5-1) 7. Sports Competitive race-walkers move at a speed of about 9 miles per hour. Write a direct variation equation for the distance y that a race-walker will cover in x hours. Then graph. (Lesson 5-6) 2. The Chang family lives 400 miles from Denver. They drive to Denver at a constant speed of 50 mi/h. The function f (x) = 400 - 50x gives their distance in miles from Denver after x hours. (Lesson 5-2) 8. A bicycle rental costs $10 plus $1.50 per hour. The cost as a function of the number of hours is shown in the graph. (Lesson 5-7) VÞViÊ,iÌ>Ê ÃÌà a. Graph this function and find the intercepts. ÃÌÊf® Îä b. What does each intercept represent? 3. History The table shows the number of nations in the United Nations in different years. Find the rate of change for each time interval. During which time interval did the U.N. grow at the greatest rate? (Lesson 5-3) Year Ó { È /iÊ ® 1960 1975 a. Write an equation that represents the cost as a function of the number of hours. 51 60 99 144 b. Identify the slope and y-intercept and describe their meaning. "ÛiÊ/i«iÀ>ÌÕÀi /i«iÀ>ÌÕÀiÊc® ä 1950 4. The graph shows the temperature of an oven at different times. Find the slope of the line. Then tell what the slope represents. (Lesson 5-4) {xä £ä]Ê{£ä® Îxä c. Find the cost of renting a bike for 6 hours. 9. A hot-air balloon is moving at a constant rate. Its altitude is a linear function of time, as shown in the table. Write an equation in slope-intercept form that represents this function. Then find the balloon’s altitude after 25 minutes. (Lesson 5-8) Balloon’s Altitude {ä]ÊÓä® Óxä ä Óä {ä /iÊ® 5. A straight highway connects the towns of Dale and Winslow. On a map, the coordinates of Dale are (5, 16), and the coordinates of Winslow are (9, 24). A rest area is located on the highway at the midpoint between the towns. What are the map coordinates of the rest area? (Lesson 5-5) 6. On a map, a campground is at (3, 5), and a fishing area is at (8, 4). Each unit on the map represents 0.1 mile. To the nearest tenth of a mile, what is the distance between the campground and the fishing area? (Lesson 5-5) Extra Practice £ä 1945 Number of Nations S32 Óä Time (min) Altitude (m) 0 250 7 215 12 190 10. Geometry Show that the points A(2, 3), B(3, 1), C (-1, -1), and D(-2, 1) are the vertices of a rectangle. (Lesson 5-9) 11. A phone plan for international calls costs $12.50 per month plus $0.04 per minute. The monthly cost for x minutes of calls is given by the function f (x) = 0.04x + 12.50. How will the graph change if the phone company raises the monthly fee to $14.50? if the cost per minute is raised to $0.05? (Lesson 5-10) Chapter 6 Applications Practice 1. Net Sounds, an online music store, charges $12 per CD plus $3 for shipping and handling. Web Discs charges $10 per CD plus $9 for shipping and handling. For how many CDs will the cost be the same? What will that cost be? (Lesson 6-1) 2. At Rocco’s Restaurant, a large pizza costs $12 plus $1.25 for each additional topping. At Pizza Palace, a large pizza costs $15 plus $0.75 for each additional topping. For how many toppings will the cost be the same? What will that cost be? (Lesson 6-1) Use the following information for Exercises 3 and 4. The coach of a baseball team is deciding between two companies that manufacture team jerseys. One company charges a $60 setup fee and $25 per jersey. The other company charges a $200 setup fee and $15 per jersey. (Lesson 6-2) 3. For how many jerseys will the cost at the two companies be the same? What will that cost be? 4. The coach is planning to purchase 20 jerseys. Which company is the better option? Why? 5. Geometry The length of a rectangle is 5 inches greater than the width. The sum of the length and width is 41 inches. Find the length and width of the rectangle. (Lesson 6-2) 6. At a movie theater, tickets cost $9.50 for adults and $6.50 for children. A group of 7 moviegoers pays a total of $54.50. How many adults and how many children are in the group? (Lesson 6-3) 7. Business A grocer is buying large quantities of fruit to resell at his store. He purchases apples at $0.50 per pound and pears at $0.75 per pound. The grocer spends a total of $17.25 for 27 pounds of fruit. How many pounds of each fruit does he buy? (Lesson 6-3) 8. Bricks are available in two sizes. Large bricks weigh 9 pounds, and small bricks weigh 4.5 pounds. A bricklayer has 14 bricks that weigh a total of 90 pounds. How many of each type of brick are there? (Lesson 6-3) 9. Sports The table shows the time it took two runners to complete the Boston Marathon in several different years. If the patterns continue, will Shanna ever complete the marathon in the same number of minutes as Maria? Explain. (Lesson 6-4) Marathon Times (min) 2003 2004 2005 2006 Shanna 190 182 174 166 Maria 175 167 159 151 10. Jordan leaves his house and rides his bike at 10 mi/h. After he goes 4 miles, his brother Tim leaves the house and rides in the same direction at 12 mi/h. If their rates stay the same, will Tim ever catch up to Jordan? Explain. (Lesson 6-4) 11. Charmaine is buying almonds and cashews for a reception. She wants to spend no more than $18. Almonds cost $4 per pound, and cashews cost $5 per pound. Write a linear inequality to describe the situation. Graph the solutions. Then give two combinations of nuts that Charmaine could buy. (Lesson 6-5) 12. Luis is buying T-shirts to give out at a school fund-raiser. He must spend less than $100 for the shirts. Child shirts cost $5 each, and adult shirts cost $8 each. Write a linear inequality to describe the situation. Graph the solutions. Then give two combinations of shirts that Luis could buy. (Lesson 6-5) 13. Nicholas is buying treats for his dog. Beef cubes cost $3 per pound, and liver cubes cost $2 per pound. He wants to buy at least 2 pounds of each type of treat, and he wants to spend no more than $14. Graph all possible combinations of the treats that Nicholas could buy. List two possible combinations. (Lesson 6-6) 14. Geometry The perimeter of a rectangle is at most 20 inches. The length and the width are each at least 3 inches. Graph all possible combinations of lengths and widths that result in such a rectangle. List two possible combinations. (Lesson 6-6) Extra Practice S33 Chapter 7 Applications Practice 1. The eye of a bee is about 10 -3 m in diameter. Simplify this expression. (Lesson 7-1) 2. A typical stroboscopic camera has a shutter speed of 10 -6 seconds. Simplify this expression. (Lesson 7-1) 3. Space Exploration During a mission that took place in August, 2005, the Space Shuttle Discovery traveled a total distance of 9.3 × 10 6 km. The Space Shuttle’s velocity was 28,000 km/h. (Lesson 7-2) a. Write the total distance that the Space Shuttle traveled in standard form. b. Write the Space Shuttle’s velocity in scientific notation. 4. There are approximately 10,000,000 grains in a pound of salt. Write this number in scientific notation. (Lesson 7-2) 5. A high-speed centrifuge spins at a speed of 2 × 10 4 rotations per minute. How many rotations does it make in one hour? Write your answer in scientific notation. (Lesson 7-3) 6. Astronomy Earth travels approximately 5.8 × 10 8 miles as it makes one orbit of the Sun. How far does Earth travel in 50 years? (Note: One year is one orbit of the Sun.) Write your answer in scientific notation. (Lesson 7-3) 7. Geography In 2005, the population of Indonesia was 2.4 × 10 8. This was 8 times the population of Afghanistan. What was the population of Afghanistan in 2005? Write your answer in standard form. (Lesson 7-4) 8. The Golden Gate Bridge weighs about 1.8 × 10 9 lb. The Eiffel Tower weighs about 2.25 × 10 7 lb. How many times heavier is the Golden Gate Bridge than the Eiffel Tower? Write your answer in standard form. (Lesson 7-4) 10. A rock is thrown off a 220-foot cliff with an initial velocity of 50 feet per second. The height of the rock above the ground is given by the polynomial -16t 2 - 50t + 220, where t is the time in seconds after the rock has been thrown. What is the height of the rock above the ground after 2 seconds? (Lesson 7-6) 11. The sum of the first n natural numbers is given by the polynomial __12 n 2 + __12 n. Use this polynomial to find the sum of the first 9 natural numbers. (Lesson 7-6) 12. Biology The population of insects in a meadow depends on the temperature. A biologist models the population of insect A with the polynomial 0.02x 2 + 0.5x + 8 and the population of insect B with the polynomial 0.04x 2 - 0.2x + 12, where x represents the temperature in degrees Fahrenheit. (Lesson 7-7) a. Write a polynomial that represents the total population of both insects. b. Write a polynomial that represents the difference of the populations of insect B and insect A. 13. Geometry The length of the rectangle shown is 1 inch longer than 3 times the width. a. Write a polynomial that represents the area of the rectangle. b. Find the area of the rectangle when the width is 4 inches. (Lesson 7-8) Ý ÎÝÊÊ£ 14. A cabinet maker starts with a square piece of wood and then cuts a square hole from its center as shown. Write a polynomial that represents the area of the remaining piece of wood. (Lesson 7-9) 9. Carl has 4 identical cubes lined up in a row and wants to find the total length of the cubes. He knows that the volume of one cube is 1 _ 343 in3. Use the formula s = V 3 to find the length of one cube. What is the length of the row of cubes? (Lesson 7-5) S34 Extra Practice ÝÊÊÎ ÝÊÊÈ Chapter 8 Applications Practice 1. Ms. Andrews’s class has 12 boys and 18 girls. For a class picture, the students will stand in rows on a set of steps. Each row must have the same number of students, and each row will contain only boys or girls. How many rows will there be if Ms. Andrews puts the maximum number of students in each row? (Lesson 8-1) 2. A museum director is planning an exhibit of Native American baskets. There are 40 baskets from North America and 32 baskets from South America. The baskets will be displayed on shelves so that each shelf has the same number of baskets. Baskets from North and South America will not be placed together on the same shelf. How many shelves will be needed if each shelf holds the maximum number of baskets? (Lesson 8-1) 3. The area of a rectangular painting is (3x 2 + 5x) ft 2. Factor this polynomial to find possible expressions for the dimensions of the painting. (Lesson 8-2) 4. Geometry The surface area of a cylinder with radius r and height h is given by the expression 2πr 2 + 2πrh. Factor this expression. (Lesson 8-2) 5. The area of a rectangular classroom in square feet is given by x 2 + 9x + 18. The width of the classroom is (x + 3) ft. What is the length of the classroom? (Lesson 8-3) 8. A rectangular poster has an area of (6x 2 + 19x + 15) in 2. The width of the poster is (2x + 3) in. What is the length of the poster? (Lesson 8-4) 9. Physics The height of an object thrown upward with a velocity of 38 feet per second from an initial height of 5 feet can be modeled by the polynomial -16t 2 + 38t + 5, where t is the time in seconds. Factor this expression. Then use the factored expression to find the object’s height after __12 second. (Lesson 8-4) 10. A rectangular pool has an area of (9x 2 + 30x + 25) ft 2. The dimensions of the pool are of the form ax + b, where a and b are whole numbers. Find an expression for the perimeter of the pool. Then find the perimeter when x = 5. (Lesson 8-5) 11. Geometry The area of a square is 9x 2 - 24x + 16. Find the length of each side of the square. Is it possible for x to equal 1 in this situation? Why or why not? (Lesson 8-5) Architecture Use the following information for Exercises 12–14. An architect is designing a rectangular hotel room. A balcony that is 5 feet wide runs along the length of the room, as shown in the figure. (Lesson 8-6) ÓÝÊvÌ xÊvÌ Gardening Use the following information for Exercises 6 and 7. A rectangular flower bed has a width of (x + 4) ft. The bed will be enlarged by increasing the length, as shown. (Lesson 8-3) ÝÊÊ{®ÊvÌ 6. The original flower bed has an area of (x 2 + 9x + 20) ft 2. What is its length? 12. The area of the room, including the balcony, is (4x 2 + 12x + 5) ft 2. Tell whether the polynomial is fully factored. Explain. 13. Find the length and width of the room (including the balcony). 14. How long is the balcony when x = 9? 7. The enlarged flower bed will have an area of (x 2 + 12x + 32) ft 2. What will be the new length of the flower bed? Extra Practice S35 Chapter 9 Applications Practice 1. The table shows the height of a ball at various times after being thrown into the air. Tell whether the function is quadratic. Explain. (Lesson 9-1) Time (s) 0 0.5 1 1.5 2 Height (ft) 4 20 28 28 20 2. The height of the curved roof of a camping tent can be modeled by f (x) = -0.5x 2 + 3x, where x is the width in feet. Find the height of the tent at its tallest point. (Lesson 9-2) 3. Engineering A small bridge passes over a stream. The height in feet of the bridge’s curved arch support can be modeled by f (x) = -0.25x 2 + 2x + 1.5, where the x-axis represents the level of the water. Find the greatest height of the arch support. (Lesson 9-2) 4. Sports The height in meters of a football that is kicked from the ground is approximated by f (x) = -5x 2 + 20x, where x is the time in seconds after the ball is kicked. Find the ball’s maximum height and the time it takes the ball to reach this height. Then find how long the ball is in the air. (Lesson 9-3) 5. Physics Two golf balls are dropped, one from a height of 400 feet and the other from a height of 576 feet. (Lesson 9-4) a. Compare the graphs that show the time it takes each golf ball to reach the ground. b. Use the graphs to tell when each golf ball reaches the ground. 6. A model rocket is launched into the air with an initial velocity of 144 feet per second. The quadratic function y = -16x 2 + 144x models the height of the rocket after x seconds. How long is the rocket in the air? (Lesson 9-5) 7. A gymnast jumps on a trampoline. The quadratic function y = -16x 2 + 24x models her height in feet above the trampoline after x seconds. How long is the gymnast in the air? (Lesson 9-5) S36 Extra Practice 8. A child standing on a rock tosses a ball into the air. The height of the ball above the ground is modeled by h = -16t 2 + 28t + 8, where h is the height in feet and t is the time in seconds. Find the time it takes the ball to reach the ground. (Lesson 9-6) 9. A fireworks rocket is launched from the edge of a rooftop. The height of the rocket above the ground is modeled by h = - 16t 2 + 40t + 24, where h is the height in feet and t is the time in seconds. Find the time it takes the rocket to hit the ground. (Lesson 9-6) 10. Geometry The base of the triangle in the figure is five times the height. The area of the triangle is 400 in 2. Find the height of the triangle to the nearest tenth. (Lesson 9-7) Ý xÝ 11. The length of a rectangular swimming pool is 8 feet greater than the width. The pool has an area of 240 ft 2. Find the length and width of the pool. (Lesson 9-8) 12. Geometry One base of a trapezoid is 4 ft longer than the other base. The height of the trapezoid is equal to the shorter base. The trapezoid’s area is 80 ft 2. Find the height. Hint: A = __12 h(b 1 + b 2) (Lesson 9-8) ( ) Ý Ý ÝÊÊ{ 13. A referee tosses a coin into the air at the start of a football game to decide which team will get the ball. The height of the coin above the ground is modeled by h = -16t 2 + 12t + 4, where h is the height in feet and t is the time in seconds after the coin is tossed. Will the coin reach a height of 8 feet? Use the discriminant to explain your answer. (Lesson 9-9) Chapter 10 Applications Practice Geography Use the following information for Exercises 1–3. The bar graph shows the areas of the Great Lakes. (Lesson 10-1) 7. Use the data to make a box-and-whisker plot. 8. The weekly salaries of five employees at a restaurant are $450, $500, $460, $980, and $520. Explain why the following statement is misleading: “The average salary is $582.” (Lesson 10-4) Ài>ÃÊvÊÌ iÊÀi>ÌÊ>ià 9. The graph shows the sales figures for three sales representatives. Explain why the graph is misleading. What might someone believe because of the graph? (Lesson 10-4) >iÊ"Ì>À >iÊV }> ->iÃÊvÀÊ"VÌLiÀ à iÀ Ü À 1. Estimate the difference in the areas between the lake with the greatest area and the lake with the least area. ` Îä]äää Óä]äää Ài>ÊÓ® £ä]äää > ä £n]äää £Ç]äää £È]äää £x]äää £{]äää >iÊ-Õ«iÀÀ à ->iÃÊf® >iÊÕÀ 7 >i >iÊÀi ->iÃÊ,i«ÀiÃiÌ>ÌÛi 2. Estimate the total area of the five lakes. 3. Approximately what percent of the total area is Lake Superior? 4. The scores of 18 students on a Spanish exam are given below. Use the data to make a stemand-leaf plot. (Lesson 10-2) Exam Scores 65 94 92 75 71 83 77 73 91 82 63 79 80 77 99 76 80 88 5. The numbers of customers who visited a hair salon each day are given below. Use the data to make a frequency table with intervals. (Lesson 10-2) Number of Customers Per Day 32 35 29 44 41 25 35 40 41 32 33 28 33 34 Sports Use the following information for Exercises 6 and 7. The numbers of points scored by a college football team in 11 games are given below. (Lesson 10-3) 10. A manager inspects 120 stereos that were built at a factory. She finds that 6 are defective. What is the experimental probability that a stereo chosen at random will be defective? (Lesson 10-5) Travel Use the following information for Exercises 11–13. A row of an airplane has 2 window seats, 3 middle seats, and 4 aisle seats. You are randomly assigned a seat in the row. (Lesson 10-6) 11. Find the probability that you are assigned a window seat. 12. Find the odds in favor of being assigned a window seat. 13. Find the probability that you are not assigned a middle seat. 14. A class consists of 19 boys and 16 girls. The teacher selects one student at random to be the class president and then selects a different student to be vice president. What is the probability that both students are girls? (Lesson 10-7) 10 17 17 14 21 7 10 14 17 17 21 6. Find the mean, median, mode, and range of the data set. Extra Practice S37 Chapter 11 Applications Practice 1. Scientists who are developing a vaccine track the number of new infections of a disease each year. The values in the table form a geometric sequence. To the nearest whole number, how many new infections will there be in the 6th year? (Lesson 11-1) Year Number of New Infections 1 12,000 2 9000 3 6750 a. Find the investment’s value after 5 years. b. Approximately how many years will it take for the investment to be worth $3100? 3. Chemistry Cesium-137 has a half-life of 30 years. Find the amount left from a 200-gram sample after 150 years. (Lesson 11-3) 4. The cost of tuition at a dance school is $300 a year and is increasing at a rate of 3% a year. Write an exponential growth function to model the situation and find the cost of tuition after 4 years. (Lesson 11-3) 5. Use the data in the table to describe how the price of the company’s stock is changing. Then write a function that models the data. Use your function to predict the price of the company’s stock after 7 years. (Lesson 11-4) Stock Prices Price ($) 0 1 2 3 10.00 11.00 12.20 13.31 6. Use the data in the table to describe the rate at which Susan reads. Then write a function that models the data. Use your function to predict the number of pages Susan will read in 6 hours. (Lesson 11-4) Total Number of Pages Read S38 Time (h) 1 2 3 4 Pages 48 96 144 192 Extra Practice 8. Geometry Given the surface area, S, of a S sphere, the formula r = ___ can be used to 4π find the sphere’s radius. What is the radius of a sphere with a surface area of 100 m 2? Use 3.14 for π. Round your answer to the nearest hundredth of a meter. (Lesson 11-5) √ 2. Finance For a savings account that earns 5% interest each year, the function x f (x) = 2000(1.05) gives the value of a $2000 investment after x years. (Lesson 11-2) Year gives the 7. The function f (x) = √1.44x approximate distance in miles to the horizon as observed by a person whose eye level is x feet above the ground. Jamal stands on a tower so that his eyes are 180 ft above the ground. What is the distance to the horizon? Round your answer to the nearest tenth. (Lesson 11-5) 9. Cooking A chef has a square baking pan with sides 8 inches long. She wants to know if an 11-inch fish can fit in the pan. Find the length of the diagonal of the pan. Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of an inch. Tell whether the fish will fit in the pan. (Lesson 11-6) 10. Alicia wants to put a fence around the irregular garden plot shown. Find the perimeter of the plot. Give your answer as a radical expression in simplest form. (Lesson 11-7) ÊȖе £ÓÊÊ е ÊȖе ÓÇÊÊ е е ÊȖÎÊÊ ÊȖе ÇxÊÊ е 11. Physics The velocity of an object in meters √ 2 √ E per second is given by _____ , where E is kinetic √ m energy in Joules and m is mass in kilograms. What is the velocity of an object that has 40 Joules of kinetic energy and a mass of 10 kilograms? Give the answer as a radical expression in simplest form. Then estimate the velocity to the nearest tenth. (Lesson 11-8) 12. A rectangular window has an area of 40 ft 2. The window is 8 feet long and its height is √ x + 2 ft. What is the value of x? What is the height of the window? (Lesson 11-9) Chapter 12 Applications Practice 1. The inverse variation xy = 200 relates the number of words per minute x at which a person types to the number of minutes y that it takes to type a 200-word paragraph. Determine a reasonable domain and range and then graph this inverse variation. Use the graph to estimate how many minutes it would take to type the paragraph at a rate of 60 words per minute. (Lesson 12-1) 7. A committee consists of five more women than men. The chairperson randomly chooses one person to serve as secretary and a different person to serve as treasurer. Write and simplify an expression that represents the probability that both people who are chosen are men. What is the probability of choosing two men if there are 6 men on the committee? (Lesson 12-4) 2. Business The owner of a deli finds that the number of sandwiches sold in one day varies inversely as the price of the sandwiches. When the price is $4.50, the deli sells 60 sandwiches. How many sandwiches can the owner expect to sell when the price is $3.60? (Lesson 12-1) 8. Transportation A delivery truck makes a delivery to a town 150 miles away traveling r miles per hour. On the return trip, the delivery truck travels 20% faster. Write and simplify an expression for the truck’s roundtrip delivery time in terms of r. Then find the round-trip delivery time if the truck travels 55 mi/h on its way to the delivery. (Lesson 12-5) 3. A gardener has $30 in his budget to buy packets of seeds. He receives 3 free packets of seeds with his order. The number of packets 30 y he can buy is y = __ x + 3, where x is the price per packet. Describe the reasonable domain and range values. Then graph the function. (Lesson 12-2) 4. Ashley wants to save $1000 for a trip to Europe. She puts aside x dollars per month, and her grandmother contributes $10 per month. The number of months y it will take to save 1000 $1000 is y = _____ . Describe the reasonable x + 10 domain and range values. Then graph the function. (Lesson 12-2) 5. Geometry Find the ratio of the area of a circle to the circumference of the circle. (Hint: For a circle, A = πr 2 and C = 2πr). For what radius is this ratio equal to 1? (Lesson 12-3) 6. Geometry For a cylinder with radius r and height h, the volume is V = πr 2h, and the surface area is S = 2πr 2 + 2πrh. What is the ratio of the volume to the surface area for a cylinder? What is this ratio when r = h = 1? (Lesson 12-3) À 9. Recreation Jordan is hiking 2 miles to a vista point at the top of a hill and then back to his campsite at the base of the hill. His downhill rate is 3 times his uphill rate, r. Write and simplify an expression in terms of r for the time that the round-trip hike will take. Then find how long the hike will take if Jordan’s uphill rate is 2 mi/h. (Lesson 12-5) 10. Geometry The volume of a rectangular prism is the area of the base times the height. A rectangular prism has a volume given by (2x 2 + 7x + 5) cm 3 and a height given by (x + 1) cm. What is the area of the base of the rectangular prism? (Lesson 12-6) 11. Tanya can deliver newspapers to all of the houses on her route in 1 hour. Her brother, Nick, can deliver newspapers along the same route in 2 hours. How long will it take to deliver the newspapers if they work together? (Lesson 12-7) 12. Agriculture Grains are harvested using a combine. A farm has two combines—one that can harvest the wheat field in 9 hours and another that can harvest the wheat field in 11 hours. How long will it take to harvest the wheat field using both combines? (Lesson 12-7) Extra Practice S39 Problem Solving Handbook Draw a Diagram Problem Solving Strategies You can draw a diagram to help you visualize what the words of a problem are describing. Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern EXAMPLE Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List A gardener wants to plant a 2.5-foot-wide border of flowers around a rectangular herb garden. The herb garden is 12 feet long and 7.5 feet wide. What is the area of the border? 1 Understand the Problem You need to find the area of the garden’s border. You are given the garden’s dimensions and the width of the border. 2 Make a Plan Draw and label a diagram of the herb garden with the surrounding border. Find the dimensions of the outer rectangle. Then find the area of the inner rectangle and subtract to find the area of the border. 3 Solve length of outer rectangle: 2.5 ft + 12 ft + 2.5 ft = 17 ft width of outer rectangle: 2.5 ft + 7.5 ft + 2.5 ft = 12.5 ft Ó°xÊvÌ Ó°xÊvÌ £ÓÊvÌ Ó°xÊvÌ Find the area of each rectangle: area of outer rectangle: 17 ft × 12.5 ft = 212.5 area of inner rectangle: 12 ft × 7.5 ft = 90 ft2 ft2 Ç°xÊvÌ Subtract: area of border: 212.5 ft2 - 90 ft2 = 122.5 ft2 4 Look Back Ó°xÊvÌ To check your answer, solve the problem in a different way. Divide the border into four parts and find the area of each part. Then add the areas. 17 ft × 2.5 ft = 42.5 ft 2 7.5 ft × 2.5 ft = 18.75 ft 2 17 ft × 2.5 ft = 42.5 ft 7.5 ft × 2.5 ft = 18.75 ft £ÇÊvÌ Ó°xÊvÌ Ó°xÊvÌ 2 2 42.5 ft 2 + 42.5 ft 2 + 18.75 ft 2 + 18.75 ft 2 + = 122.5 ft2 PRACTICE 1. A circular fish pond is surrounded by a circular border of stones that is 18 inches wide. The fish pond is 4 feet in diameter. What is the area of the border? (Use 3.14 for π.) 2. Thirty-two teams are in the first round of a softball tournament. A team is eliminated as soon as it loses a game. How many games need to be played to determine the winner? (Hint: Use a tree diagram.) PS2 Problem Solving Handbook Ç°xÊvÌ Make a Model You can make a model, or representation of the objects in a problem, to help you solve it. EXAMPLE Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Mr. Duncan is using blue and white square tiles to create a pattern on his kitchen wall. The entire design will have 8 rows with 15 tiles in each row. The bottom row alternates colors starting with blue, and the row above that alternates colors starting with white. He will continue this alternating pattern so that the same two colors are never next to each other. How many of each color tile does Mr. Duncan need to complete the entire design? 1 Understand the Problem You need to find how many of each color tile are needed. You know the number of rows and the number of tiles in each row. The colors alternate so that the same two colors are never next to each other. 2 Make a Plan Use blocks (preferably blue and white, but any two colors would work) to make a model of the first two rows. Count how many of each color you use. Then multiply to find how many of each color would be used in the entire design. 3 Solve Create the bottom row. Start with a blue block and alternate colors across the row until you have used 15 blocks. Create the row above the bottom row. Start with a white block. You could build all 8 rows and just count the number of each color, but each group of two rows will be the same, so this way is quicker. There will be a total of 8 rows: 4 that start with blue and 4 that start with white. Count how many of each color are used above and multiply each number by 4. blue: 15 × 4 = 60 white: 15 × 4 = 60 Mr. Duncan needs 60 blue tiles and 60 white tiles. 4 Look Back The grid is 15 units by 8 units, so there are 15 × 8 = 120 squares in the grid. Add the number of blue and white tiles to see if the sum is 120: 60 + 60 = 120. PRACTICE 1. Mr. Duncan decides to tile another area of his kitchen wall. This design will have 12 rows with 10 tiles in each row. The bottom row will repeat this pattern: blue, white, blue, blue, white. The row above the bottom row will repeat this pattern: white, green, white, white, green. He will use these two patterns for each of the remaining rows so that the first colors of each row always alternate. How many of each color tile will Mr. Duncan need? Problem Solving Handbook PS3 Guess and Test Problem Solving Strategies The guess and test strategy can be used when you cannot think of another way to solve the problem. Begin by making a reasonable guess, and then test it to see whether your guess is correct. If not, adjust the guess accordingly and test again. Keep guessing and testing until you correctly solve the problem. Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List EXAMPLE The manager of a college computer lab purchased 24 printers at a total cost of $3120. Some of the printers were laser, and some were ink jet. The laser printers cost $250 each, and the ink jet printers cost $70 each. How many of each type of printer did the manager purchase? 1 Understand the Problem You know the cost of each type of printer and the total number of printers. You need to find the number of each type of printer purchased. 2 Make a Plan Make reasonable first guesses for each type of printer. The sum must be 24. Then multiply each guess by the cost of each printer. Find the total and compare it to $3120. Adjust the guess as needed and continue until you find the solution. 3 Solve Use a table to organize your guesses. Laser Printers Ink Jet Printers Total Priners 1st guess 12 12 24 12($250) + 12($70) $3000 + $840 = $3840 Too high—try fewer laser printers. 2nd guess 6 18 24 6($250) + 18($70) $1500 + $1260 = $2760 Too low—try more laser printers. 3rd guess 8 16 24 8($250) + 16($70) $2000 + $1120 = $3120 Correct! Total Cost The manager purchased 8 laser printers and 16 ink jet printers. 4 Look Back The total spent is $3120, and the total number of printers is 24. The solution is correct. PRACTICE 1. All 350 seats were sold for a concert in the park. Adult tickets cost $15, and child tickets cost $5. Ticket sales totaled $4350. How many of each type of ticket were sold? 2. Jane is 3 times as old as Theo. Luke is 5 years older than Theo. Zoe is 8 years younger than twice Theo’s age, and Cassie is 6 years younger than Theo. The sum of their ages is 71. How old is each person? PS4 Problem Solving Handbook Work Backward You can work backward to solve a problem when you know the ending value and are asked to find the initial value. EXAMPLE Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Lee Ann is taking a vacation in Paris, France. Her flight arrived in Paris at 9:35 A.M. on Tuesday. The plane left New York City and flew for 7 hours and 55 minutes to Nice, France, where there was a layover of 1 hour 12 minutes. From Nice the plane flew 1 hour and 25 minutes to Paris. Paris time is 6 hours ahead of New York City time. What time did the plane leave New York City? 1 Understand the Problem You are asked to find the time that the plane left New York City. You know when the flight arrived in Paris, the length of the stops that were made along the way, and the time difference between New York City and Paris. 2 Make a Plan Work backward from the time the plane arrived in Paris, using inverse operations. Then apply the time difference between the two cities. 3 Solve Subtract the length of time it took to fly from Nice to Paris from the time Lee Ann arrived in Paris. 9:35 A.M. - 1 hour 25 minutes = 8:10 A.M. Subtract the length of the layover in Nice. 8:10 A.M. - 1 hour 12 minutes = 6:58 A.M. Subtract the length of the flight from New York to Nice. 6:58 A.M. – 7 hours 55 minutes = 11:03 P.M. Monday Since Paris time is ahead of New York time, subtract the time difference. 11:03 P.M., Monday - 6 hours = 5:03 P.M. Monday Lee Ann’s flight left New York City on Monday at 5:03 P.M. 4 Look Back Work forward to check your answer. 5:03 P.M. Monday + 6 h + 7 h 55 min + 1 h 12 min + 1 h 25 min = 5:03 P.M. Monday + 16 h 32 min = 9:35 A.M. Tuesday This matches the information given in the problem. PRACTICE 1. A bus arrives in Dallas, Texas, at 10:59 A.M. on Friday. The bus left Atlanta, Georgia, and took 12 hours and 15 minutes to arrive in Shreveport, Louisiana, where there was a 45-minute layover. From Shreveport it took 4 hours and 29 minutes to get to Dallas. Dallas time is 1 hour behind Atlanta time. What time did the bus leave Atlanta? 2. Carolina bought a DVD player that was on sale for 90% of the original price. The total amount she paid was $135.72, which included a sales tax of $5.22. What was the original price of the DVD player? Problem Solving Handbook PS5 Find a Pattern Problem Solving Strategies If a problem involves a sequence of numbers or figures, it is often necessary to find a pattern to solve the problem. EXAMPLE Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Darian created the following sequence of stars: How many stars will be in the 6th figure? 1 Understand the Problem You need to find the number of stars in the 6th figure. You can find the number in the first four figures by counting. 2 Make a Plan Count the number of stars in each of the first four figures. Use the information to find a pattern and determine a general rule. 3 Solve Look for a pattern between the position of each figure in the sequence and the number of stars in that figure. Position 1 2 3 4 Stars 2 6 12 20 The number of stars is the square of the position number plus the position number. This rule written algebraically is n 2 + n. Evaluate the expression for n = 6: n 2 + n 6 2 + 6 = 36 + 6 = 42 There will be 42 stars in the 6th figure. 4 Look Back Look for another pattern. The number of stars in each position increases by 4, then by 6, then by 8. That is, the amount of increase always increases by 2. So the number of stars in the 5th position will be 20 + 10, or 30, and the number of stars in the 6th position will be 30 + 12, or 42. PRACTICE 1. The first three figures of a pattern are shown. How many circles will be in the 10th figure? PS6 Problem Solving Handbook 2. Lily drew the first four figures of a pattern. How many squares will be in the 7th figure? Make a Table Problem Solving Strategies You can make a table to solve problems because the rows and columns can help you arrange information. Sometimes this also allows you to discover relationships that might otherwise be hard to notice. Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List EXAMPLE A scientist begins a culture with 500 bacteria. The number of bacteria triples every 1 30 minutes. How many bacteria are in the culture after 2 __ hours? 2 1 Understand the Problem You are asked to find the number of bacteria in the culture after 2 __12 hours. You know the initial number of bacteria, and you know that the population triples every half hour. 2 Make a Plan Make a table with rows for time and number of bacteria. Start with the initial number in the culture. Increase the time in 30-minute increments and triple the number of bacteria with each increase. Keep extending the table until the time is 2 __12 hours (150 minutes). 3 Solve Time (min) Bacteria 0 30 60 90 120 150 500 1500 4500 13,500 40,500 121,500 There are 121,500 bacteria in the culture after 2 __12 hours. 4 Look Back Check your answer by solving a simpler problem. The number of bacteria in the culture triples five times (150 min ÷ 30 min = 5). Start with 5 instead of 500 and triple the number five times. 5 × 3 = 15 15 × 3 = 45 45 × 3 = 135 135 × 3 = 405 405 × 3 = 1215 Multiply by 100 to find the total if you had started with 500; 1215 × 100 = 121,500 PRACTICE 1. A dietician’s report states that a 125-pound woman needs to eat about 1750 Calories a day to maintain her weight. It also states that a 132-pound woman needs 1848 Calories and a 139-pound woman needs 1946 Calories a day. Based on these values, how many Calories does a 160-pound woman need to eat each day to maintain her weight? 2. Simon opened a savings account with an initial deposit of $200. At the end of each year, the account earns 4% interest. If he does not deposit or withdraw any additional money, what would his balance be at the end of 6 years? Problem Solving Handbook PS7 Solve a Simpler Problem Sometimes a problem contains numbers that make it seem difficult to solve. You can solve a simpler problem by rewriting the numbers so they are easier to compute. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List EXAMPLE During a skating competition, Jules skated around the track 35 times. One lap is 0.9 mile. If Jules finished in 1 hour 30 minutes, what was his average speed? 1 Understand the Problem You are asked to find Jules’s average speed for 35 laps. You know the distance of each lap and the amount of time it took him to finish the competition. 2 Make a Plan Solve a simpler problem by using easier numbers to do the computations. 3 Solve Find the total distance skated. 35(0.9) There were 35 laps that measured 0.9 mile. 35(1 - 0.1) Write 0.9 as 1 - 0.1 35(1) - 3.5(0.1) Use the Distributive Property. 35 - 3.5 31.5 Use the distance formula to find the average speed. d = rt 1 hour 30 minutes = 1.5 hours 31.5 = r × 1.5 31.5 = r _ Solve for r. 1.5 315 = r Multiply the numerator and denominator by 10 _ 15 to eliminate the decimals. 1 (315) = r _ 15 1 (300 + 15) = r _ Write 315 as 300 + 15. 15 1 (300) + _ 1 (15) = r _ Use the Distributive Property. 15 15 20 + 1 = r 21 = r Jules skated at an average speed of 21 miles per hour. 4 Look Back Each lap is a little less than 1 mile, so 35 laps is a little less than 35 miles. Round this distance to 30 miles and use d = rt to find the rate when the time is 1.5 hours: 30 mi = (1.5 h)r r = 20 mi/h. This is close to 21 mi/h. PRACTICE 1. Diana swam 24 laps in the pool today. One lap is 200 feet. She swam at an average rate of 4 feet per second. How many minutes did Diana swim? PS8 Problem Solving Handbook Use Logical Reasoning Problem Solving Strategies Use logical reasoning to solve problems when you are given several facts and can use common sense to find a missing fact. EXAMPLE 1 Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Five players on a baseball team wear the numbers 2, 12, 15, 34, and 42. Their positions are pitcher, catcher, first base, left field, and center field. The pitcher’s number is less than the left fielder’s number. The center fielder’s number is greater than 25, and the left fielder wears an even number. The catcher wears number 34. What is the pitcher’s number? 1 Understand the Problem You want to find the jersey number of the pitcher. You know there are five positions and five jersey numbers. Some information about who wears which number is given. 2 Make a Plan Organize the information in a table. Start with the fact that the catcher wears number 34 and use logical reasoning to determine the numbers of the other positions. 3 Solve The catcher wears number 34. No other player wears 34, and the catcher wears no other number. Enter Y’s and N’s in the chart as shown. The center fielder’s number is greater than 25, so he must wear number 42. The left fielder cannot wear number 15 (because it is odd), and he cannot have the least number (the pitcher’s number is less than his). The left fielder must wear 12. The pitcher’s number is less than 12 (the left fielder’s), so he must wear number 2. 2 12 15 34 42 Pitcher Y N N N N Catcher N N N Y N First Base N N Y N N Left Fielder N Y N N N Center Fielder N N N N Y Y = yes; N = no Once you enter Y in a cell, enter N in the remaining cells for the row and the column that include it. The pitcher wears number 2. 4 Look Back Complete the chart if needed. Read the problem while looking at the chart to make sure there are no contradictions. PRACTICE 1. Rose, Jill, Gaby, and Chloe bowled the scores 110, 125, 144, and 150. Jill did not bowl the 110. The person who bowled the 150 is Rose’s sister and Jill’s aunt. Chloe bowled the 125. What score did Jill bowl? Problem Solving Handbook PS9 Use a Venn Diagram You can use a Venn diagram to display relationships among sets of numbers. Circles are used to represent the individual sets. EXAMPLE Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List At a local supermarket, 194 people were given samples of two brands of orange juice. Their opinions were as follows: 120 people liked brand A, 101 people liked brand B, and 15 people did not like either brand. How many people liked only brand A? 1 Understand the Problem The total number of people was 194, and 15 of them did not like either brand. The statement “120 people liked brand A” means some of the 120 people liked only brand A and some liked brand A and brand B. The statement “101 people liked brand B” means some of the 101 people liked only brand B and some liked brand A and brand B. 2 Make a Plan Use a Venn diagram to show the relationship among the groups of people. 3 Solve Draw and label two intersecting circles to show the sets of people who liked brand A and brand B. Write 15 in the area labeled “Neither.” Out of 194 people, 15 liked neither brand. Subtract 15 from 194 to find how many people liked at least one brand: 194 - 15 = 179. Add the number of people who liked brand A to the number of people who liked brand B: 120 + 101 = 221. You know there are only 179 people who liked at least one brand, so subtract 179 from 221: 221 - 179 = 42. This means 42 people were counted twice, and that 42 people liked both brands. Write 42 in the area labeled both. Out of 120 people who liked brand A, 42 also liked brand B. Subtract 42 from 120 to find the number of people who liked only brand A: 120 - 42 = 78. So 78 people liked only brand A. 4 Look Back À>`Ê À>`Ê Ì iÌ iÀ\Ê£x À>`Ê À>`Ê Ì Çn iÌ iÀ\Ê£x Find the number of people who liked brand B only: 101 - 42 = 59. Add all the numbers in the Venn diagram. The sum of the number who liked only brand A, the number who liked only brand B, the number who liked both brands, and the number who liked neither brand should be the total number of people surveyed: 78 + 59 + 42 + 15 = 194. PRACTICE In a group of 138 people, 55 own a cat, 27 own a cat and a dog, and 42 own neither pet. 1. How many people own only a cat? 2. How many people own a dog? PS10 Problem Solving Handbook {Ó x Make an Organized List Problem Solving Strategies If a problem asks you to find all the possible ways in which something can happen, you can make an organized list to keep track of the outcomes. Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List EXAMPLE A fair coin is tossed 4 times. What is the probability that it lands heads up at least 3 times? 1 Understand the Problem You need to find the probability that a coin tossed 4 times lands heads up 3 or 4 times. 2 Make a Plan The formula for probability is: number of favorable outcomes probability = ___ total number of outcomes The total number of outcomes is the number of items in the list. The number of favorable outcomes is the number of times the coin lands heads up 3 or 4 times. Make an organized list of the coin tosses to find the total number of outcomes. 3 Solve Start with heads for all 4 tosses, then heads for the first 3 tosses, then heads for the first 2 tosses, and then heads for the first toss. Repeat the pattern for tails. HHHH HTHH TTTT THTT HHHT HTHT TTTH THTH HHTH HTTH TTHT THHT HHTT HTTT TTHH THHH There are 16 total outcomes. There are 5 favorable outcomes. 5 The probability that the coin lands heads up 3 or 4 times is __ . 16 4 Look Back Double-check that each combination is listed and that no combination is written more than once. You can also use the Fundamental Counting Principle to check the total number of outcomes. For each of the 4 coin tosses, there are 2 possible outcomes, so the total number of outcomes is 2 × 2 × 2 × 2 = 16. PRACTICE 1. A beagle, a fox terrier, an Afghan hound, and a golden retriever are competing in the finals of a dog show. How many ways can the dogs finish in first, second, and third place? 2. Two number cubes are rolled. What is the probability that the sum of the numbers rolled is an odd number? Problem Solving Handbook PS11 Selected Answers Chapter 1 81. c divided by d; the quotient of c and d 83. __52 85. 280 1-3 Check It Out! 1a. -7 1b. 44 1-1 Check It Out! 1a. 4 decreased by n; n less than 4 1b. the quotient of t and 5; t divided by 5 1c. the sum of 9 and q; q added to 9 1d. the product of 3 and h; 3 times h 2a. 65t 2b. m + 5 2c. 32d 3a. 6 3b. 7 3c. 3 4. a. 63s, b. 756 bottles; 1575 bottles; 3150 bottles Exercises 1. variable 3. the quotient of f and 3; f divided by 3 5. 9 decreased by y; y less than 9 7. the sum of t and 12; t increased by 12 9. x decreased by 3; the difference of x and 3 11. w + 4 13. 12 15. 6 17. the product of 5 and p; 5 groups of p 19. the sum of 3 and x; 3 increased by x 21. negative 3 times s; the product of negative 3 and s 23. 14 decreased by t; the difference of 14 and t 25. t + 20 27. 1 29. 2 31a. h - 40, b. 0; 4; 8; 12 33. 2x 35. y + 10 37. 9w; 9 in 2; 72 in 2; 81 in 2; 99 in 2 39. 13; 14; 15; 16 41. 6; 10; 13; 15 43a. 47.84 + m; b. 58.53 - s 45. x + 7; 19; 21 47. x + 3; 15; 17 49. F 51. 36 53. 1 55. 45° 57. 90° 59. __12 61. 1 63. Multiply the previous term by 3; 729, 2187, 6561. 1-2 Check It Out! 1a. 4 1b. -10 1c. 1.5 2a. -12 2b. -35.8 2c. -16 3a. -8 3b. 4 3c. -2 4. 13,018 ft Exercises 1. opposite 3. -8.5 9 1 5. 9 __14 7. 1 9. __ 11. -4.1 13. 1__ 16 10 15. 4 17. -11 __34 19. -30 21. 14 23. -10 25. 13.4 27. 23 ˚F 29. 0.75 31. -12 __25 33. -12 35. 37 37. 0 1 39. __ 41. > 43. > 45. < 10 47. 11,331 ft 49. never 51. A 55. F 57. -9 59. 2 61. Subtract 4; -2, -6, -10 63. 12,660.5 ft − −− 65. 0.2 67. 0.36 69. 720˚ SA2 Selected Answers 1 1c. -42 2a. __ 2b. - __14 2c. - __12 12 3a. 0 3b. undefined 3c. 0 4. 7.875 mi Exercises 3. -121 5. 7 7. 2 9. undefined 11. 0 13. about 9 $210,000,000 15. -32 17. __ 10 19. -3 21. 0 23. 0 25. -15 °F 27. -4 29. -62 31. 18.75 33. 1 35. -12 37. __32 39. negative 41. negative 43. positive 45. undefined 47. 1 49. __12 51. - __15 53. __98 55. 15 h per semester 57. < 59. > 61. = 63a. positive b. negative c. The product of two negative numbers is positive. The product of that positive number and a negative 1 number is negative. d. no 65. 75 __ 15 1 67. -121 __ 69. never 73. C 11 25 75. 16 quarter notes 77. __ 49 27 __ 79. 5 81. 1 83. - 64 85. Multiply by -2; -16, 32, -64. 87. The numbers are alternating positive and negative multiples of 5; 30, -35, 40. 89. $85 91. 1510 in2 93. < 95. = ( ) ( ) 1-4 Check It Out! 1a. 2 2 1b. x 3 27 2a. -125 2b. -36 2c. __ 3a. 8 2 64 3 8 3b. (-3) 4. 2 = 256 Exercises 1. the number of times to use the base as a factor 3. 2 3 5. 49 7. -32 9. 9 2 11. (-4)3 13. 3 4 15. 3 5 = 243 17. 3 3 19. 27 21. -16 23. 7 2 25. (-2) 3 27. 4 3 29. 2 4 = 16 31. < 33. = 35. = 1 37. > 39. 8 41. -64 43. -1 45. __ 27 2 2 2 47a. 36 in b. 9 in c. 27 in 49. 6 2 3 51. (-1)4 53. __19 55. between ( ) 8000 cm 3 and 15,625 cm 3 57. 2 59. 4 61. 2 63. 4 65a. 100, 1000, 10,000 b. The exponent is the same as the number of zeros in the answer. 67. C 69. B 71. 64 73. 65,536 75a. 4 · 4; 4 · 4 · 4 b. 4 · 4 · 4 · 4 · 4 = 4 5 c. 2 + 3 = 5; the sum of the exponents in 4 2 and 4 3 is the exponent in the product 4 5. 77. 5 79. 5 minus x; x less than 5 1-5 Check It Out! 1a. 2 1b. -5 1c. 3 2a. __23 2b. __12 2c. -__27 3. 3.0 ft 4a. rational number, repeating decimal 4b. rational number, terminating decimal, integer 4c. irrational number 4d. natural number, whole number, integer, rational number, terminating decimal Exercises 3. -15 5. 5 7. -3 9. -4 11. __23 13. __38 15. __14 17. -__15 19. rational number, terminating decimal, integer 21. irrational number 23. 11 25. -10 31. 4.1 cm 33. rational number, terminating decimal, integer, whole number, natural number 35. irrational number 37. > 39. < 41. 45; rational number, terminating decimal, integer, whole number, natural number 43. 34.625; rational number, terminating decimal 45. always 47. always 51. 18 53. A 55. D 57. 0.9 59. -0.1 3 61. 4 63. 65 65a. no 67. __12 69. -__ 16 8 71. -3.5 73. -___ 75. 64 125 1-6 Check It Out! 1a. 48 1b. 2.6 1c. 2 2a. 15 2b. 3 3a. 1 3b. -3 3c. 21 4. 6.2(9.4 + 8) 5. 400 Exercises 3. 15 5. -9 7. 14 9. 1 11. 14 13. 92 15. 1.5 17. -3 19. -22 21. 12(-2 + 6) 23. 188.4 ft2 25. 19 27. -15 29. 3 31. -5 33. 24 35. 17 37. -9 39. 17 41. -7 43. 0 45. __14 47. 1 49. 6 51. 3 - __25 53. 8 - ⎪3 · 5⎥ 55a. 55 b. 498 c. 250 d. 10 e. 30 √7 f. 70 57. 2⎡⎣9 + (-x)⎤⎦ 59. ____ 3 · 10 63. 3 · 5 - 6 · 2 = 3 69. H 71. -3 73. 6 77. 20 79. acute 81. 100 83. -11 85. 8 87. __67 1-7 Check It Out! 1a. 21 1b. 560 1c. 28 2a. 9(50) + 9(2) = 468 2b. 12(100) - 12(2) = 1176 2c. 7(30) + 7(4) = 238 3a. 100p 3b. -28.5t 3c. 3m 2 + m 3 4a. 6x - 15 4b. 3a - 16x Exercises 1. Associative Property of Addition 3. 24 5. 56 7. 118,000 9. 304 11. 456 13. 763 15. 20x 17. -9r 19. 7.9x 21. 9a - 31 23. 7x - 3x 2 25. 2a + 2 39. -3x - 14 43. 13y - 10 45a. Amy: 98:21; Julie: 81:12; Mardi: 83:39; Sabine: 63:47 b. Sabine, Julie, Mardi, Amy 47. Commutative Property of Addition 49. Distributive Property 51. Distributive Property 53. 6p + 9 57a. equal b. 96π cm2 c. 2(16π) + 96π = 128π cm2 59. J 61. 12x + 116 63. -3b - 7 65a. Commutative Property of Addition b. Associative Property of Addition c. Distributive Property d. Rule for subtraction 67. 36 ft 2 69. 64 71. -__18 73. 3 75. -2 1-8 Check It Out! 1a. 1b. 1c. I (-2, 6) 8 4 -8 -4 0 -4 n H (0, 2) 4 m 8 G (2, -3) -8 2a. none 2b. I 2c. III 2d. II 3. y = 10 + 20x; (1, 30), (2, 50), (3, 70), (4, 90) 4a. (-4, -6), (-2, -5), (0, -4), (2, -3), (4, -2); line 4b. (-3, 30), (-1, 6), (0, 3), (1, 6), (3, 30); U shape 4c. (0, 2), (1, 1), (2, 0), (3, 1), (4, 2); V shape Exercises 7. none 9. none 11. I 13. (-2, 0), (-1, 1), (0, 2), (1, 3), (2, 4); line 15. (-2, -4), (-1, -2), (0, 0), (1, -2), (2, -4); V shape 21. none 23. none 25. II 27. y = 500 + 0.10x; (500, 550), (3000, 800), (5000, 1000), (7500, 1250) 29. (-2, -4), (-1, -1), (0, 0), (1, -1), (2, -4); U shape 31. (-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7); U shape 33. triangle 35. rectangle 37a. c = 2.90f b. f is input; c is output. c. f c 1 2 3 4 5 6 7 8 2.90 5.80 8.70 11.60 14.50 17.40 20.30 23.20 d. 7 yards 39. y = __12 x + (-3); (-4,-5), (-2, -4), (0, -3), (2, -2), (4, -1); line 41a. y = 50 + 1.5x b. (100, 200); (150, 275); (200, 350); (250, 425); (300, 500) 43. line 45. line 51. G 53. H 57. (-4, 4) 59. The points make a horizontal line at y = 6. 61. (-4, 5); 42 square units 63. cylinder 65. pentagon 67. irrational 69. rational, terminating decimal, integer 71. x 2 + 3x Study Guide: Review 1. constant 2. whole numbers 3. coefficient 4. origin 5. 1.99g 6. t + 3 7. 5 8. 5 9. 6 10. 150 ÷ m; 30; 25; 15 11. -14 12. -4.6 13. 4 __12 14. -1 15. -24 16. 14.3 17. 2231 ft 18. 90 19. 0 20. -15.2 21. -8 22. 0 23. undefined 24. 9 15 25. - __23 26. __ 27. 3,650,000 steps 7 28. 64 29. -27 30. 81 31. -25 8 16 32. __ 33. __ 34. 2 4 35. (-10)3 27 25 2 36. (-8) 37. 12 1 38. 729 in3 39. 6 40. 4 41. -7 42. -12 43. __56 44. __13 45. rational number, terminating decimal, integer, whole number, natural number 46. rational number, terminating decimal, integer, whole number 47. rational number, terminating decimal, integer 48. rational number, terminating decimal 49. irrational number 50. rational number, repeating decimal 51. 3.6 ft 52. 23 53. 8 54. 6 55. __12 56. -18 57. 0 58. 62 59. 10 60. 8 61. 10 12 62. 8 + 7(-2) 63. ____ 64. 4 √ 20 - x 8+3 65. 168 ft 66. 40 67. 270 68. 13(100) + 13(3) = 1339 69. 18(100) - 18(1) = 1782 70. 4x 71. 7y 2 72. 4x + 24 73. 2x 2 + 2 74. -4y + 3y 2 75. 8y - a 76. $9 77–80. B y C A x D 81. I 82. IV 83. I 84. II 85. III 1 86. IV 87. y = p + __ p; ($2, $2.10); 20 ($15, $15.75); ($30, $31.50); ($40, ( ) $42.00) 88. (-4, 4), -1, __14 , (0, 0), 1, __14 , (4, 4); U shape ( ) Chapter 2 2-1 Check It Out! 1a. 8.8 1b. 0 1c. 25 2a. __12 2b. -10 2c. 8 3a. 9.3 3b. 2 3c. 44 4. 35 years old Exercises 3. 21 5. 16.3 7. __12 9. 0 17 11. 2.3 13. 1.2 15. 32 17. 3.7 19. __ 6 4 __ 21. 9 23. 17 25. 7 27. 10.5 29. 9 31. 0 33. -17 35. -3100 37. -0.5 39. 0.05 41. 15 43. 1545 45. 30 47. __13 49. a + 500 = 4732; $4232 51. x - 10 = 12; x = 22 53. x + 8 = 16; x = 8 55. 5 + x = 6; x = 1 57. x - 4 = 9; x = 13 59. m + 560 = 1680; $1120 61. 63 + x = 90; x = 27 63. x + 15 = 90; x = 75 12 65. h - 47 = 28; 75 69. J 71. - __ 5 13 __ 73. - 12 75. 10 77. 90 79. 9 81. 72 83. 6 ft 85. -80 87. -3 2-2 Check It Out! 1a. 50 1b. -39 1c. 56 2a. 4 2b. -20 2c. 5 3a. - __54 3b. 1 3c. 612 4. 15,000 ft Exercises 1. 32 3. 14 5. 19 7. 7 9. 5 11. 2.5 13. 14 15. -9 17. __18 19. 16c = 192; $12 21. 24 23. -36 25. -150 27. 55 29. -3 31. 1 33. 13 35. 0.3 37. 2 39. -16 41. -3.5 7 43. -2 45. __ s = 392; $560 10 49. 4s = 84; 21 in. 51. 4s = 16.4; 4.1 cm 53. -3x = 12; x = -4 55. __3x = -8; x = -24 57. 6.25h = 50; 8 h 59. 0.05m = 13.80; 276 min 61. -2 63. 0; 8y = 0; 0 65a. number of data values c. 185,300 acres 3 67. 7 69. 605 71. __ 73. 5.7 16 2 __ 75. 3 g = 2; 3 g 77. D 79. B Selected Answers SA3 81a. 6c = 4.80 b. c = $0.80 83. 2 85. 9 87. 2 89. -20 91. -132 93. Multiply both sides by a. 95. 12 97. 25 99. 6 years old 101. 6 103. 16 2-3 Check It Out! 1a. 1 1b. 6 1c. 0 55 2a. __ 2b. __12 2c. 15 3a. - __56 3b. 5 4 3c. 8 4. $60 5. -42 Exercises 1. 2 3. -18 5. 2 7. 66 9. __54 11. -12 13. 16 15. -3.2 17. 4 19. 15 passes 21. 4 23. -4 25. 4 27. 5 29. -9 31. __14 33. 1 35. 3 28 37. __ 39. 3 41. 8 43. 7 45. - __12 5 47. x = 40 49. x = 35 51. 8 - 3n = 2; n = 2 53a. 1963 - 5s = 1863; s = 20 53b. 3 55. 8 57. 4.5 59. -10 61. 10 63. 5k - 70 = 60; 26 in. 65. Stan: 36; Mark: 37; Wayne: 38 67a. 45,000; 112,500; 225,000; 337,500; 225n 67b. c = 225n 71. H 73. 27 75. 6 __15 77. 14.5 79. -6 81. irrational 83. repeating decimal, rational 85. 8(60) + 8(1) = 488 87. 11(20) + 11(8) = 308 89. 13 91. -18 2-4 Check It Out! 1a. -2 1b. 2 2a. 4 2b. -2 3a. no solution 3b. all real numbers 4. 10 years old Exercises 3. 1 5. 40 7. - __23 9. 3 11. no solution 13. all real numbers 15. 6 17. 6 19. 2.85 21. 10 23. 6 25. 14 27. __34 29. -4 31. no solution 33a. 15 weeks 33b. 180 lb 35. x - 30 = 14 - 3x; x = 11 37. -4 39. 7 41. -3 43. 2 45. 1 47. - __75 49. 4 51. no solution 53. 9 59. F 61. H 63. 2 65. no solution 67. -20 69. 6, 7, 8 71. $1.68 73. 3y cm 75. -63 77. 4 79. 2 81. -125 83. 15 85. 3 2-5 Check It Out! 1. about 1.46 h 5-b m 2. i = f + gt 3a. t = ____ 3b. V = __ 2 D V 3. w = __ 5. m = 4n + 8 Exercises h 10 7. a = ____ 9. I = A - P b+c k+5 x-2 ____ 13. ____ =y 11. x = y z y-b 15. x = 5(a + g) 17. x = ____ m PV 21. T = M + R 19. T = ___ nR c - 2a _____ 25. r = 7 - ax 23. b = 2 SA4 Selected Answers 5 - 4y t-g 27. x = _____ 31. a = ______ 3 -0.0035 ( ) is equal to the ratio of the corresponding sides. 35. C 37. D 39. a = __52 c + __34 b ( ) v -u 41. d = 500 t - __12 43. s = ______ 2a 2 2 45. 120 s 47. 12 49. -6 51. 20 53. 12 2-6 Check It Out! 1a. -7, 7 1b. -5.5, 10.5 2a. no solutions 2b. 4 3. |x - 134 = 0.18; minimum height: 133.82 m; maximum height: 134.18 m Exercises 1. – 6, 6 3. -2, 2 5. -__32 , __12 7. no solutions 9. no solutions 11. 2.8 13. ⎪x - 207⎥ = 2; mile markers 205 and 209 15. -9, 13 ft 17. -2, 2 19. 18.8, 65.28 14 21. -__ , 4 23. 0 25. 0 27. __23 3 29. ⎪x - 5⎥ = 0.001; 4.999 mm; 5.001 mm 31. ⎪x - 7⎥ = 2; 5, 9 33. ⎪x - 1500⎥ = 75; 1575 bricks; 1425 bricks 35. ⎪x⎥ = 3 37. ⎪x - 2⎥ = 3 39. sometimes 41. always 43a. ⎪t - 24⎥ = 5 43b. 19; 29 43c. yes 43d. The measurements are correct to within 5 mi/h. 47. C 49. B 51. Division Property of Equality; Subtraction Property of Equality; Division Property of Equality 53. 1__34 ft 55. -8__12 57. -1.4 59. T = S – R 61. w = xz - 3 63. N = 6M – S Exercises 3. 10 ft 7. 7 in. 11. 480 ft2 13. 4 15. 2.8 ft 17. 4 cm 1.5 4.5 2 ___ 21. ___ x = 36 ; 12 m 23. k 25. G 27. w = 4; x = 7.5; y = 8 29. 16.6 cm 31. -12 33. -46 35. (-2, 4); (-1, 1); (0, 0); (1, 1); (2, 4) 37. (-2, -7); (-1, -4); (0, -1); (1, 2); (2, 5) 39. 32 41. 3.5 2-9 Check It Out! 1a. 12 1b. 16.8 1c. 1.44 2a. 20% 2b. 300% 3a. 75 3b. 320 4. 10 karats Exercises 3. 21 5. 5.6 7. 80% 9. 12.5% 11. 175 13. 36 15. 48 17. 2.5 19. 25% 21. 50% 23. 40 25. 511.1 27. 100 mg 29. 2% 31. 8% 33. 64% 35. 85% 37. 85% 39. 0.52; 13 90 28 __ 41. 90.0; ___ 43. 1.12; __ 25 100 25 3 3 47. 0.006; ___ 49. 0.5 is 45. 0.06; __ 50 500 greater than __12 % because __12 % = 0.005. 1 51. 0.001, 1%, __ , 11%, 1.1 53. 0.49, 10 5 __ 4 __ , , 82%, 0.94 55a. 40% 9 5 b. action c. 3% d. 36.9% 57. box 1: 200; 100; 50 box 2: 12; 24; 148; 96 box 3: 25; 50; 100; 200 x 40 59a. __ = ___ ; $36 b. $54 61. F 90 100 63. G 65. 17.2% 67. 88.5 71. 120 73. 160 75. 6 in. 77. 3 2-7 Check It Out! 1. 12 2. $7.50/h 3. 20.5 ft/s 4a. -20 4b. 5.75 5. 6 in. Exercises 1. The ratios are equivalent. 3. 682 trillion 5. 18,749 lb/cow 7. 0.075 page/min 9. 18 mi/gal 11. __35 13. 39 15. 6.5 h ; 2.94 m 21. 72 17. 23 19. __35 = ___ 4.9 23. $403.90/oz 25. 2498.4 km/h 27. 10 29. -1 31. 13 33. 1.2 35. __19 37. 45 39. $84 43. 1.625 45. 3 11 47. - __27 49. __ 51. 3 53. 24 3 55. -120 59. A 61. D 63. 40°; 50° 65. 0.0006722 people/m2 67. -27 1 69. - __ 71. 10 2 73. -5 75. 8 32 nRT ____ 77. V = P 2-8 Check It Out! 1. 2.8 in. 150 45 5.5 3.5 ___ ___ ___ 2a. ___ x = 195 ; 650 cm 2b. x = 28 ; 44 ft 3. The ratio of the perimeters 2-10 Check It Out! 1. $462.80 2a. $270 2b. $7650 3a. about $3.30 3b. about $5.60 Exercises 3. $41,775 5. 4 __12 yr 7. about $6.45 9. $462.50 11. $266.75 13. 5 yr 15. about $30 17. $50,400 19. 2 yr 21. $2.89 25. A 27. D 29. 900 31. $47.17 33. $93 14 35. > 37. > 39. 12, 2 41. 2, __ 3 43. 50% 45. 80 2-11 Check It Out! 1a. 45% decrease 1b. 20% increase 1c. 43.75% increase 2a. 90 2b. 6 3a. $88 3b. 20% 4a. $15.30 4b. 130% Exercises 3. 20% decrease 5. 12.5% increase 7. 20% decrease 9. 61.8 11. 8 13. 70% 15. 90% 17. 25% decrease 19. 400% increase 21. 30% increase 23. 15% decrease 25. 20% increase 27. 8 __13 % decrease 29. 252 31. 7.6 33. 15% 35. 650% 37. 50% 41. 18 43. 200% increase 45. 20 47. 60 49. 25% decrease 60 18 51a. 60% b. ___ = __ x ; x = $30 100 53. H 55. G 57. 200 59. 625 61. 64 fl oz 63. $9.43 65. 80°; 170° 67. 60°; 150° 69. -20 71. 57 73. 36 75. -100 77. about $4.20 Study Guide: Review 1. literal equation 2. ratio 3. 36 4. -2 5. -21 6. 18 7. __98 8. __73 10 9. 27 + s = 108; 81 10. 7 11. - __ 3 12. -90 13. 13 14. 0 15. -2 16. 17.5 17. -5 18. 40 19. -3 20. - __12 21. 15 22. 18 23. 1 24. 41; 123°; 57° 25. -2 26. -2 27. 1 28. - __23 29. no solution 30. all real 360 numbers 31. 9 32. n = ___ c 2S 33. a = __ n - 34. 3.7 gal 35. x = 15, -27 36. y = 7, 3 37. y = 9, -9 38. x = 17.4, -6.6 39. g = -4, -8 40. x = __57 , - __57 41. |x - 55| = 5; minimum speed: 50 mi/h; maximum speed: 60 mi/h 42. 3 __13 c 43. 1080 m/h 44. 0.85 mi/ min 45. 1.6 46. 54 47. 5 48. -3 49. 2.5 cm 50. 16 ft 51. The ratio of the areas is the square of the ratio of the radii. 52. 5.29 53. 3105 54. 66.7% 55. 400% 56. 133.3 57. 240 58. 80% 59. $48,500 60. $9000 61. about $5.60 62. about $2.20 63. 37% increase 64. 33% decrease 65. 91 66. 127.5 67. $3.75; $6.25 68. 37.5% Chapter 3 3-1 Check It Out! 1. all real numbers greater than 4 2a. 2 2.5 3 3.5 4 2b. 2c. 3. x < 2.5 4. d = amount employee can earn per hour; d ≥ 8.25 Exercises 1. A solution of an inequality makes the inequality true when substituted for the variable. 3. all real numbers greater than -3 5. all real numbers greater than or equal to 3 11. b > -8 __12 13. d < -7 15. f ≤ 14 17. r < 140 where r is positive 19. all real numbers less than 2 21. all real numbers less than or equal to 12 27. v < -11 29. x > -3.3 31. z ≥ 9 33. y = years of experience; y ≥ 5 35. h is less than -5. 37. r is greater than or equal to -2. 39. p ≤ 17 41. f > 0 43. p = profits; p < 10,000 45. e = elevation; e ≤ 5000 51. D 53. C 59. D 61. C 65. < 71. 10 73. 7 75. 3x + 3 77. g = 2b; 16 79. b = 9 81. no solutions 3-2 Check It Out! 1a. s ≤ 9 Ê?? 1b. t < 5 __12 Ê?? 1c. q < 11 2. 11 + m ≤ 15; m ≤ 4 where m is nonnegative; Sarah can consume 4 mg or less without exceeding the RDA. 3. 250 + p > 282; p > 32; Josh needs to bench press more than 32 additional pounds to break the school record. Exercises 1. p > 6 3. x ≤ -15 5. 102 + t ≤ 104; t ≤ 2 where t is nonnegative 7. a ≥ 5 9. x < 15 11. 1400 + 243 + w ≤ 2000; w ≤ 357 where w is nonnegative 13. x - 10 > 32; x > 42 15. r - 13 ≤ 15; r ≤ 28 17. q > 51 19. p ≤ 0.8 21. c > -202 23. x ≥ 0 25. 21 + d ≤ 30; d ≤ 9 where d is nonnegative 27. x < 3; B 29. x ≤ 3; D 31. p ≤ 40,421 where p is nonnegative 35. a. 411 + 411 = 882 miles b. 822 + m ≤ 1000 c. m ≤ 178, but m cannot be negative. 1 37. F 39. J 41. r ≤ 5 __ 10 43. sometimes 45. always c-2 47. y = 3 - __23 x 49. a = ____ b 51. k = 2s - 11 53. x = 10 55. x ≥ -1 3-3 Check It Out! 1a. k > 6 1b. q ≤ -10 1c. g > 36 2a. x ≥ -10 2b. h > -17 3. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 servings Exercises 1. b > 9 3. d > 18 5. m ≤ 1.1 7. s > -2 9. x > 5 11. n > -0.4 13. d > -3 15. t > -72 17. 0, 1, 2, 3, 4, 5, or 6 nights 19. j ≤ 12 21. d < 7 23. h ≤ __87 25. c ≤ 1 -12 27. b ≥ __ 29. b ≤ -16 10 31. r < - __32 33. y < 2 35. t > 4 37. z < -11 39. k ≤ -7 41. p ≥ -12 43. x > -3 45. x < 20 47. p ≤ -6 49. b < 2 51. 7x ≥ 21; x ≥ 3 53. - __45 b ≤ -16; b ≥ 20 57. C 14 4 59. A 67. B 71. g ≤ - __ 73. m > __ 5 15 75. x = 5 79. 2 3 81. $1.89/gal 83. 35 words/min 85. t < 1 3-4 Check It Out! 1a. x ≤ -6 1b. x < -11 1c. n ≤ -10 2a. m > 10 2b. x > -4 2c. x > 2 __13 Ê?? Selected Answers SA5 95 + x ≥ 90; 95 + x ≥ 180; x ≥ 85; 3. _____ 2 Jim’s score must be at least 85. Exercises 1. m > 6 3. x ≤ -2 5. x > -16 7. x ≥ -9 9. x > - __12 11. x ≤ 19 13. x > 1 15. sales of more than $9000 17. x ≤ 1 19. w < -2 21. x < -6 23. f < -4.5 25. w > 0 27. v > __23 29. x > -5 31. x < -2 33. a ≥ 11 35. x > 3 37. starting at 29 min 39. x ≤ 2 41. x < 4 43. x < -6 45. r < 8 47. x < 7 49. p ≥ 18 51. __12 x + 9 < 33; x < 48 53. 4(x + 12) ≤ 16; x ≤ -8 55. B 57. A 59. 24 months or more 61a. Number Process Cost 1 350 + 3 353 2 350 + 3(2) 356 3 350 + 3(3) 359 10 350 + 3(10) 380 n 350 + 3n 350 + 3n b. c = 350 + 3n c. 350 + 3n ≤ 500; n ≤ 50; 50 CDs or fewer 65. G 67. 59 69. x > 5 71. x > 0 73. x ≥ 0 75. -3x > 0 77. 7 79. __23 81. -9 83. 25 + 2m = 10 + 2.5m; m = 30 85. a ≥ 6 3-5 Check It Out! 1a. x ≤ -2 1b. t < -1 2. more than 160 flyers 3a. r ≤ 2 3b. x < 3 4a. no solutions 4b. all real numbers Exercises 1. x < 3 3. x < 2 5. c < -2 7. at least 34 pizzas 9. p < -17 11. x > 3 13. t < 6.8 15. no solutions 17. all real numbers 19. no solutions 21. y > -2 23. b ≥ -7 25. m > 5 27. x ≥ 2 29. w ≥ 6 31. r ≥ -4 33. no solutions 35. all real numbers 37. all real numbers 39. t < -7 41. x > 3 43. x < 2 45. x > -2 SA6 Selected Answers 47. x ≤ -6 49. 27 s 51a. 400 + 4.50n b. 12n c. 400 + 4.50n < 12n; n > 53 __13 ; 54 CDs or more 53. 5x - 10 < 6x - 8; x > -2 55. __34 x ≥ x - 5; x ≤ 20 59. x can never be greater than itself plus 1. 61. D 63. A 67. x < -3 69. w ≥ -1 __67 73. 26 in. 75. y = years; y ≥ 14 Check It Out! 1. 1.0 < c < 3.0 3a. r < 10 OR r > 14 3b. x ≥ 3 OR x < -1 4a. -9 < y < -2 4b. x ≤ -13 OR x ≥ 2 Exercises 1. intersection 3. -5 < x < 5 5. 0 < x < 3 7. x < -8 OR x > 4 9. n < 1 OR n > 4 11. -5 ≤ a ≤ -3 13. c < 1 OR c ≥ 9 15. 16 ≤ k ≤ 50 17. 3 ≤ n ≤ 6 19. 2 < x < 6 21. x < 0 OR x > 3 23. x < -3 OR x > 2 25. q < 0 OR q ≥ 2 27. -2 < s < 1 29a. 225 + 80n gives the cost of the studio and technicians; the band will spend between $200 and $550. b. -0.3125 ≤ n ≤ 4.0625; n cannot be a negative number c. $155 31. 1 ≤ x ≤ 2 33. -10 ≤ x ≤ 10 35. t < 0 OR t > 100 37. -2 < x < 5 39. a < 0 OR a > 1 41. n < 2 OR n > 5 43. 7 ≤ m ≤ 60 47. D 49. B 51. 0.5 < c < 3 53. s ≤ 6 OR s ≥ 9 55. -1 ≤ x ≤ 3 57. 4x - 5 59. 3a + 3 61. (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3); U-shaped 63. m < 2 65. x ≤ -2 3. ⎪p - 125⎥ ≤ 75; 50 ≤ p ≤ 200 Exercises 1. -3 ≤ x ≤ 3 3. -2 < x < 2 5. 4 < x < 6 7. x < -22 OR x > 22 9. x ≤ -4 OR x ≥ 4 11. x ≤ 1 OR x ≥ 5 13. ⎪x - 55⎥ ≤ 25; 30 ≤ x ≤ 80 15. no solutions 17. all real numbers 19. no solutions 21. 2 < x < 4 23. -3 < x < 3 25. -6 < x < 0 27. x ≤ -10 OR x ≥ 10 29. x ≤ -10 OR x ≥ 6 31. x < -1 OR x > 2 33. no solutions 35. all real numbers 37. no solutions 39. always 41. sometimes 43. ⎪x - 2⎥ ≤ 3; -1 ≤ x ≤ 5 45. ⎪a⎥ ≤ 2 47. ⎪c⎥ ≥ 6__12 49a. 10,010 Hz 49b. ⎪x - 10,010⎥ ≤ 9990 51. ⎪n - 23⎥ > 12 53. k ≤ 1; the inequality is equivalent to ⎪x⎥ < k - 1, and this has no solutions when the expression on the right side is less than or equal to 0 (i.e., when k - 1 ≤ 0 or k ≤ 1). 55. B 57. B 61. 1__12 63. 2__12 65. all real numbers less than 2 67. all real numbers greater than or equal to -6 69. 0 < x < 4 71. x < 1 OR x > 4 Study Guide: Review 1. inequality 2. union 3. compound inequality 4. intersection 5. solution of an inequality 6. 7. 8. Check It Out! 1a. -3 ≤ x ≤ 3 9. 2b. x ≤ -6 OR x ≥ 1 3-7 4a. all real numbers 4b. no solutions 2b. -3 ≤ n < 2 2a. x ≤ -2 OR x ≥ 2 2a. 1 < x < 5 3-6 1b. -15 ≤ x ≤ 9 10. 11. 2b. continuous; 12. a < 2 13. k ≥ -3.5 14. q < -10 15. t = temperature; t ≥ 72 16. s = students; s ≤ 12 where s is a natural number 17. m = minutes; m < 30 where m is nonnegative 18. t < 7 19. k ≤ 2 20. m > -5 21. x ≥ 4.5 22. w < 9.5 23. a < 5 24. h < 1 25. v < -2 26. 5.5 mi or more 27. $18 or less 28. a ≤ 5 29. t > -3 30. p > 8 31. x ≤ -25 32. n > 6 33. g < -12 34. k > -7 35. r < -9 36. h < -3 37. g < 2.5 38. 0, 1, 2, 3, 4, 5, 6, 7 39. at least 334 lanyards 40. x < 5 41. t ≥ 6 42. m > -11 43. x < -1 44. h > -3 45. x > 1 __12 46. b ≤ 10 47. y > 3 __12 48. n > -15 49. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, or 13 50. m < -1 51. y ≥ -2 52. c < -3 53. q ≤ -4 54. x > 2 55. t < 3 56. no solutions 57. all real numbers 58. p > - __12 59. all real numbers 60. k > 2 61. no solutions 62. 8.75 > m 63. -10 < t < 4 64. -6 < k ≤ 7 65. r > 7 OR r < -2 66. no solutions 67. -2 < p ≤ 5 68. all real numbers 69. 68 ≤ t ≤ 84 70. 102 ≤ n ≤ 183.6 71. -22 ≤ x ≤ 22 72. x < -12 OR x > 4 73. -4 ≤ x ≤ 4 74. -18 < x < 0 75. x ≤ -3 OR x ≥ 3 76. -3 < x < 3 77. x < -13.9 OR x > 13.9 78. -12.5 < x < 2.1 79. x ≤ 5 OR x ≥ 9 80. x ≤ -4 OR x ≥ 4 81. no solutions 82. -16.8 ≤ x ≤ 5.8 83. |d 72| ≤ 4; 68 ≤ d ≤ 76 Chapter 4 4-1 -1 -1 4IME 3. Possible answer: When the number of students reaches a certain point, the number of pizzas bought increases. Exercises 1. continuous 3. B 5. C 11. A 13. continuous 19. The point of intersection represents the time of day when you will be the same distance from the base of the mountain on both the hike up and the hike down. 23. C 27. C 29. -8 31. __19 33. (-2, 5), (-1, 3), (0, 1), (1, -1), (2, -3); line 35. (-2, 6), (-1, 3), (0, 2), (1, 3), (2, 6); U-shape 37. n - 5 = -2; 3 7ORDSPERMINUTE 7EEKS 0 0 1 -1 2 -4 17. D: {3}; R: 1 ≤ y ≤ 5 19. D: -2 ≤ x ≤ 2; R: 0 ≤ y ≤ 2; yes 21. yes 23. yes 25. yes 27. no 29a. D: 0 ≤ t ≤ 5; R: 0 ≤ v ≤ 750 b. yes c. (2, 300); (3.5, 525) 33. G 35a. {(-3, 5), (-1, 7), (0, 9), (1, 11), (3, 13)} b. D: {-3, -1, 0, 1, 3}; R: {5, 7, 9, 11, 13} c. yes 37. all real numbers 39. 27 cm 41. x ≥ 19 4-3 Check It Out! 1. y = 3x 4-2 Check It Out! 1. x y 1 3 2 4 3 5 2a. independent: time; dependent: cost 2b. independent: pounds; dependent: cost 3a. independent: pounds; dependent: cost; f (x) = 1.69x 3b. independent: people; dependent: cost; f (x) = 6 + 29.99x 4a. 1; -7 4b. -5; 101 5. f (x) = 500x; D: {0, 1, 2, 3}; R: {0, 500, 1000, 1500} y x 2a. D: {6, 5, 2, 1}; R: {-4, -1, 0} 2b. D: {1, 4, 8}; R: {1, 4} 3a. D: {-6, -4, 1, 8}; R: {1, 2, 9}; yes; each domain value is paired with exactly one range value. 3b. D: {2, 3, 4}; R: {-5, -4, -3}; no; the domain value 2 is paired with both -5 and -4. 3. +EYBOARDING 36, 81} 11. D: {1}; R: {-2, 0, 3, 8}; no 13. D: {-2, -1, 0, 1, 2}; R: {1}; yes 15. x y -2 -4 Exercises Check It Out! 1. C 2a. discrete; 7ATERLEVEL 7ATER4ANK 5. x y x y 1 1 -7 7 1 2 -3 3 -1 1 5 -5 Exercises 1. dependent 3. y = x - 2 5. independent: size of bottle; dependent: cost of water 7. independent: hours; dependent: cost; f (h) = 75h 9. 2; 9 11. -1; -15 13. y = -2x 15. independent: size of lawn; dependent: cost 17. independent: days late; dependent: total cost; f(x) = 3.99 + 0.99x 19. independent: gallons of gas; dependent: miles; f(x) = 28x 21. 7; 10 23. f(n) = 2n + 5; D: {1, 2, 3, 4}; R: {$7, $9, $11, $13} 25. z 1 2 3 4 g(z) -3 -1 1 3 7. D: {-5, 0, 2, 5}; R: {-20, -8, 0, 7} 9. D: {2, 3, 5, 6, 8}; R: {4, 9, 25, Selected Answers SA7 27. f (-6.89) ≈ -16; f (1.01) ≈ 8; f (4.67) ≈ 20 33. D 35. 3.5 37. 44.1 m 39. y = -3 41. x = 2 43. D: x ≥ 0; R: all real numbers; no 3. 21. y 4-4 Check It Out! 1a. 5. y x 25. y = 5 y y 29. y x x 9. y x 31. y x x y x 13. y 33. 11. y = -1 y 2b. y 15. x x 17. 4IMEH x Exercises 1. 19. y y x x SA8 Selected Answers 37. x = 1 39. y = -8 41. yes; yes 43. no; yes 45. no; yes; yes 47. yes; no; yes 55a. v = 10,000 1500h b. 8500 gal c. y y y !VERAGE3PEEDOF,AVA&LOW 35. 3. x = 3 4. Possible answer: about 32.5 mi x $ISTANCEMI x x 2a. 7. 23. x 1b. y x x y y Time (h) Volume (gal) 0 10,000 1 8,500 2 7,000 3 5,500 4 4,000 59. J 61. J 63. y = 4x + 64 65. 2 3 67. p < -4 69. b ≥ 20 71. -3; 7 41. Check It Out! 1. 5. continuous y x (EIGHTOFBALL 4-5 4IME 4-6 1b. no 2a. -343 2b. 19.6 3. 750 lb 'AME 2. positive 3a. No correlation; the temperature in Houston has nothing to do with the number of cars sold in Boston. 3b. Positive; as the number of family members increases, more food is needed, so the grocery bill increases too. 3c. Negative; as the number of times you sharpen your pencil increases, the length of the pencil decreases. 4. Graph A; it cannot be graph B because graph B shows negative minutes; it cannot be graph C because graph C shows the temperature of the pie increasing, a positive correlation. 5. about 75 rolls Exercises 3. no 5. positive 7. negative 9. positive 11. A 15. positive 17. positive 19. A 23. positive 25. B *UANS4RIP $ISTANCEMI 27a. 4IMEMIN b. positive 29. C 35. 5(n + 2) = 2n - 8; n = -6 37. no solution y Exercises 1. common difference 3. yes; -0.7; -0.7, -1.4, -2.1 5. no 7. -53 9. no 11. yes; -9; -58, -67, -76 13. 5.9 15. 9500 mi 17. __14 19. -2.2 21. 0.07 23. - __38 , - __1, - __5, - __43 25. -0.2, 2 8 -0.7, -1.2, -1.7 27. -0.3, -0.1, 0.1, 0.3 29. 22 31. 122 33b. $9, $11, $13, $15; a n = 2n + 7 c. $37 20 d. no 35. -104.5 37. __ 39a. a n = 3 6 + 3(n - 1) b. 48 c. $7800 d. a n = 7 + 3(n - 1); $8200 41a. Time Interval Mile Marker 1 520 2 509 3 498 4 487 5 476 6 465 b. a n = 520 + (n - 1)(-11) c. number of miles per interval d. 421 43. F 45. 173 and 182; 20th and 21st terms 47a. session 16; yes b. Thursday 49. 20 51. t < -2 OR t > 2 53. negative Study Guide: Review (EIGHT Check It Out! 1a. yes; __12 ; __54 , __74 , _94_ 39. 6. continuous x 1. domain 2. negative correlation 3. term 4. continuous 4IME 7. Possible answer: A family buys a fish tank and some fish. After two weeks, they buy some more fish. After two more weeks, they buy more fish. 8. Possible answer: A monkey swings from a high branch to a lower branch. He climbs along the branch. Then he jumps to a higher branch and takes a nap. 9. 10. x -1 0 2 y 0 1 1 x -2 -1 2 3 y -1 3 4 1 11. D: {-4, -2, 0, 2}; R: { -1, 1, 3, 5} 12. D: {-2, -1, 0, 1, 2}; R: {-1, 0} 13. D: {0, 1, 4}; R: {-2, -1, 0, 1, 2} 14. D: -4 ≤ x ≤ 3; R: -3 ≤ y ≤ 5 15. D: {-5, -3, -1, 1}; R: {-3, -2, -1, 0}; yes 16. D: {-4, -2, 0, 2}; R: {-2, 1}; yes 17. D: {1, 2, 3 ,4}; R: {-1, 0, 1, 2, 3}; no 18. {(1, 5.00), (2, 6.50), (3, 8.00), (4, 9.50), (5, 11.00)}; yes 19. yes 20. y is 7 less than x; y = x - 7. 21. y is 9 times x; y = 9x. 22. independent: number of cakes; dependent: cost; f (c) = 6c 23. independent: number of CDs Raul will buy; dependent: number of CDs Tim will buy; g(n) = 2n 24. 14 25. -11 26. 6; -1 27. Distance walked 0OINTSSCORED &OOTBALL4EAM3CORES y x Time Selected Answers SA9 28. 3a. y x notebooks are purchased; y-intercept: number of notebooks that can be purchased if no pens are purchased 3a. yes y 29. x y x y y 3b. yes 3b. x y x x y x 31. x y 3c. no 4. Rental Payment 24 16 8 0 2 4 6 Manicures D: {0, 1, 2, 3, …} R: {$10, $13, $16, $19, …} 32. x 33. Possible answer: $44 34. negative 35. Possible answer: 33 36. yes; -6; -4, -10, -16 37. no 38. no 39. yes; 2.5; 2, 4.5, 7 40. 105 41. -62 42. 20 43. $408 44. -15.5 °C Chapter 5 Exercises 1. No; it is not in the form Ax + By = C. 3. yes; yes 5. yes 7. yes 9. yes 11. no 15. yes; no 17. yes; no 19. yes 23. no 27. yes; yes 29. yes; yes 31. yes; -4x + y = 2; A = -4; B = 1; C = 2 33. no 35. yes; x = 7; A = 1; B = 0; C = 7 37. yes; 3x - y = 1; A = 3; B = -1; C = 1 39. yes; 5x - 2y = -3; A = 5, B = -2, C = -3 41. no 55. no 57. C 63. not linear 65. -1 67. __19 69. 2 71. 9 domain value is paired with exactly one range value; yes 1b. Yes; each domain value is paired with exactly one range value; yes 1c. No; each domain value is not paired with exactly one range value. 2. Yes; a constant change of +2 in x corresponds to a constant change of -1 in y. Selected Answers y-intercept: 3 1b. x-intercept: -10; y-intercept: 6 1c. x-intercept: 4; y-intercept: 8 2a. 3CHOOL3TORE0URCHASES .OTEBOOKS Check It Out! 1a. Yes; each Exercises 1. y-intercept 3. x-intercept: 2; y-intercept: -4 5. x-intercept: 2; y-intercept: -1 7. x-intercept: 2; y-intercept: 8 13. x-intercept: -1; y-intercept: 3 15. x-intercept: -4; y-intercept: 2 17. x-intercept: -4; y-intercept: 2 19. x-intercept: 2; y-intercept: 8 21. x-intercept: __18 ; y-intercept: -1 35. A 37. B 41. F 47. x-intercept: 950; y-intercept: -55 49. c > 4 51. m ≥ -6 53. yes 5-3 Check It Out! 1. day 1 to day 6: -53; day 6 to day 16: -7.5; day 16 to day 22: 0; day 22 to day 30: -4.375; from day 1 to day 6 2. "ANK"ALANCE DAY DAY DAY DAY 5-2 Check It Out! 1a. x-intercept: -2; 5-1 SA10 "ALANCE Rental payment ($) 30. 0ENS x-intercept: 30; y-intercept: 20 2b. x-intercept: number of pens that can be purchased if no $AY 3. - __25 4a. undefined 4b. 0 5a. undefined 5b. positive Exercises 1. constant 5. - __34 7. undefined 9. undefined 11. positive 15. 1 17. 0 17 19. positive 23. __ 29. C 31. G 18 35. -2 37. D: {3}; R: {4, 2, 0, -2}; no 39. x-intercept: 3; y-intercept: 6 41. x-intercept: __14 ; y-intercept: __12 Exercises 1. direct variation 5-4 Check It Out! 1a. m = 0 1b. m = 3 1c. m = 2 2a. m = __12 2b. m = -3 2c. m = 2 2d. m = - __3 3. m = __1 ; 2 3. yes; -4 5. no 7. 18 9. y = 7x 11. yes; __14 13. yes 15. -16 17. y = 2.50x 19. no 21. y = -3x 2 9. - __59 11. -4 13. undefined 9 13 19. - __ 15. - __34 17. - ____ 5 5000 23a. Car 1; 20 mi/h b. The speed and the slope are both equal to the distance divided by time. c. 20 mi/h 25a. y = 220 - x 27. G 29. - __ab 5-5 ( ) Check It Out! 1. __32 , 0 2. (4, 3) 3. 6.71 4. 17.7 mi Exercises 3. (1__12 , -4) 5. (3, 3) 7. 5.39 9. 10.82 11. (-2, -3) 13. (17, -23) 15. (2, -8) 17. 8.94 19. 9.22 21. 6.1 mi 25. 13.42 27. 14.32 29a. 7.2 b. (0, -2) c. 23 −− −− −− 31. CD, EF, AB 35. B 37. D 39. 12.5 square units 41. ±8 43. 26 45. x > -2 47. z > -4 49. __14 5-6 4. y = 4x 0ERIMETEROFA3QUARE y 0ERIMETER x 3b. y = -3x + 5 y x x The value of k is 2, and the graph shows that the slope of the line is 2. 29. k = - __29 x 3c. y = -4 y x y x The value of k is - __29 , and the graph shows that the slope of the line is - __29 . 33. y = -6x 4a. y = 18x + 200 4b. slope: 18; cost per person; y-intercept: 200; fee 4c. $3800 Exercises 1. y x y x The value of k is -6, and the graph shows that the slope of the line is -6. 41. C 43. B 47. p = 7 - 4q 4 - 2y 5. y = x - 2 7. y = -3 9. y = -2x - 1 11. y = 3x - 1 13a. y = 18x + 10 b. slope: 18; Helen’s speed; y-intercept: 10; distance she has already biked c. 46 mi 15. y x 5-7 Check It Out! y 49. x = _____ 51. y = -2x 53. -4 y 55. __12 The value of k is -3, and the graph shows that the slope of the line is -3. 25. y = 2x Check It Out! 1a. no 1b. yes; - __34 1c. yes; -3 2a. No; possible y answer: the value of __x is not the same for each ordered pair. 2b. Yes; y possible answer: the value of __x is the same for each ordered pair. 2c. No; possible answer: the value y of __x is not the same for each ordered pair. 3. 90 x 2a. y = -12x - __12 2b. y = x 2c. y = 8x - 25 3a. y = __23 x x 31. __32 - y 33. x = __12 35. x = -3 37. x = 0 39. -11 < x < -5 41. x ≤ -7 OR x ≥ 7 43. x ≤ -8 OR x ≥ 6 45. yes y the height of the plant is increasing at a rate of 1 cm every 2 days. 4. m = - __23 1 Exercises 1. 1 3. - __12 5. 10 7. ___ 540 y 1b. 1a. 3IDELENGTH y x Selected Answers SA11 17. y x 19. y = 5x - 9 21. y = - __12 x + 7 23. y = __1 x + 4 25. y = -2x + 8 2 29. possible 31. impossible 33. C 37. B 39. B 41. y = __13 x - 3 43. -6 45. h = hours; h ≤ 2 where h is nonnegative 47. n ≤ 8 49. t < -3 3 Exercises 1. parallel 3. y = __ x-1 4 3 2 __ __ and y - 3 = (x - 5) 5. y = x - 4 5-8 Check It Out! ( 1a. y - 1 = 2 x - __12 4 ) 1b. y + 4 = 0(x - 3) 2a. y x y 2 x -2 0 2 -2 3a. y = __13 x + 2 3b. y = 6x - 8 4. x-intercept: -3; y-intercept: 9 5. y = 2.25x + 6; $53.25 Exercises 1. y + 6 = __15 (x - 2) 3. y + 7 = 0(x - 3) 7. y = -__13 x + 7 9. y = -x 11. y = -__1 x + 4 2 13. x-intercept: 3; y-intercept: -3 15. x-intercept: -1; y-intercept: 3 17. y - 5 = __29 (x + 1) 19. y - 8 = 8(x - 1) 23. y = -__27 x + 1 25. y = 11 -6x + 57 27. y = - __ x + 18 2 29. y = 2x - 6 31. x-intercept: 1; y-intercept: –2 33. x-intercept: -6; 1 y-intercept: 9 35. y = -___ x + 212; 500 200 °F 41. never 43a. y - 11 = 2.5 (x - 2) b. 6 in. c. 16__58 in. 47. y = -8; x = 4 49. A 53a. (0, 12) and (6, 8) b. y = -__23 x + 12 11 c. 18 min 55. H 57. y = -3x + __ 4 59. -6 ≤ x ≤ -1 63. y = -2x + 8 5-9 Check It Out! 1a. y = 2x + 2 and y = 2x + 1 1b. y = 3x and y - 1 = −− 3(x + 2) 2. slope of AB = 0; slope −− __5 −− of BC = 3 ; slope of CD = 0; slope SA12 Selected Answers 4. and y = -6x - 8; y = 3x - 2 and 3y = -x - 11 17. y = - __67 x 19. neither 21. parallel 23. y = __12 x - 5 25. y = 2x + 5 27. y = 3x + 13 29. y = -x + 5 31. y = 4x - 23 33. y = - __34 x 35. y = -x + 1 31 2 11 37. y = __ x - __ 39. y = - __15 x - __ 5 5 5 1 1 1 __ __ __ 41. y = - 2 x - 2 43. y = 2 x + 6 45. y = x - 3 47. y = -4 51a. y = 50x b. y = 50x + 30 53. H 57. - __15 59. 94 + t > 112; t > 18 63. y = __23 x - 5 65. y = - __12 x - __12 67. y = 3 y x gx fx reflection across y-axis and translation 2 units up 5. The graph will be rotated about (0, 175) and become less steep; the graph will be translated 5 units up. Exercises 1. translation 3. y fx 3 and y = - __32 x + 2; y = -1 and x = 3 9. x = 7 and x = -9; y = - __56 x + 8 and y = - __56 x - 4 11. y = -3x + 2 and 3x + y = 27; y = __12 x - 1 and -x + 2y = 17 13. y = 6x and y = - __16 x; y = __1 x and y = -6x 15. x - 6y = 15 6 2b. −− −− −−− of AD = __53 ; AB is parallel to CD because they have the same slope. −− −− AD is parallel to BC because they have the same slope. Since opposite sides are parallel, ABCD is a parallelogram. 3. y = -4 and x = 3; y - 6 = 5(x + 4) and y = - __15 x + 2 −− −− 4. slope of PQ = 2; slope of QR = -1; −− −− 1 slope of PR = - __2 ; PQ is −− perpendicular to PR because the product of their slopes is -1. Since PQR contains a right angle, PQR is a right triangle. 5a. y = __45 x + 3 5b. y = - __15 x + 2 x gx translation 4 units down 7. y fx gx x rotation about (0, 0) (less steep) 9. y gx fx x 5-10 Check It Out! 1. y fx rotation about (0, -2) (steeper) y 13. x x gx fx g(x) = - __13 x - 6 x 17. gx y fx rotation about (0, -1) (less steep) n 3. [ (m) 4 x 2 m 0 gx translation 6 units down 2. y 2 \(m) g(x) = - __23 x + 2 rotation about (0, 0) (steeper) and translation 1 unit up 23. y fx gx x different changes in x. 8. 5x + y = 1; A = 5; B = 1; C = 1 9. x + 6y = -2; A = 1; B = 6; C = -2 10. 7x - 4y = 0; A = 7; B = -4; C = 0 11. y = 9; A = 0; B = 1; C = 9 12. #UPCAKE3ALES x y 45. !MOUNTEARNED rotation about (0, 2) (less steep) 27. y f x gx x rotation about (0, 0) (steeper) and translation 5 units down 31. rotation about (0, 0) (steeper) y fx x 46. y = __13 x + 5 47. y = 4x - 9 48. $ISTANCEFT y x FT S FT S FT S 4IMES ( 50. y = 2x + 1 51. y = -5x -26 52. y = 2x + 2 53. y = 2x + 8 54. y = - __13 x and y = - __13 x - 6 55. y - 2 = -4 (x - 1) and y = -4x - 2 56. y - 1 = -5(x - 6) and y = __15 x + 2 57. y - 2 = 3(x + 1) and y = - __13 x 58. y = 2x - 3 59. y x 60. y fx gx x reflection across y-axis 61. fx gx fx translation 4 units up ) -ALEKAS"ABYSITTING%ARNINGS y x 44. units 36. 15.65 units 37. 2.2 mi 38. yes; -6 39. yes; 1 40. no 41. yes; -__12 42. -12 43. y = 8x -ONEYEARNED 1. translation; rotation; reflection 2. y-intercept 3. slope; y-intercept 4. No; a constant change of +2 in x corresponds to different changes in y. 5. Yes; a constant change of +1 in x corresponds to a constant change of +2 in y. 6. Yes; a constant change of +1 in x corresponds to a constant change of -2 in y. 7. No; a constant change of -1 in y corresponds to x gx 20. 5 21. - __43 22. -3 23. - __12 24. 3 25. 7 26. 4 27. -5 28. -1 29. 1 30. 2 31. undefined 32. 0 33. (15, 15) 34. -__12 , -9 35. 7.81 Study Guide: Review FT S g(x) = __16 x - 4 39. translation 9 units down 41. rotation about (0, 0) (steeper) 43. rotation about (0, 0) (steeper) 45a. $300 b. 20% c. Commission changes to 25%. Base pay changes to $400. 49. D 53. 15x 55. positive 57. negative 59. y = - __35 x and y + 1 = - __35 (x - 2) 61. x = 4 and y = -3; 2y + x = 6 and y = 2x + 3 y 49. g x 37. x #UPCAKESSOLD D: whole numbers; R: nonnegative multiples of 0.5 13. x-intercept: 2; y-intercept: -4 14. x-intercept: 5; y-intercept: 6 15. x-intercept: 3; y-intercept: -9 16. x-intercept: -__12 ; y-intercept: 1 17. x-intercept: -18; y-intercept: 3 18. x-intercept: __13 ; y-intercept: -__14 19. 2ATE They have different slopes and the same y-intercept. y 4IMEH reflection across y-axis 62. translation 2 units up; rotation about (0, 3) (steeper) y x Selected Answers SA13 Chapter 6 6-1 2a. (-2, 3) 2b. (3, -2) 3. 5 movies; $25 Exercises 1. an ordered pair that satisfies both equations 3. yes 5. (2, 1) 7. (-4, 7) 9. no 11. yes 13. (3, 3) 15. (3, -1) ⎧y = 2x ⎩ y = 16 + 0.50x 17a. ⎨ #ARNATION3ALES b. #OST Add the two equations: -7x + y = -2 +7x - y = _ 2 _ 0+ 0 = 0 6-3 Check It Out! 1a. yes 1b. no &LORISTSPRICE 3CHOOLBANDSPRICE #ARNATIONS It represents how many carnations need to be sold to break even. c. No, because the solution is not a whole number of carnations; 11 carnations. 19. (-2.4, -9.3) 21. (0.3, -0.3) 23. 45 white; 120 pink 25. 8 yr 29. C 31. month 11; 400 33. 42 35. 2.2 37. numbers less than 5 39. numbers greater than 6 41. c ≤ -9 6-2 Check It Out! 1. (-2, 4) 2. (4, 1) 0 = 3a. (2, 0) 3b. (3, 4) 4. 9 lilies; 4 tulips Exercises 1. (-4, 1) 3. (-2, -4) 5. (-6, 30) 7. (3, 2) 9. (4, -3) 11. (-1, -2) 13. (1, 5) 15. 6, - __12 17. (-1, 2) 19. (-1, 2) ( ) ⎧ - w = 2 ; length: 11 units; 21. ⎨ ⎩ 2 + 2w = 40 width: 9 units ( ) ( 1b. (0, 2) 1c. (3, -10) 2. (-1, 6) 3. 10 months; $860; the first option; the first option is cheaper for the first 9 months; the second option is cheaper after 10 months. Exercises 1. (9, 35) 3. (3, 8) 5. (-3, -9) 7a. 3 months; $136 b. Green Lawn 9. (-4, 2) 11. (-1, 2) 13. (1, 5) 15. (3, -2) 17. 6 months; $360; the second option 19. (2, -2) 21. (8, 6) 23. (-9, -14.8) 25. 12 nickels; 8 dimes ⎧x + y = 1000 27. ⎨ ; $200 at 5%; ⎩ 0.05x + 0.06y = 58 $800 at 6% 29. x = 60°; y = 30° 35. Possible estimate: (1.75, -2.5); (1.8, -2.4) 37. F 39. r = 5; s = -2; t = 4 41. a = 9; b = 5; c = 0 45. x-intercept: 2; y-intercept: -6 Selected Answers 0✔ 9. inconsistent; no solutions ) 46 __ 15 __ 25. (3, 3) 27. __ , 8 29. __ ,9 7 7 7 7 ⎧3A + 2B = 16 31a. ⎨ b. A = 4; B = 2 ⎩ 2A + 3B = 14 c. Buying the first package will save $8; buying the second package will save $7. 33. A 35a. s = number of student tickets; n = number of nonstudent tickets; ⎧s + n = 358 ⎨ ⎩ 1.50s + 3.25n = 752.25 b. s = 235; n = 123; 235 student tickets, 123 nonstudent tickets 37. x = 4; y = -1; z = 10 ⎧x + y = 5 39. ⎨ ; x = 1; y = 4; ⎩ 3(10x + y) = 42 the number is 14. 41. y = 3x 43. yes; __12 45. no 47. (4, 9) 6-4 Check It Out! 1a. (-2, 1) SA14 47. x-intercept: 8; y-intercept: 10 49. yes Check It Out! 1. Possible answer: Substitute -2x + 5 for y in the second equation: 2x + (-2x + 5) = 1; 5 = 1 ✘ 2. Possible answer: Substitute x - 3 for y in the second equation: x - (x - 3) 3 = 0; 3 - 3 = 0; 0 = 0 ✔ 3a. consistent, dependent; infinitely many solutions 3b. consistent, independent; one solution 3c. inconsistent; no solution 4. Yes; the graphs of the two equations have different slopes so they intersect. 11. yes 13. Possible answer: Substitute -x - 1 for y in the first equation: x + (-x - 1) = 3; -1 = 3 ✘ 15. Possible answer: Compare slopes and intercepts. -6 + y = 2x → y = 2x - 6; y = 2x - 36; the lines have the same slope and different y-intercepts. Therefore the lines are parallel. 17. Possible answer: Substitute x - 2 for y in the second equation: x - (x 2) - 2 = 0; 2 - 2 = 0; 0 = 0 ✔ 19. Possible answer: Compare slopes and intercepts. -9x – 3y = -18 → y = -3x + 6; 3x + y = 6 → y = -3x + 6; the lines have the same slope and the same y-intercepts. Therefore the graphs are one line. 21. consistent, independent; one solution 23. Yes; the graphs of the two equations have different slopes, so they intersect. 27. They will always have the same number; both started with 2 and add 4 every year. 29. The graph will be 2 parallel lines. 31. A 33. D 35. p = q; p ≠ q 37. 11 km 39. no 41. d = -1__12 ; -6, -7__12 , -9 43. (-2, -4) 6-5 Check It Out! 1a. no 1b. yes 2a. x 2b. y x Exercises 1. consistent 3. Possible answer: Substitute -3x + 2 for y in the first equation: 3x + (-3x + 2) = 6; 2 = 6 ✘ 5. Possible answer: Substitute -x + 3 for y in the second equation: x + (-x + 3) 3 = 0; 0 = 0 ✔ 7. Possible answer: y 2c. y x 3a. 2.5b + 2g ≤ 6 hot dogs) 21. y ≤ - __15 x + 3 3b. 23. /LIVE#OMBINATIONS y x 'REENOLIVES Possible answer: solutions: (3, 3), (4, 4); not solutions: (-3, 1), (-1, -4) y 2b. x 25. x Possible answer: solutions: (0, 0), (3, -2); not solutions: (4, 4), (1, -6) y 3a. 29. y no solutions y 3b. x x y 31. y x 7. x y x Exercises 3. yes 5. y "LACKOLIVES 3c. Possible answer: (1 lb black, 1 lb green), (0.5 lb black, 2 lb green) 4a. y < -x 4b. y ≥ -2x - 3 all points between and on the parallel lines 33. y 3c. y x 9a. r + p ≤ 16 b. Punch Combinations x 35. 12 y x 8 4 4. 4 8 12 16 Orange juice (c) c. Possible answer: (2 c orange, 2 c pineapple), (4 c orange, 10 c pineapple) 11. y ≥ x + 5 13. yes 37. 7a + 4s ≥ 280 41. A 43. B 45. C 47. y 19a. 3x + 2y ≤ 30 b. (OTDOGSLB &OOD#OMBINATIONS c. Possible answer: (3 lb hamburger, 2 lb hot dogs), (5 lb hamburger, 6 lb 2 2 4 6 8 Pepper jack cheese (lb) 49. y ≥ __12 x + 3 51. yes 53. yes Possible answer: (3 lb pepper jack, 2 lb cheddar), (2.5 lb pepper jack, 4 lb cheddar) 55. y = __34 x + __74 57. y = 3x + 1 Exercises 1. all 3. yes 59. y = -__12 x + __12 61. (-2, 15) 63. (2, 5) 65. (12, 3) 5. y 6-6 Check It Out! 1a. yes 1b. no 2a. x y x (AMBURGERMEATLB 4 0 x 6 8 x y Cheese Combinations 0 15. same as solutions of y > -2x + 3 Cheddar cheese (lb) Pineapple juice (c) 16 Possible answer: solutions: (3, 3), (4, 3); not solutions: (0, 0), (2, 1) Selected Answers SA15 7. y 23. y x 49. x Possible answer: solutions: (0, 4), (1, 4); not solutions: (2, -1), (3, 1) 9. 25. 51. 25 cm 2 53. 12.5 cm 2 55. no 57. yes y x x All points are solutions. no solutions 27. y y x x y x 24 (6, 13) 16 (10, 10) (OURSATPHARMACY Possible answer: (0 h at pharmacy, 9 h babysitting), (8.5 h at pharmacy, 10 h babysitting) 31. 8 0 19. y x 8 16 24 Lemonade (c) Possible answer: (6 lemonade, 13 cupcakes), (10 lemonade, 10 cupcakes) 17. yes 33. x x y Possible answer: solutions: (-2, 0), (-3, 1); not solutions: (0, 0), (1, 4) ) 17. (-5, 2) 18. (6, 6) 19. 10 h; $1350; Motor Works 20. (-1, 3) 21. (5, -3) 22. (11, 1) 23. (0, 3) 24. (-2, 8) 25. (3, -5) 26. (4, -6) 27. (2, 2) 28. no solution 29. infinitely many solutions 30. (-2, -4) 31. infinitely many solutions 32. infinitely many solutions 33. (-1, -3) 34. no 35. consistent, independent; one solution 36. inconsistent; no solution 37. consistent, dependent; infinitely many solutions 38. inconsistent; no solution 39. consistent, independent; one solution 40. consistent, dependent; infinitely many solutions 41. inconsistent; no solution 42. no 43. yes 44. yes 45. no 46. y x 21. 7. yes 8. yes 9. no 10. (-1, -1) 11. (3, 4) 12. 8 h; $10 13. (-9, -6) y y same solutions as y > 2x - 1 Sales Goals 15. Cupcakes ,INDAS7ORK(OURS (OURSBABYSITTING 13. 29. 1. independent system 2. system of linear equations 3. solution of a system of linear inequalities 4. inconsistent system 5. independent system 6. no ( same solutions as y > 2 all points between the parallel lines and on the solid line Study Guide: Review 14. __12 , -2 15. (-1, 6) 16. (4, -5) no solutions y x 11. y ⎧y > x + 1 35. ⎨ ⎩y<x+3 ⎧y < 2 37. ⎨ ⎩ x ≥ -2 39. Student B 45. G 47. about 12 square units 47. y x y 48. x x Possible answer: solutions: (-1, 3), (0, 5); not solutions: (0, 0), (1, 4) SA16 Selected Answers 49. 58. y x Exercises 3. 0.00001 5. 100,000,000 y 50. y x Possible answer: solutions: (8, -8), (9, 0); not solutions: (0, 0), (0, -4) 59. y x 51. y y 60. x x 52. x = slices of pizza; y = bottles of soda; 2x + y ≥ 450 "OTTLESOFLEMONADE &UNDRAISING.EEDS Check It Out! 1a. 7 12 1b. 3 × 5 10 m 1 1c. ___ 1d. __ 2. 6.696 × 10 8 mi 4 7 5 n 3LICESOFPIZZA Possible answer: (200, 50), (150, 150) 53. no 54. yes 55. 1 1 mi 7. y 32 9. __ , or __ 11. x 7 y 13 4 81 3 1 ___ 125 1 1 2c. - __ 2d. 2b. __ 16 32 1 m 2a. _____ 10,000 1 1 - __ 3a. __ 32 64 2 1 4b. ___ 4c. g 4h 6 3b. 2 4a. ___ 3 3 m 7r 1 1 Exercises 1. ________ m 3. 1 5. __ 10,000,000 27 1 1 1 9. 1 11. __ 13. ___ 7. - ___ 512 16 256 3 1 17. __ 19. x 10d 3 21. __4 15. - __ 4 32 p7 f 1 1 __ __ 23. __ q 25. 1 27. 81 29. - 36 31. 1 2g 10 b 5 1 45. __ 47. - __ 49. ____ 43. __ 4 3 7 2 3 Possible answer: solutions: (-6, 6), (-10, 0); not solutions: (0, 0), (4, -4) y 51. s 5t 12 53. 1 55. x 1 __ q2 h 57. _____ 2 3 6m k a 1 61. - __16 63. 3 65. 3 67. __ 59. __ 2 16 2 y x Possible answer: solutions: (0, 0), (-5, 0); not solutions: (8, 0), (3, -3) y 3 b 1 _____ 3x 8y 12 79. never 81. sometimes 83. sometimes 87. 81 89. 1 91. -3 93. -1 95. D 97. A 103. -2 105. 4 107. 28 111. y = __13 x + 5 113. y = -4x + 9 7-2 Check It Out! 1a. 0.01 1b. 100,000 x k w 5 3 __ 69. ___ y 71. - 6 73. 2a b 75. g6 k 1 1 39. 1 41. ___ 33. - __13 35. 4 37. ___ 256 144 57. y Exercises 1. 2 5 3. n 8 5. 7.5 × 10 8 y x Possible answer: solutions: (-6, 2), (-8, 1); not solutions: (0, 0), (4, 1) x 3a. 3 20 3b. 1 3c. a 18 4a. 64p 3 1 4b. 25t 4 4c. __ 4 Chapter 7 Check It Out! 1. 56. 7-3 7-1 7. 10 -6 9. 650,300,000 11. 0.092 13. 5.85 × 10 -3, 2.5 × 10 -1, 8.5 × 10 -1, 3.6 × 10 8, 8.5 × 10 8 15. 0.000000001 17. 100,000,000,000,000 19. 10 6 21. 92,000 23. 0.00042 25. 10,000,000,000,000 27. 1.23 × 10 -3, 1.32 × 10 -3, 3.12 × 10 -3, 2.13 × 10 -1, 2.13 × 10 1, 3.12 × 10 2 29. 2.7 × 10 7 31. 2.35 × 10 5 33. 6 × 10 -7 35. 4.12 × 10 -2 37. yes 39. no; 2.5 × 10 2 41. yes 43. yes 47. 10 -3 51. F 55. Let m = number of minutes; m ≥ 45 3 1 59. (-2, 1) 61. __ 63. ___ 16 125 1c. 10,000,000,000 2a. 10 8 2b. 10 -4 2c. 10 -1 3a. 85,340,000 3b. 0.00163 4a. 1.43 × 10 5 km 4b. 13,000 m/s 5. 2 × 10 -12, 4 × 10 -3, 5.2 × 10 -3, 3 × 10 14, 4.5 × 10 14, 4.5 × 10 30 13. 36k 2 15. -8x 15 17. b 10 19. 6 8 x a 12 1 21. __ 23. 2 9, or 512 25. __ 27. ___ 5 3 2 y b x 29. 27x 3 31. p 28q 14 33. -256x 12 35. 6 37. 3 39. 8 41. 2x 3 43. 2m 10n 6 45. 108x 13 47. 125x 6 49. 3a 6 a7 51. 10 3, or 1000 57. __ 59. 15m 12n 9 b5 2 7 7 61. 9s t 63. t 67. yes 69. 17k 2 71. 6x 4 73. 15a 2b 3 75a. 6 × 10 -7 m b. 3 × 10 8 m/s c. Associative and Commutative Properties of 2 2 1 Multiplication 77. (6ab) 79. _____ 2kmn 81. H 83. F 85. 3 2x 87. x + 1 2 89. x 3y + 3z 91. x x 93. x = 4 95. x = 4 97. 1.728 × 10 -6 99. 15 101. no 103. 7,800,000 105. 98.3 ( ) 7-4 n n 1 1b. __ 1c. ___ Check It Out! 1a. ___ m5 m5 y3 3 3 3 2. 1.1 × 10 -2 3. $12,800 1d. __ 16 64 a b a 9 2 , or __ 4b. _____ 4c. __ 5a. __ , 4a. __ 4 10 15 3 3 81 5 20 6 or 3 729 ___ 64 3 c d b c t 5b. ____ 5c. __ 4 4 8 12 16a b 3 4 s Exercises 1. 25 3. 3 5. 7 × 10 2 7. 1 16 2b 4 1 9. __ 11. ____ 13. __ 15. ___ 17. 27 6 4 2 25 9 2 a b 3a 19. x 5 21. 5 × 10 -7 23. 7 × 10 -3 y3 a 29. __6 25. 2 × 10 27 kg 27. ___ 6 12 b 31. 39. x y 25 196 3x 5 ___ 33. ___ 35. 2d 2 37. ___ 10 2 4 x 9x c4 25 1 __ 41. __ 43. ___ 45. -1 4 2 100 a p Selected Answers SA17 47. 2000: 3 × 10; 1995: 2.84 × 10; 1990: 2.65 × 10 1 51. 3 53. 3; 4 55. B 57. A 59. 3 61. m; (-n); m; -n; Definition of negative exponent; a n 63. 1 65. 12 67 x = - __12 69. 1 71. -125x 12 7-5 Check It Out! 1a. 3 1b. 15 y 2a. 8 2b. 1 2c. 81 3. 1944 4a. xy 3 4b. xy 7-7 Exercises 1. 5 3. 4 5. 3 7. 6 9. 5 11. 10 13. 4 15. 32 17. 125 19. 256 21. 0 23. x 2y 25. x 3y 3 27. a 2 29. 1 31. 10 33. 8 35. 2 37. 2 39. 14 41. 8 43. 8 45. 64 47. 1000 49. 243 51. 2g 53. 2m 55. 3x 2 57. ab 4 59. a 8b 61. 1 63. 0 65. 625 8 67. 3 69. __23 71. __14 73. __49 75. ___ 343 2 __ 16 1 79. ___ 81. 1.86 in. 83. n 3 77. __ 27 625 will be less than n because __23 < 1. 3 __ n 2 will be greater than n because 3 __ > 1. 85a. 10 in. 85b. The distance 2 doubles (20 in.). 87. B 89. C 91. a 93. x 3 95. 3 97. 36π cm 2; both volume and surface area are described by 36π (although the units are different). 99. -1 101. n < 3 103. y ≤ -2 105. D: {-2, -1, -0, 1}; R: {0, 1, 2, 3}; yes; each domain value is paired with exactly one range value. 107. D: 1 ≤ x ≤ 4; R: 2 ≤ y ≤ 4; yes; each domain value is paired with exactly one range value. 7-6 Check It Out! 1a. 3 1b. 1 1c. 3 2a. 1 2b. 5 3a. x 5 + 9x 3 - 4x 2 + 16; 1 3b. -3y 8 + 18y 5 + 14y; -3 4a. cubic polynomial 4b. constant monomial 4c. 8th degree trinomial 5. 1606 ft Exercises 1. d 3. a 5. 3 7. 0 9. 8 11. 3 13. 4 15. -8a 9 + 9a 8; -8 17. 3x 2 + 2x - 1; 3 19. 5c 4 + 5c 3 + 3c 2 - 4; 5 21. linear binomial 23. quartic polynomial 25. quartic trinomial 27. 4 29. 6 31. 7 33. 1 35. 4 37. 2 39. 3 41. 4.9t 3 - 4t 2 + t + 2.5; 4.9 43. x 10 + x 7 - x 5 + x 3 - x; 1 45. 5x 3 + 3x 2 + 5x - 4; 5 47. -d 3 + 3d 2 + 4d + 5; -1 49. -x 5 - x 3 + 4x 2 + 1; -1 51. linear monomial 53. quadratic SA18 Selected Answers trinomial 55. quartic trinomial 57. quadratic monomial 59. always 61. never 63a. 58.5 in3 b. 66 in3 c. 0 d. yes 65. -48; 0; 3270 75. A is incorrect 77. J 79a. 58 cm; 65 cm b. 50.310 cm 81. 90 - m 83. inconsistent; no solutions 85. consistent and independent; p8 x2 one solution 87. __ 89. __ 5 16 Check It Out! 1a. 5s 2 + s 1b. 20z 4 - 6 1c. x 8 + 6y 8 1d. b 3c 2 2. 12a 3 + 15a 2 - 16a 3. -2x 2 - x 4. -0.05x 2 + 46x - 3200 Exercises 1. -3a 2 + 9a 3. 0.26r 4 + 0.32r 3 5. 3b 3c 7. 23n 3 + 3n + 15 9. 9x 2 - x - 6 11. 4c 4 + 8c + 6 13. -3r + 11 15. 8a 2 + 5a + 9 17. 12n 2 + 6n - 3m 19. d 5 + 1 21. 5x 23. 2x 3 - 5 25. 10t 2 + t 27. x 5 + x 4 29. -2t 3 + 8t 2 31. -6m 3 + 2m 2 + 5m + 3 33. 4w 2 + 6w + 4 35. t - 5 37. 2n -2 39. 6x 2 - x - 1 41. -u 3 + 3u 2 + 3u + 6 43. x = __32 , or 1.5 45. B is incorrect. 47. 3x + 6 49. 6x + 14 51. 2x 2 + x - 5 55. G 57. 3x 2 - 2 69. b 11 71. 9z 12 7-8 Check It Out! 1a. 18x 5 1b. 10r 2t 4 1c. 4x 5y 5z 7 2a. 8x 2 + 2x + 6 2b. 15a 3b + 3ab 2 2c. 5r 3s 2 - 15r 2s 3 3a. a 2 - a - 12 3b. x 2 - 6x + 9 3c. 2a 2 + 7ab 2 - 4b 4 4a. x 3 - x 2 6x + 18 4b. 3x 3 - 4x 2 + 11x + 10 5a. x 2 - 4x 5b. 12 m 2 Exercises 1. 14x 6 3. 3r 5s 5t 5 5. 21x 7y 3 7. 4x 2 + 8x + 4 9. 6a 5b 2 + 2a 4b 3 11. 10x 3y 4 - 5x 2y 2 13. x 2 - x - 2 15. x 2 - 4x + 4 17. 4a 4 - 2ab 12a 3b 2 + 6b 3 19. x 3 + 3x 2 - 7x + 15 21. -6x 4 + 12x 3 + 4x 2 - 18x + 20 23. x 3 - 4x 2 - 4x - 5 25a. 2x 2 - 3x b. 20 in 2 27. -12r 5s 5 29. 10a 4 31. -6a 5b 6 33. -12a 7b 7c 8 35. 9s 2 + 54s 37. 27x 3 - 12x 2 39. 10s 3t 3 - 15s 2t 5 41. -10x 3 + 15x 2 + 5x 43. -14x 6y 3 + 7x 5y 4 45. x 2 + 8x + 16 47. 5x 2 + 13x - 6 49. 10x 2 - x - 2 51. 7x 2 - 52x - 32 53. x 3 - x 2 - x + 10 3 55. -10x 4 + 2x + 20x 2 - 19x + 3 57. 8x 5 - 12x 3 - 2x 4 + 17x 2 - 21 59. x 3 - 3x - 2 61. -x 3 + 3x 2 - 3x + 1 63. 16x 2 - 48x + 36 65a. 3; 2; 10x 5 + 5x 3; 5 b. 2; 2; x 4 - x 3 + 2x 2 - 2x; 4 c. 1; 3; x 4 - 5x 3 + 6x 2 + x 3; 4 d. m + n 67. 12x 2 + 12x + 3 69a. 2x 2 b. 800 m 2 71. 2x 2 - 7x 30 73. 8x 2 - 16xy + 6y 2 75. 6x 2 9x - 6 77. x 3 + 3x 2 79. 2x 3 7x 2 - 10x + 24 81. 8p 3 - 36p 2q + 54pq 2 - 27q 3 87. C 89. D 91. -x 2 - 6 93. a. x 2 - 1 b. 8x + 16 95. x 3 + 3x 2 + 2x 97. a = 2 99. 3.61 101. 9.49 7-9 Check It Out! 1a. x 2 + 12x + 36 1b. 25a2 + 10ab + b2 1c. 1 + 2c 3 + c6 2a. x 2 - 14x + 49 2b. 9b 2 - 12bc + 4c 2 2c. a 4 - 8a 2 + 16 3a. x 2 - 64 3b. 9 - 4y 4 3c. 81 - r 2 4. 25 Exercises 3. 4 + 4x + x 2 5. 4x 2 + 24x + 36 7. 4a 2 + 28ab + 49b 2 9. x 2 - 4x + 4 11. 64 - 16x + x 2 13. 49a 2 - 28ab + 4b 2 15. x 2 - 36 17. 4x 4 - 9 19. 4x 2 - 25y 2 21. x 2 + 6x + 9 23. x 4 + 2x 2y 2 + y 4 25. 4 + 12x + 9x 2 27. s 4 - 14s 2 + 49 29. a 2 - 16a + 64 31. 9x 2 - 24x + 16 33. a 2 - 100 35. 49x 2 - 9 37. 25a 4 - 81 39. π x 2 + 8π x + 16π 41. x 2 + 2xy + y 2 43. x 4 - 16 45. x 4 - 8x 2 + 16 47. 1 + 2x + x 2 49. x 6 - 2a 3x 3 + a 6 51. 36a 2 25b 2 53. 4; 4 55. 25; 25 57. 9; 9 59. -5; -5 61. 840 65. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 67. B 69. D 71. x 3 + 4x 2 - 16x - 64 73. b = ± 2 √c 75. 13 cm Study Guide: Review 1. cubic 2. standard form of a polynomial 3. monomial 4. trinomial 5. scientific notation 1 1 1 6. __ in. 7. 1 8. 1 9. ___ 10. _____ , 32 125 10,000 27 1 1 12. ___ 13. __ or 0.0001 11. __ 4 16 256 1 1 15. b 16. -____ 17. 2b 6c 4 14. ___ 2 4 2 m 2x y 3a s 19. ___ 20. 10,000,000 18. ___ 2 2 2 3 4c qr 21. 0.00001 22. 10 2 23. 10 -11 24. 325,000 25. 1800 26. 0.17 27. 0.000299 28. 5.8 × 10 -7, 6.3 × 10 -3, 2.2 × 10 2, 1.2 × 10 4 29. $38,500,000,000 30. 5 9 31. 2 3 · 3 4 32. b 10 33. r 5 34. x 12 1 1 1 35. 1 36. __ , or __18 37. __ , or ___ 4 3 625 2 5 1 38. ____ 39. g 12h 8 40. x 4y 2 6 16b 44. __1 41. -x y 42. x y 43. j k 5 45. m 8n 30 46. 8 × 10 11 47. 9 × 10 7 48. 1 × 10 10 49. 2.8 × 10 15 50. 6 × 10 1 51. 1.8 × 10 -8 7 52. 3.55 × 10 7 53. 64 54. m 5 55. __ 32 3 4 1 56. 6b 57. t v 58. 16 59. 5 ×10 60. 2.5 × 10 7 61. 9 62. 7 63. 16 64. 8 65. z 2 66. 5x 2 67. x 4y 3 68. m 2n 4 69. 0 70. 3 71. 6 72. 1 73. 3n 2 + 2n - 4; 3 74. -a 6 - a 4 + 3a 3 + 2a; -1 75. linear binomial 76. quintic monomial 77. quartic trinomial 78. constant monomial 79. -4t + 3 80. -6x 6 - x 5 81. 3h 3 3h 2 + 5 82. 2m 2 - 5m - 1 83. p 2 + 5p + 8 84. -7z 2 - z + 10 85. 3g 2 + 2g + 4 86. -x 2 + 4x + 8 87. 8r 2 88. 6a 6b 89. 18x 3y 2 90. 3s 6t 14 91. 2x 2 - 8x + 12 92. -3a 2b 2 + 6a 3b 2 - 15a 2b 93. a 2 - 3a - 18 94. b 2 - 6b - 27 95. x 2 - 12x + 20 96. t 2 - 1 97. 8q 2 + 34q + 30 98. 20g 2 - 37g + 8 99. p 2 - 8p + 16 100. x 2 + 24x + 144 101. m 2 + 12m + 36 102. 9c 2 + 42c + 49 103. 4r 2 - 4r + 1 104. 9a 2 - 6ab + b 2 105. 4n 2 - 20n + 25 106. h 2 26h + 169 107. x 2 - 1 108. z 2 - 225 109. c 4 - d 2 110. 9k 4 - 49 4 2 6 15 6 9 Chapter 8 8-1 Check It Out! 1a. 2 3 · 5 1b. 3 · 11 1c. 7 2 1d. 19 2a. 4 2b. 5 3a. 9g 2 3b. 1 3c. 1 4. 7 Exercises 3. 3 2 · 2 2 5. 3 3 · 2 7. 7 (prime) 9. 3 · 5 2 11. 7 13. x 2 15. 1 17. 2 · 3 2 19. 2 2 · 3 21. 17 23. 7 2 25. 9 27. 10 29. 9s 31. 5 33. 4x 2 35. 2n 39. 15 rows 41. 8 and 20; 4 43. 63 and 105; 21 45. 54 and 72; 18 47. 36; 2; 9; 3 49. 105; 5; 7 51. 2; 2; 27; 3 53. 24; 2; 6; 3 55. 2; 2; 10, 5 57. D 61. 9y 63. 2p 2r 65. 4a 2b 3 69. 3.12 h 71. ≈ 0.10 mi/yr 73. 40 75. 3x 2 + 14x - 3 1d. 2x 2(4x 2 + 2x - 1) 2. 2x cm; (x + 2) cm 3a. (4s - 5)(s + 6) 3b. (7x + 1)(2x + 3) 3c. cannot be factored 3d. (5x - 2)2 4a. (2b 2 + 3)(3b + 4) 4b. (4r + 1)(r 2 + 6) 5a. (5x 2 - 4)(3 - 2x) 5b. (8 + x)(y - 1) Exercises 1. 5a(3 - a) 3. 7(-5x + 6) 5. 2h(6h 3 + 4h - 3) 7. m(9m + 1) 9. 3(12f + 6f 2 + 1) 11. 16t(-t + 20) 13. (2b + 5)(b + 3) 15. (x 2 + 2)(x + 4) 17. (2b 2 + 5)(2b - 3) 19. (7r 2 + 6)(r - 5) 21. 2(r - 2)(r - 3) 23. (7q - 2)(2q - 3) 25. (2m 2 - 3)(m - 3) 27. 9y(y + 5) 29. x 2(-14x 2 + 5) 31. -d 2(4d 2 - d + 3) 33. 7c(3c + 2) 35. -5g 2(g + 3) 37. cannot be factored 39. (6y + 1)(y - 7) 41. (-3 + 4b)(b + 2) 43. (2a 2 + 3)(a - 4) 45. (6x 2 + 1)(x + 3) 47. (n 2 + 5)(n - 2) 49. (2m 2 - 3)(m - 1) 51. (2f 2 - 5)(3f - 4) 53. (b 2 - 2)(b + 4) 55. 3v 57. 2k 59. 2; binomial; x(x + 5) 61. 3; trinomial; a 2(a 2 + a + 1) 63a. 100x 3; 200x 2; 400x b. 100x 3 + 200x 2 + 400x + 800 c. 100(x 2 + 4)(x + 2); $1603.12 67. The sum of opposite binomials is 0. 69a. Commutative Property of Addition b. Associative Property of Addition c. Distributive Property d. Distributive Property 71. D 73. C 75. -9ab(8ab + 5) 77. (a + c)(b + d) 79. (x 2 + 3)(x - 4) 81. 11 83. 5 8-3 Check It Out! 1a. (x + 4)(x + 6) 1b. (x + 4)(x + 3) 2a. (x + 6)(x + 2) 2b. (x - 6)(x + 1) 2c. (x + 6)(x + 7) 2d. (x - 8)(x - 5) 3a. (x + 5)(x - 3) 3b. (x - 4)(x - 2) 3c. (x - 10)(x + 2) 4. n n 2 - 7n + 10 0 0 2 - 7 (0) + 10 = 10 Check It Out! 1a. b(5 + 9b 2) 1 1 2 - 7(1) + 10 = 4 1b. cannot be factored 1c. -y 2(18y + 7) 2 2 2 - 7(2) + 10 = 0 3 3 2 - 7(3) + 10 = -2 4 4 2 - 7(4) + 10 = -2 8-2 n 0 1 2 3 4 (n - 5)(n - 2) (0 - 5)(0 - 2) = 10 (1 - 5)(1 - 2) = 4 (2 - 5)(2 - 2) = 0 (3 - 5)(3 - 2) = -2 (4 - 5)(4 - 2) = -2 Exercises 1. (x + 4)(x + 9) 3. (x + 4)(x + 10) 5. (x + 2)(x + 8) 7. (x - 1)(x - 6) 9. (x - 3)(x - 8) 11. (x + 9)(x - 3) 13. (x - 2)(x + 1) 15. (x - 9)(x + 5) 17. (x + 3)(x + 10) 19. (x + 4)(x + 12) 21. (x + 2)(x + 14) 23. (x - 1)(x - 5) 25. (x - 4)(x - 8) 27. (x + 7)(x - 3) 29. (x - 13)(x + 1) 31. (x - 7)(x + 5) 33. C 35. D 37. They are inverse operations. 39. (x - 2)(x - 9) 41. (x + 1)(x + 9) 43. (x + 6)(x + 7) 45. (x + 2)(x + 9) 47. (x - 3)(x + 8) 49. (x - 5)(x + 9) 51. approximately 1.5 55. x 2 + 6x + 8; (x + 4)(x + 2) 57. Positive; - , - ; Both negative 59. Negative; + , - ; Positive; Negative 61a. d = t 2 b. d = 4t c. t(t - 4) 63. true 65. false 67. 4 69. 4 71a. (x + 10) ft b. = (x + 14) ft; w = (x + 6) ft c. A = (x 2 + 20x + 84) ft 2 73. D 75. C 77. (x 2 + 9)(x 2 + 9) 79. (d 2 + 21)(d 2 + 1) 81. (de - 5)(de + 4) 83. 16; 11; 29 85a. (x + 7) ft b. (4x + 26) ft c. $92.00 d. $36.96 e. $128.96 87. x 5 89. t 12 91. (x + 2)(x 2 + 5) 93. (p - 2)(2p 3 + 7) 8-4 Check It Out! 1a. (3x + 1)(2x + 3) 1b. (3x + 4)(x - 2) 2a. (2x + 5)(3x + 1) 2b. (3x - 4)(3x - 1) 2c. (3x + 4)(x + 3) 3a. (3x - 1)(2x + 3) 3b. (4n + 3)(n - 1) 4a. -1(2x + 3)(3x + 4) 4b. -1(3x + 2)(x + 5) Exercises 1. (2x + 5)(x + 2) 3. (5x - 3)(x + 2) 5. (3x + 4)(x - 6) 7. (x + 2)(5x + 1) 9. (4x - 5)(x - 1) 11. (5x + 4)(x + 1) 13. (2a - 1)(2a + 5) 15. (2x - 3)(x + 2) 17. (10x + 1)(x - 1) 19. (2x + 3)(4 - x) Selected Answers SA19 21. -1(5x + 3)(x - 2) 23. -1(2x - 1)(2x + 5) 25. (3x + 2)(3x + 1) 27. (n + 2)(3n + 2) 29. (4c - 5)(c - 3) 31. (2x + 5)(4x + 1) 33. (5x - 6)(x + 3) 35. (10n - 7)(n - 1) 37. (7x + 1)(x + 2) 39. (3x - 4)(x - 5) 41. (x - 7)(4x - 3) 43. (4y - 1)(3y + 5) 45. (2x - 1)(2x + 3) 47. (3x + 5)(x - 3) 49. -1(2x - 3)(2x + 5) 51. -1(3x - 2)(x + 1) 53. 2x 2 - 5x + 2; (x - 2)(2x - 1) 55. (9n + 8)(n + 1) 57. (2x - 1)(2x - 5) 59. (3x + 8)(x + 2) 61. (3x + 4)(2x - 3) 63. (2x - 3)(2x - 3) 65. (2x + 3)(3x + 2) 69a. -16t 2 + 20t + 6 b. -2(4t + 1)(2t - 3) c. 10 ft 71. D 73. B 77. B 79. A 81. (2x + 1)(2x + 1) 83. (9x + 1)(9x + 1) 85. (5x + 2)(5x + 2) 87. -7; -5; 5; 7 89. -6; 6 95. (x + 1)(x - 9) 8-5 Check It Out! 1a. yes; (x + 2)2 1b. yes; (x - 7) 2 1c. no; -6x ≠ 2(3x)(2) 2. 4(3x + 1) m; 40 m 3a. yes; (1 - 2x)(1 + 2x) 3b. yes; (p 4 + 7q 3)(p 4 - 7q 3) 3c. No; 4y 5 is not a perfect square. Exercises 1. yes; (x - 2)2 3. yes; (3x - 2)2 5. yes; (x - 3) 2 7. 4(x + 12); 88 yd 9. yes; (s + 4) (s - 4) 11. yes; (2x 2 + 3y)(2x 2 - 3y) 13. yes; x 3 + 3x 3 - 3 15. No; the last term must be positive. 17. no; 10x ≠ 2(5x)(2) 19. yes; (4x - 5)2 21. yes; (1 + 2x)(1 - 2x) 23. No; 4x and 9y are not perfect squares. 25. yes; (9 - 10x 2)(9 + 10x 2) 27. 49 29. 4y 2 31. (10x + 9y)(10x - 9y); difference of 2 squares 33. (2r 3 + 5s 3) (2r 3 - 5s 3); difference of 2 squares 35. (x 7 + 12)(x 7 - 12); difference of 2 squares 39. c = 32 41a. 5z - 4 b. 20z - 16 c. 11; 44; 121 43a. 0; 0; 100; 100; 0 b. 16; 16; 36; SA20 Selected Answers 36; -24 c. 25; 25; 25; 25; -25 d. 36; 36; 16; 16; -24 e. 100; 100; 0; 0; 0 45. a - b; a + b 47. C 49. 1 51a. a = 2; b = v + 2 b. [2 + (v + 2)][2 - (v + 2)] = (v + 4)(-v) = -v 2 - 4v 53. a = 3y; b = y; (3y - 4)(9y 2 + 12y + 16) 55. D: {5, 4, 3, 2}; R: {2, 1, 0, -1}; yes 57. D: {2}; R: {-8, -2, 4, 10}; no 59. 6a 3 + 14a 2 - 10a 61. t 2 - 8t + 16 63. 8 8-6 Check It Out! 1a. yes 1b. no; 4(x + 1) 2 2a. 4x(x + 2) 2 2b. 2y(x - y)(x + y) 3a. (3x + 4)(x + 1) 3b. 2p 3(p + 6)(p - 1) 3c. 3q 4(3q + 4)(q + 2) 3d. 2(x 4 + 9) Exercises 1. yes 3. yes 5. no; 4(2p 2 + 1)(2p 2 - 1) 7. 3x 3(x + 2)(x - 2) 9. 2p(2q + 1) 2 11. mn(n 2 + m)(n 2 - m) 13. 3x 2(2x - 3)(x + 1) 15. (p 3 + 1)(p 2 + 3) 17. unfactorable 19. no; 2xy(y 2 - 4y + 5) 21. no; 3n 2(n + 5)(n - 5) 23. yes 25. -4x(x - 3)2 27. 5(d - 3)(d - 9) 29. 2x(7x + 5y)(7x - 5y) 31. unfactorable 33. (p 2 + 4)(p + 2)(p - 2) 35. (k 2 + 3)(2k + 3) 37. x 2 + 12x + 36 = (x + 6)2 39. s 2 - 16s + 28 = (s - 2)(s - 14) 41. b 2 - 49 = (b + 7)(b - 7) 45. (3x - 1)(x + 7) 47. (3x + y - 3)(3x - y - 7) 53. 8 55. C 57. C 59a. V = 8p⎡⎣π(3p + 1)2⎤⎦ b. r = (3p + 1) cm c. h = 8 cm; V = 128π cm 3 61. h 2(h 4 + 1)(h 2 + 1) 63. x n + 3(x 2 + x + 1) 65. -2n 23 67. 12.3r 69. __ = __3x ; 34.5 cm 2 71. (2x - 1)(2x + 3) Study Guide: Review 1. prime factorization 2. greatest common factor 3. 2 2 · 3 4. 2 2 · 5 5. 2 5 6. prime 7. 2 3 · 5 8. 2 6 9. 2 · 3 · 11 10. 2 · 3 · 19 11. 5 12. 12 13. 1 14. 27 15. 4 16. 3 17. 2x 18. 9b 2 19. 25r 20. 6 boxes; 13 rows 21. 5x(1 - 3x 2) 22. 16(-b + 2) 23. -7(2v + 3) 24. 4(a 2 - 3a - 2) 25. 5g(g 2 - 3)(g 2 + 1) 26. 10(4p 2 - p + 3) 27. (6x + 5) ft by x ft 28. (2x + 9)(x - 4) 29. (t - 6)(3t + 5) 30. (5 - 3n)(6 - n) 31. (b + 2)(b + 4) 32. (x 2 + 7)(x - 3) 33. (n 2 + 1)(n - 4) 34. (2b + 5)(3b - 4) 35. (2h 2 - 7)(h + 7) 36. (3t + 1)(t + 6) 37. (5m - 1)(2m + 3) 38. (4p - 3)(2p 2 + 1) 39. -1(r - 5)(r - 2) 40. (b 2 - 5)(b - 3) 41. (t + 4)(-t 2 + 6) 42. -1(3h - 1)(h - 4) 2 43. -1(d - 1) 44. (2 - b)(5b - 6) 45. (t + 1)(5 - t) 46. (2b 2 + 5)(4 - b) 47. -1(3r - 1)(r - 1) 48. left rectangle: 2x 2 + 3x; right rectangle: 8x + 12; combined: 2x 2 + 8x + 3x + 12; (2x + 3)(x + 4) 49. (x + 1)(x + 5) 50. (x + 2)(x + 4) 51. (x + 3)(x + 5) 52. (x - 6)(x - 2) 53. (x + 5) 2 54. (x - 2)(x - 11) 55. (x + 4)(x + 20) 56. (x - 6)(x - 20) 57. (x + 12)(x - 7) 58. (x + 3)(x - 8) 59. (x + 4)(x - 7) 60. (x - 1)(x + 5) 61. (x + 3)(x - 2) 62. (x + 5)(x - 4) 63. (x - 8)(x + 6) 64. (x - 9)(x + 4) 65. (x - 12)(x + 6) 66. (x - 10)(x + 7) 67. (x + 20)(x - 6) 68. (x + 7)(x - 1) 69. (y + 3) m 70. (2x + 1)(x + 5) 71. (3x + 7)(x + 1) 72. (2x - 1)(x - 1) 73. (3x + 2)(x + 2) 74. (5x + 3)(x + 5) 75. (2x - 3)(3x - 5) 76. (4x + 5)(x + 2) 77. (3x + 4)(x + 2) 78. (7x - 2)(x - 5) 79. (3x + 2)(3x + 4) 80. (2x + 1)(x - 1) 81. (3x + 1)(x - 4) 82. (2x - 1)(x - 5) 83. (7x + 2)(x - 3) 84. (5x + 1)(x - 2) 85. -1(2x - 1)(3x + 2) 86. (6x + 5)(x - 1) 87. (3x - 2)(2x + 7) 88. -1(2x + 1)(2x - 5) 89. -1(2x - 3)(5x + 2) 90. 12x 2 - 11x - 5; (4x - 5)(3x + 1) 2 91. yes; (x + 6)2 92. no; 5x ≠ 2(x)(5) 93. no; -2x ≠ 2(2x)(1) 94. yes; (3x + 2)2 95. no; 8x ≠ 2(4x)(2) 96. yes; (x + 7)2 97. yes; (10x - 9)(10x + 9) 98. No; 2 is not a perfect square. 99. No; 5 and 10 are not perfect squares. 100. yes; (-12 + x 3)(-12 - x 3) 101. no; terms must be subtracted 102. yes; (10p - 5q)(10p + 5q) 103. (x - 5)(x + 5); difference of 2 squares 104. (x + 10) 2; perfectsquare trinomial 105. (j - k2)(j + k2); difference of 2 squares 106. (3x - 7)2; perfect-square trinomial 107. (9x + 8) 2; perfect-square trinomial 108. (4b 2 - 11c 3)(4b 2 + 11c 3); difference of 2 squares 109. no; 2(2x + 3)(x + 1) 110. yes 111. no; (b 2 + 9)(b - 3)(b + 3) 112. yes 113. 4(x - 4)(x + 4) 114. 3b 3(b - 4)(b + 2) 115. a 2b 3(a - b)(a + b) 116. t 4 (t 8 + 1)(t 4 + 1)(t 2 + 1) (t + 1)(t - 1) 117. 5(x + 3)(x + 1) 118. 2x 2(x - 5)(x + 5) 119. 2(s + 4)(t + 4) 120. 5m(5m + 2)(m - 4) 121. 4x(4x 2 + 1)(2x - 3) 122. 6s 2t (s + t 2) 123. 2(m + 3)(m - 3)(5m + 2) Chapter 9 4a. vertex: (-2, 5); maximum: 5 4b. vertex: (3, -1); minimum: -1 5a. D: all real numbers; R: y ≥ 4 5b. D: all real numbers; R: y ≤ 3 1b. x Exercises 1. minimum 3. yes 5. no 7. Exercises 1. 9. 3. x 2b. y x 11. upward; a > 0 13. upward; a > 0 15. downward; a < 0 17. (-3, -4); minimum: -4 19. D: all real numbers; R: y ≤ 4 21. D: all real numbers; R: y ≥ -4 23. yes 25. yes 31. upward; a > 0 33. vertex: (0, -5); minimum: -5 35. D: all real numbers; R: y ≤ 0 37. D: all real numbers; R: y ≥ -2 39. never 41. always 43. sometimes 45. no 47. yes 49. yes 53. quadratic 55. quadratic 57. neither 59. linear 61b. t ≥ 0 c. 16 ft d. 2 s 65. C 67. yes 71. (-2)4 73. 42 __34 mi 5. 7. x = 2 9. x = -2 11. x = - __34 13. (1, 8) 15. (-2, -2) 17. (-2, -9) 19. no zeros 21. -8, -2 23. x = 6 25. x = - __12 27. x = 5 29. (-3.5, -12.25) 31. (-2, 5) 33. (1, 4.5) 35. x = 0. 37. 0 39. 2 41. B 43. 2 45. 25 ft; 100 ft 47. y = -2x + 3 49. y = -4x + 2 51. yes 9-3 1a. y y x 7. maximum height: 144 ft at 3 s; time in the air: 6 s y 9. x 11. 9-2 x 3a. Because a < 0, the parabola opens downward. 3b. Because a > 0, the parabola opens upward. x Check It Out! y Exercises 3. -1 5. no zeros x differences are constant. 1b. Yes; the function can be written in the form y = ax 2 + bx + c. y y x x 1b. 3 2a. x = -3 2b. x = 1 3. x = - __14 4. (2, -14) 5. 7 ft y 9-1 Check It Out! 1a. Yes; the second 2. maximum height: 16 ft at 0.5 s; time it takes to reach the pool: 1.5 s y y x Check It Out! 1a. no zeros 2a. y 13. y x 15. x = 4; (4, -16) 17. x = 0; (0, 4) 15 19. x = - __12 ; - __12 , - __ 21. vertex: 4 (0, 0); axis of symmetry: x = 0 23. vertex: (3, -5); axis of symmetry: x = 3 25. vertex: (0, -4); axis of symmetry: x = 0 27b. D: 0 ≤x ≤ 3.16; R: 0 ≤ y ≤ 50 c. about 3.16 s 31. 12 cm/s 35a. h(t) = -16t 2 + 45t + 50 b. approximately (1.4, 81.6) 37. A 39. D 41. -1 43. x-intercept: 3; y-intercept: 6 ( ) Selected Answers SA21 45. no x-intercept; y-intercept: 3 47. (3, -1) 49. (-1, -16) 51. (-1, 4) 9-4 Check It Out! 1a. f (x), g(x) 1b. g(x), f (x), h (x) 2a. same width, same axis of symmetry, opens upward, translated 4 units down 2b. narrower, same axis of symmetry, opens upward, translated 9 units up 2c. wider, same axis of symmetry, opens upward, translated 2 units up 3a. The graph of the ball that is dropped from a height of 100 ft is a vertical translation of the graph of the ball that is dropped from a height of 16 ft. The y-intercept of the graph of the ball that is dropped from 100 ft is 84 units higher. 3b. The ball that is dropped from 16 ft reaches the ground in 1 s. The ball that is dropped from 100 ft reaches the ground in 2.5 s. Exercises 1. f (x), g (x) 3. h (x), g (x), f (x) 5. same width, same axis of symmetry, opens upward, translated 6 units up 7. wider, same axis of symmetry, opens upward, same vertex 9a. h 1(t) = -16t 2 + 16, h 2(t) = -16t 2 + 256; the graph of h 2 is a vertical translation of the graph of h 1. The y-intercept of h 2 is 240 units higher. b. The baseball that is dropped from 256 ft reaches the ground in 4 s. The baseball that is dropped from 16 ft reaches the ground in 1 s. 11. f (x), g(x) 13. f(x), h (x), g (x) 19. always 21. never 25. f (x) = 3x 2 - 6 29. B 31. A 39. D 41. 0 43a. f (x) = x 2 - 7 b. f (x) = -x 2 + 2 c. f (x) = __12 x 2 + 1 45. no correlation 9-5 Check It Out! 1a. x = -4 1b. no real solutions 1c. x = -2 or x = 2 2. 2 s Exercises 1. zeros, x-intercepts 3. -4, 4 5. 6 7. -2, 4 9. -1 11. no real solutions 13. no real solutions 15. x = -4 or x = 4 17. x = __12 or x = 5 19. x = -3 21. no real solutions 23. no real solutions 25. always 27. always SA22 Selected Answers 29. never 31a. 4 s b. 10 ft 33. 1.4 s 35. -1, 1 41. C 45. no real solutions 47. x ≈ -1.6 or x ≈ 0.86 49. y - 4 = -3(x + 2) 51. 27 53. x 11 55. a 3b 3 27b 2 57. ____ 6 8a 9-6 Check It Out! 1a. 0, -4 1b. -4, 3 2a. 3 2b. 1, -5 2c. - __53 2d. __13 , 1 3. 1.5 s Exercises 1. -2, 8 3. -7, -9 5. -11, 0 7. -6, 2 9. 2, 3 11. -8, -2 13. 4 15. 6 17. -2, -__32 19. 1 s 21. -4, -7 23. 0, 9 25. - __12 , __13 27. -2, 4 29. -__13 , 1 31. -2 33. 1 35. 1 37. 1 39. B 41. 6 m 43. 6 s 45. no 47a. 3 s b. 64 ft c. yes 49. F 51. -5, -1 53. __12 , -3 55. -2, 5 57. -1, 0 59. x 2 - x - 12 61. x 2 - 4x - 12 = 48; x = 10 63. -8 65. 2 67. (-9, -16) 69. ±7 71. 3, 5 23. -13, -2 25. -6, 8 27. -2, 3 -1 ± √ 5 -15 ± √ 105 31. _________ 33. 4 in. 29. _______ 2 2 7 37. -3, __12 39. -10, 2 35. 1 ± √ 49 41. 81 43. __ 45. 9 4 47a. (10 + 2x)(24 + 2x) = 640 b. 3 ft 51. -6 ± √ 26 53. -6 55. no real solutions 57. no real solutions 61a. -16t 2 + 64t + 32 = 0 b. 4 c. 4.4 s 63. B 65. B 67. - __32 , __23 √ 7 √ 7 , - __23 + ___ 71. 0, - __ab 69. - __23 - ___ 3 3 2 2 77. x - 8x + 16 79. t - 8t + 16 81. 64b 4 - 4 83. ±1 85. ±4 87. ±15 89. ±1.55 91. ±5.10 93. ±1.48 9-9 Check It Out! 1a. 2, - __13 1b. 2, -__15 8 - √ 56 8 + √ 56 ≈ 0.13, ______ ≈ 3.87 2. ______ 4 4 3a. 0 3b. 1 3c. 2 4. No; for the equation 45 = -16t 2 + 20t + 0, the discriminant is negative, so the weight will not ring the bell. 5a. -2, -5 5b. -2, 7 -4 - √ 184 9-7 Check It Out! 1a. ±11 1b. 0 1c. no real solutions 2a. no real solutions 2b. 9, 1 3a. ±9.49 3b. ±5.66 3c. no real solutions 4. 45 ft Exercises 1. ±15 3. no real solutions 5. no real solutions 7. ±5 9. no real solutions 11. 11, -5 13. ±5.20 15. ±4.47 17. ±13 19. no real solutions 21. no real 13 solutions 23. ± __29 25. ± __58 27. ± __ 7 29. ±4.69 31. ±10.20 33. ±7.07 2d __ 35. 6.1 s 37. t = √ 39. a = -6 a and b = -3 or a = 6 and b = 3 41. about 4.2 ft by 8.4 ft 43. A 45. sometimes 47a. a > 0 b. a = 0 c. a < 0 49. no; x = 1 ± √__ , irrational 51. yes; x = ± __12 , 2 8 rational 55. H 57. ± __14 59. ± __ 11 61. 13 63. 2, 4 65. -2, 7 9-8 -4 + √ 184 5c. ________ ≈ -4.39, ________ ≈ 4 4 2.39 Exercises 1. no 3. __12 , 3 5. -4, -10 -6 - √ 24 1 __ 7. - __ , 3 9. ________ ≈ -5.45, 2 2 2 -6 - √ 24 ________ ≈ -0.55 2 -1 - √ 61 -1 + √ 61 ≈ 1.14, ________ ≈ 11. ________ 6 6 √ -1 - 41 ≈ -1.85, -1.47 13. ________ 4 -1 + √ 41 ________ ≈ 1.35 15. 1 17. 0 4 19. 2 21. 0 23. yes 25. -3 27. - __12 29. -3, __32 31. 1, 9 √ 4 - 12 ≈ 0.27, 33. ______ 2 -7 - √ 17 4 + √ 12 ______ ≈ 3.73 35. ________ ≈ 2.78, 4 2 -7 + √ 17 ________ ≈ -0.72 37. 2 4 39. no 41. -5, 3 43. no real solutions 45. 2 solutions; -2, __14 47. 1 solution; __23 49. 2 x-intercepts; 7 __ , -3 51. 1 x-intercept; 5 57. A 2 59a. 1 b. -1 c. -1 d. -1 63. -10, 4 65. (r 3 + t)(s 2 + 5) 67. (n 4 - 2)(n - 6) 69. f (x), g (x) 25 Check It Out! 1a. 36 1b. __ 1c. 16 4 Study Guide: Review Exercises 3. 4 5. -5, -1 7. -6, 5 1. vertex 2. minimum; maximum 3. zero of a function 4. discriminant 5. completing the square 6. Yes; it is in standard form. 7. No; a = 0 8. Yes; it is in standard form. 9. No; a quadratic function does not have a power of x greater than 2. 2a. -9, -1 2b. 4 ± √ 21 3a. - __13 , 2 3b. no real solutions 4. 16.4 ft by 24.4 ft -5 ± 3 √5 13. no real 9. 1, 9 11. ________ 2 solutions 15. 4 ± √ 10 17. 7.2 m; 11.2 m 19. 1 21. -2, 12 10. 25. y Chapter 10 y 10-1 x 11. x x y x y 27. x 13. y x 14. upward 15. downward 16. (-2, -4); minimum: -4 17. -5 and 2 18. -1 and 2 19. x = 6; (6, 4) 20. x = -1; (-1, -18) 21. y x 22. x y y x 24. y -4 -4 -8 x /THER 3LEEPING 3PORTS (OMEWORK Check It Out! 1a. bread 1b. cheese and mayonnaise 2. 2001, 2002, and 2005; about 13,000 3. about 18 °F 4. Prices increased from January through July or August, and then prices decreased through November. 5. 31.25% 6. 6ERAS$AY 7ATER&OUNTAIN (EIGHTM 12. 26. y 23. 4IMES In 2 seconds, the water reaches its maximum height of 20 meters. It takes a total of 4 seconds for the water to reach the ground. 28. g(x), f (x) 29. The graphs have the same width. 30. h (x), f (x), g (x) 31. same width, same axis of symmetry, opens upward, vertex translated 5 units up 32. narrower, same axis of symmetry, opens upward, vertex translated 1 unit down 33. narrower, same axis of symmetry, opens upward, vertex translated 3 units up 34. x = -3 or x = -1 35. x = -3 36. no real solutions 37. x = 1 or x = 5 38. x = 4 39. x = 1 or x = -1 40. no real solutions 41. x = -5 or x = -1 42. x = -7 or x = -2 43. x = -3 or x = 5 44. x = -1 or x = 2 45. x = -5 46. x = 4.5 47. x 2 + 2x = 48; 6 ft 48. x = -8 or x = 8 49. x = -12 or x = 12 50. no real solutions 51. x = 0 52. x = -5 or x = 5 53. x = - __52 or x = __52 54. 4 ft 55. x = -8 or x = 6 56. x = -7 or x = 3 57. x = 1 or x = 5 58. x = 5 ± √ 5 59. 16 ft by 12 ft 60. x = -1 or x = 6 61. x = - __12 or x = 5 62. x = 1 6 ± √ 8 64. 1 65. 0 66. 2 67. 2 63. x = ______ 2 3CHOOL %ATING A circle graph shows parts of a whole. Exercises 1. one part of a whole 3. 82 animals 5. $15 7. Prices at stadium A are greater than prices at stadium B. 9. between weeks 4 and 5 11. 18% 13. purple 15. blue and green 17. 225,000 19. Friday 21. 3.5 times 23. games 3, 4, and 5 25. Stock Y changed the most between April and July of 2004. 27. 8 __13 % 31. double line 33. circle 35a. Greece; about 40% b. United States; about 15% 37. D 41. 19 girls 43. D: {-3, -1, 0, 1, 3}; R: {0, 1, 3}; yes 47. quadratic binomial 10-2 Check It Out! 1. 4EMPERATUREª# 3TEM ,EAVES Key: 1]9 means 19 2. Interval Frequency 4–6 5 7–9 4 10–12 4 13–15 2 -12 Selected Answers SA23 3. 11a. 6ACATION .UMBER Interval Frequency Cumulative Frequency 36–38 4 4 39–41 6 10 42–44 5 15 45–47 1 16 b. 10 n 4a. n n n ,ENGTHDAYS 15a. Interval Frequency 160–169.9 2 170–179.9 4 Interval Frequency Cumulative Frequency 180–189.9 3 28–31 2 2 190–199.9 1 32–35 7 9 200–209.9 2 36–39 5 14 210–219.9 1 40–43 3 17 4b. 9 Exercises 1. stem-and-leaf plot 3. !USTIN 3TEM .EW9ORK "REATHING)NTERVALS 5. &REQUENCY n n n n 4IMEMIN 3UMMER 3TEM 7INTER Key: ]2]1 means 21 Ê Ê 7]2] means 27 SA24 Check It Out! 1. mean: 14 lb; median: 14 lb; modes: 12 lb and 16 lb; range: 4 lb 2. 3; the outlier decreases the mean by 3.7 and increases the range by 18. It has no effect on the median and mode. 3a. mode: 7 3b. Median: 81; the median is greater than either the mean or the mode. 9. 10-3 4. 7. 19. G 21. 8; 8; 41; 66 23. -2.3 25. 0.5 in. 27. books Interval Frequency 2.0–2.4 2 2.5–2.9 7 3.0–3.4 5 3.5–3.9 3 Selected Answers 5a. The data set for 2000; the distance between the points for the least and greatest values is less for 2000 than for 2007. 5b. about $40 million Exercises 3. mean: 31.5; median: 33.5; mode: 44; range: 32 5. mean: 78.25; median: 78; mode: 78; range: 15 7. 13; the outlier decreases the mean by 11.15 and the median by 4. It increases the range by 51 and has no effect on the mode. 9. Median: 83; the median is greater than the mean, and there is no mode. 13. Simon; about 3000 points 15. mean: 2.5; median: 2.5; modes: 2 and 3; range: 3 17. mean: 60; median: 60; mode: 60; range: 5 19. 23; the outlier increases the mean by 3, the median by 2.5, and the range by 15. It has no effect on the mode. 21. Mean: 153; the mean is greater than the median, and there is no mode. 25. Sneaks R Us; the middle half of the data doesn’t vary as much at Sneaks R Us as at Jump N Run. 27. mean: 5.5; median: 5.5; mode: none; range: 9 29. mean: 3.5; median: 3.4; mode: none; range: 5.3 31. mean: 24.4; median: 25; modes: 23 and 25; range: 3 33. mean: 15__16 ; median: 12__12 ; mode: none; range: 35 37. sometimes 39. always 41. Median; the mean is affected by the outlier of 1218, and there is no mode. 43. Median or mode; the store wants their prices to seem low, and the median and mode are both $2.80 less than the mean. 49. Mean: $32,000; median: $25,000; median; the outlier of $78,000 increases the mean significantly. 51. 96 53. increase the mean; decrease the mean 55. G 57. The mean decreases by 6.6 lb. 61. 32; typing speed is 32 words per minute. 63. 2 65. 5 67. length: 5 yd; width: 3 yd 10-4 Check It Out! 1. Possible answer: company D; the fertilizer from company D appears to be more effective than the other fertilizers. 2. Possible answer: taxi drivers; the drivers could justify charging higher rates by using this graph, which seems to show that gas prices have increased dramatically. 3. Possible answer: Smith; Smith might want to show that he or she got many more votes than Atkins or Napier. 4. The sample size is much too small. Exercises 3a. The vertical scale does not start at 0, and the categories on the horizontal scale are not at equal time intervals. b. Possible answer: Tourism is decreasing rapidly. 5. The sample size is too small. 7a. The vertical scale does not start at 0. b. Possible answer: Single men pay significantly more than single women. 10-5 Check It Out! 1. sample space: {1, 2, 3, 4, 5, 6}; outcome shown: 3 7 13 2. certain 3a. __ 3b. __ 4a. 99.8% 20 20 4b. 34,930 Exercises 3. sample space: {blue, red, yellow, green}; outcome shown: red 5. impossible 7. unlikely 9. __35 11a. 30% b. 54 13. sample space: {blue, red, yellow}; outcome shown: blue 15. as likely as not 6 17. likely 19. __ 21a. 5% b. 21 25 27. about 1%; about 57 29. B 31. B 33. as likely as not 35. unlikely 37a. 7 b. 8 39. $14 41. reflected across the x-axis 43. mean: 6; median: 5; mode: 5 10-8 9. Check It Out! 1. 15,600 2a. permutation; 6 2b. combination; 6 3. 362,880 4. 792 Exercises 1. combination 3. 8 5. combination; 6 7. 20,160 9. 35 11. 15,504 13. 441,000 15. combination; 10 17. 120 19. 133,784,560 21. nP r 1 25a. _________ b. about 85,810 h 308,915,776 27. J 29. 260,130 31. 168 33a. 28 b. 28 37. 5x + 5 39. independent: minutes; dependent: volume of water in tub; f (x) = 15x 41. independent: hours; dependent: total fee; f (x) = 300 + 80x 1. outcome 2. interquartile range 3. independent events 4. 2003 5. 14 more boys 6. 3TEM ,EAVES Check It Out! 1a. 50% 1b. 33 __13 % 1 2. 0.4 3. __ 25 Key: 1]2 means 12 7. #OMEDY #AMP 9 23. __15 25. __45 17. __59 19. 70% 21. __ 10 1 __ 27. 4:1 31. 2 35. D 37. B 10-7 Check It Out! 1a. Independent; Exercises 1. dependent 3. independent 5. independent 1 7. __18 9. __16 11. __ 13. __27 16 1 15. dependent 17. __18 19. __ 12 27 1 b. __ 25. dependent 21. __29 23a. __ 64 64 3 b. __16 27. independent 29a. __ 20 9 1 2 ___ __ __ c. 100 d. 15 31. 15 35. D 37. A 39. 72.9% 41. 80% 43. 48% 5 45. 24 47. wider 49. __35 51. __ 26 $AYSAND $AYS 13 1 45. __ 41. (4x - 3)(x - 1) 43. __ 10 20 the result of rolling the number cube the first time does not affect the result of the second roll. 1b. Dependent; choosing the first student leaves fewer students to choose from the second time. 2. __14 8 3. __ 87 n n n n #APACITYGAL 10. mean: 14; median: 12; mode: 12; range: 28 11. Median; the mean is higher than 4 of the 5 prices; the mode is the lowest price. 12. Study Guide: Review 10-6 Exercises 1. complement 3. 25% 9 5. __12 7. __23 9. __ 11. 1:12 13. __14 15. 50% 10 'AS4ANK#APACITIES &REQUENCY 9a. The sectors of the graph do not add to 100%. b. Possible answer: Nearly half of the state’s spending was for welfare. 15. B 19. w ≤ 1500 21. b ≤ 20 23. t ≥ -4 Chapter 11 Gas Tank Capacities Capacity Key: ]12]8 means 128 Ê Ê 4]10] means 104 8. 13. The scale on the vertical axis is too large. This makes the slopes of the segments less steep. 14. Someone might believe that the price has been relatively stable when in fact it has doubled. 15. 99.5% 16. 24,875 17. 250 18. __12 19. __14 20. __58 21. independent 22. independent 23. dependent 2048 256 24. ____ 25. 0 26. ____ 27. 60 9555 2401 28. permutation; 604,800 29. combination; 220 30. combination; 1365 Tally Frequency 10−14 IIII I 6 15−19 IIII IIII 20−24 III 3 25−29 III 3 10 11-1 Check It Out! 1a. 80, -160, 320 1b. 216, 162, 121.5 2. 7.8125 3. $1342.18 Exercises 3. 25, 12.5, 6.25 5. 1,000,000,000 7. 4 9. 162, 243, 364.5 11. 2058; 14,406; 100,842 5 ___ 5 13. __ , 5 , ___ 15. 0.0000000001, or 32 128 512 1 1 × 10 -10 17. 80; 160 19. __13 21. __17 ; __ 49 1 23. 6; -48 25. 4913 27. yes; __3 29. no 31. no 33a. 1.28 cm b. 40.96 cm 35. -2, -8, -32, -128 3 37. 2, 4, 8, 16 39. 12, 3, __34 , __ 16 43a. $3993; $4392.30 b. 1.1 c. $2727.27 45. J 47. x 4, x 5, x 6 49. 1, y, y 2 51. -400 53. the 7th term 55. b > 10 57. c < - __13 61. f (x) = x 2 + 4 Selected Answers SA25 11-2 Check It Out! 1. 3.375 in. 2a. no 2b. yes 3a. y x 3b. y x 4a. y x Exercises 1. exponential growth t 3. y = 300(1.08) ; 441 5. A = 4200(1.007)4t; $4965.43 7. y = t 10(0.84) ; 4.98 mg 9. 5.5 g 11. y = t 1600(1.03) ; 2150 13. A = 30(1.078)t; 47 members 15. A = 7000(1.0075)4t; t $9438.44 17. A = 12,000(1.026) ; t $17,635.66 19. y = 58(0.9) ; $24.97 21. growth; 61% 23. decay; 33 __13 % 25. growth; 10% 27. growth; 25% t 29. y = 58,000,000(1.001) ; t 58,174,174 31. y = 8200(0.98) ; t $7118.63 33. y = 970(1.012) ; 1030 35. B 37. 18 yr 39. A; B 45. D 47. D 49. about 20 yr 51. 100 min, or 1 h 40 min 53. $225,344 55. 16 ft 11-4 Check It Out! 1a. y 4b. y y 1b. 5a. y x x Exercises 1. no 3. no 17. about 2023 19. 289 ft 21. yes 23. no 35. y = 4.8(2)x 41. -0.125 43a. $2000 b. 8% c. $2938.66 45. C 45. C 47. D 49. 3 51. The value of a is the y-intercept. 53. 25 55. 9x 2 quadratic 2. quadratic 3. The oven temperature decreases by 50 °F every 10 min; y = -5x + 375; 75 °F 3. linear 5. exponential 7. Grapes cost $1.79/lb; y = 1.79x ; $10.74 11. linear 13. exponential 15. = 6k ; linear 17. linear x 19. y = 0.2(4) 21. linear 27. C 145 29. C 33. ___ g 35. 5, -5 37. __94 , - __94 11-5 Check It Out! 1a. 40 ft/s 11-3 Check It Out! 1. y = 1200(1.08)t; $1904.25 2a. A = 1200(1.00875)4t; $1379.49 2b. A = 4000(1.0025)12t; t $5083.47 3. y = 48,000(0.97) ; 38,783 3a. SA26 Selected Answers 27. x ≥ - __32 37. 3.61 units 39. Mercury: 4214 m/s; Venus: 10,361 m/s; Earth: 11,200 m/s; Mars: 5016 m/s 45. A 47. C 49. x ≤ -5 OR x ≥ 5 51. x ≤ -4 OR x ≥ __32 53. D: x ≤ 3; R: y ≤ 4 57a. 2 and 4 b. 3, 1 59. y = -__12 x + 2 61. 9x 2 - 6x + 1 63. a 2 - 2ab 2c + b 4c 2 65. 9r 2 - 4s 2 67. A = 42,000(1.0125)4t; $48,751.69 y x Check It Out! 1a. 8 1b. 7 1c. 13 1d. ⎪3 - x⎥ 2a. 8 √ 2 2b. xy √ x y3 6 3b 3a. __23 3b. __ 3c. __ 2c. 4a √ 2 2 z 2 √z x p3 __ ft; 5. 60 √2 q5 17 6 2 9. 4x 2y √ 2y 11. ____ 13. ___ 7. 18 √ 5 7 √x 6 √3 √ √ √ 16 2 5x 2 17. ____ 19. _____ 21. _____ 15. ___ 7 3 9 13 41 mi; 32 mi 25. 20 27. 9 23. 5 √ 29. ⎪x + 1⎥ 31. ⎪x - 3⎥ 33. 20 √ 10 3 √5 √ x √x y 14 5 37. ____ 39. ____ 41. ____ 35. 8rs √ 3 2 3 8s √ 3 47. 15x √ 45. -20 √3 7 43. ____ 7 x 3 √x √ √ 49. x 51. ____ 53. 36 ; 6 55. 50 ; 3 59. √ 2 57. √3 20 ; 2 √ 5 67. C 5 √ 69. C 71. x √ x + 1 73a. ⎪x⎥ b. x 2 c. ⎪x 3⎥ d. x 4 e. ⎪x 5⎥ f. x n; ⎪x n⎥ 75. no 77. exponential 11-7 Check It Out! 1a. - √ 7 1b. 3 √3 1b. 30.98 ft/s 2a. x ≥ __12 2b. x ≥ __53 4a. 1.5625 mg 4b. 0.78125 g Exercises 1. There is no variable under the square-root sign. 3. x ≥ -6 5. x ≥ 0 7. x ≥ -3 15. 49.96 mi/h 17. x ≥ 0 19. x ≤ __32 21. x ≥ - __53 23. x ≥ 40 25. x ≥ 9 √ Exercises 1. exponential 6. after about 13 yr Exercises 1. 3x - 6 3. 7 5. 6 √ 5 x y 84.9 ft 5b. x 2 √ 5 4b. ____ 4c. 4a. ____ 7 5y y 11-6 exponential x x 3b. + 8 √ 1c. 8 √n 5s 2a. 5 √ 6 1d. √2s 2b. 12 √ 3 - 3 √ 2 2c. 5 √ 3y 3. 10 √ b in. + 5 √ Exercises 3. 10 √ 5 5. 3 √7 2 + 6 √5a 9. 13 √3 11. - √5x 7. 5 √6a - 4 √3t 15. 6 √3 17. -3 √ 13. 8 √2t 11 √ √ 19. -4 √n 21. 7 7 23. 12 2 25. 3 √ 7x - 12 √ 3x 27. 3 √ 5j 31. 12 √7 33. 0 35. 7 √3 29. 2 √3m 37. 7 √ 2 41. 2 √ 3 + 5 √ 5+5 √ 43. 8 √ 7x - 70x 45. 35 √ 5k 47. 5 √ 3 + 5 √ 5 51. 9 53. 18 in.; 8 √ 55. 36x 2 57. 16 √3 3 in.; 24 √ 3 in. 59. B 61. A 63. √ x (x + 2) 65. 0 67. (x + 2) √ x-1 1 69. 3 √ x + 1 - x √ x + 2 73. __ 12 75. x ≥ -3 11-8 Check It Out! 1a. 5 √ 2 1b. 63 - 3 √6 2a. 4 √3 1c. 2m √7 2b. 5 √ 2 + 4 √ 15 2c. 7 √ k - 5 √ 7k 3a. 21 + 5 √ 2d. 150 - 20 √5 3 3b. 83 + 18 √ 2 3c. 11 - 6 √2 √ √ 65 21a 3 4a. ____ 4b. _____ 3d. 17 - √ 5 6 8 √ 35 4c. _____ 7 Exercises 1. √ 6 3. 125 5. 3 √ 30a 9. √ 7. 2 √ 6 + √42 35 - √ 21 11. 5 √ 3y + 4 √ 5y 13. 12 + 7 √ 2 17. 81 - 30 √ 15. -5 - 2 √3 2 33. no solution 35. 4 37. 2 39. no solution 41. 48 43. -25 45. 71 47. -8 49. 36 51. -16 53. 8 55. 9 57. 2 59. -5 61. 5 63. 1 65. 1 67. x = 144; 12 in. 69. √ x - 3 = 4; 49 71. x = √ x + 6 ; 3 73. 3 in. by 1 in. 75a. 54.88 joules b. 0 joules 77. 1690 ft 79. x = 25; y = 16 81. sometimes 87. A 89. C 91. A 93. 1, 2 95. 2 97. 0 101. 51.2 mi 103. 10,000 Study Guide: Review 1. square-root function 2. exponential decay 3. common ratio 4. exponential function 5. 81, 243, 729 6. 48, -96, 192 7. 5, 2.5, 1.25 8. -256, -1024, -4096 9. 7,812,500 10. 19,131,876 11. yes 12. no 13. π √6 3x + x 69. ____ s ≈ 1.9 s 67. 3 + 2 √ 4 71. 269.5 ft 2 75. B 77. D + 4 √5 81. -5 - 2 √ 6 79. -4 √3 85. 2 √ 83. 2 - √3 6 + 2 √5 87. translation 4 units down 89. (x - 3)(x + 10) 91. (x + 4)(x - 4) 93. 2(x 2 + 3)(x 2 - 3) 7x 95. 6 √ 10 97. ___ 2 8y 38. y x 39. y x 40. y x x 41. y 14. y x 42. Check It Out! 1a. 36 1b. 3 1c. __13 y y x 43. y x 18 33. x ≥ __34 31. x ≥ __72 32. x ≥ - __ 5 34. x ≥ 1 35. x t 15. y = 9(1.15) ; 24 t 16. y = 24,500(0.96) ; 3182 17. quadratic 18. linear 19. exponential 20. exponential 21. quadratic 22. linear 23. y = 1.5x; 15 h 24. 4.74 cm 25. x ≥ 0 26. x ≥ -4 27. x ≥ 0 28. x ≥ -2 29. x ≥ __43 30. x ≥ -3 11-9 2a. 9 2b. 18 2c. 3 3a. 121 3b. 64 11 3c. 100 4a. 2 4b. __ 5a. no solution 2 5b. no solution 5c. 4 6. 8; 3 cm 7 √ 2x 51. 2 √y 53. 180 in 2 49. _____ 2 55. (6 √ 10 - 2 √ 5 ) cm 2 57. √ 30 61. 3 √ 59. -5 - 2 √3 2 63. 134 √ 3 + 96 65. x - 2 √ xy + y √ 5x 25. ____ 27. 3 √ 10 29. 8 5x x y y √ √ 2 √7 26 33 21. ____ 23. ____ 19. ____ 7 2 18 7 33. 4 √ 5 - 5 √ 2 31. 6d √ 35. 2 √ 3 - 2 √ 5 37. 3 √f + 12 √ 3f 39. 75 + 19 √ 15 41. 10 - √ 2 √ 5 √ 6 3x 43. 67 + 16 √ 3 45. ____ 47. ____ x 2 37. 44. y x x 36. 45. 11 46. n 2 47. x + 3 48. 5 49. 6d 50. y 3 √ x 51. 2 √ 3 √5 t 2 √ 55. __2 52. 4b 2ab 53. ___ 54. __ y Exercises 1. No; it does not contain a variable under the radical sign. 3. -8 5. -144 7. 27 9. 50 11. -2 13. 9 15. 64 17. 16 19. 16 21. __49 23. 100 25. 5 27. 13 29. 6 31. 2 2 10 3 4p √ 2 t √t 2b 3 √2 56. _____ 57. ____ 58. _____ s 7 5 2 x 4 61. 3 √ 7 60. 3 √3 2 + 2 √ 3 59. 9 √ 62. √ 5t 63. 2 √ 2 64. 2 √ 3 + 2 √ 5 Selected Answers SA27 67. √ 65. -2 √ 5x 66. 10 √6 14 √ 68. 3 √ 2 69. 6 7x 70. 150 4 √ 5 71. 4 √ 2 - 4 72. 71 + 16 √ 7 73. ____ 5 2 17. y = __ x 12-1 Check It Out! 1a. No; the product xy is not constant. 1b. Yes; the product xy is constant. 1c. No; the equation cannot be written in the form y = __kx . 2. y = __5x y x x 2 x 200 0 100 200 300 Members 41. C 43. C 47. approximately 888.9 watts 49. D: {-4, -2, 0, 2, 4}; R: {1, 3, 5}; yes 51. -1, 7 53. 2 √ 10 cm 3 x 25. 2a. x = 5; y = 0 2b. x = -4; y = 5 2c. x = -77; y = -15 3a. 27a. D: x > 0; R: natural numbers > 5 b. x 3b. 0ARTS 0RESSUREATM y 4a. D: x > 0; R: natural numbers > 10 x -8 9. 4 11. 16 teeth 13. yes 15. no 12-3 #OPIES Exercises 1. excluded value 3. -3 5. 4 7. x = -5; y = 0 9. x = -9; y = -10 Selected Answers !VERAGEPRICEOFPARTS 29. 7 31. -__12 37. x = -1; y = 0 39. x = 2; y = 5 41. B 43. C 51. D: x > 2 53. D: x > - __15 55. I and III; II and IV 59. J 61a. yes b. D: all real numbers c. R: 0 < y ≤ 1 d. no 3 63. y = ____ + 3 65. -2, 3 x+2 67. 0.46875 cm 69. no 4b. 8 0RICE 0 y x -6 7. y = ___ x x y y Exercises 3. no 5. yes SA28 15. 0 17. 0 19. x = 4; y = 0 21. x = 3; y = 4 23. y 100 4. 3 5. 80.625 lb -8 y Check It Out! 1a. 0 1b. 1 1c. -4 3. D: x > 0; R: y > 0; 2.5 mm x 8 12-2 6OLUMEOFGASMM 13. 300 0 -10 19. 2 21. 12 yd 23. direct; 8 25. neither 27. inverse; 12 10 29. inverse; 15 31. d = __ n ; inverse 2000 33. neither 35. y = ____ x ; D: natural numbers; R: y > 0 Contribution ($) Chapter 12 0 y 2 -4 2 √3 √ √6 10n 3a √ 2 75. ___ 76. _____ 77. ___ 74. _____ 2 3 2n 2 3 79. 64 80. 8 81. 3 82. 25 78. - √ 83. -81 84. 100 85. 3 86. no solution 87. x = 4 88. x = 6 19 89. x = 7 90. x = __ 91. x = 12 2 92. x = 3 93. x = 4 94. x = 5 11. y Check It Out! 1a. -5 1b. 0, -5 m 1c. -3, -4 2a. __ ; m ≠ 0 2b. 6p 3 3n b-5 1 ; n ≠ 2 3a. ____ 3b. ____ 2c. ____ r+5 n-2 b+5 3 3 1 4b. - ____ 4c. _____ 4a. - ____ 4+x x+1 x + 11 5. The barrel cactus with a radius of 3 inches has less of a chance to survive. Its surface-area-to-volume ratio is greater than for a cactus with a radius of 6 inches. 2 Exercises 3. 0, 8 5. __a2 ; a ≠ 0 7. ____ ; y+3 h 1 ____ ____ ; h ≠ -2 11. y ≠ -3 9. h+2 j-5 c+2 2 15. ____ 17. - ____ 13. ____ c-4 8+n j-3 b+1 2 b1 + b2 1 ; x ≠ ±4 65. - ____ x+4 p-7 z-1 2 35. ____ 37. ____ 33. ____ p-5 x-4 z+1 12-6 p+6 3w + 7 1 43. ____ 45. __13 47. _____ 39. - ____ 12 3 b+7 5+x 53a. __6 49. 1 51. - ____ s b. 3 c. 1 57. F 59. sometimes a-3 65. ±14 61. sometimes 63. ____ a+5 67. -2, 0 73. -6 5x 2y 4 Check It Out! 1a. - __94 1b. ____ 6 p 2 - p - 20 n+4 3m - 15 ______ ______ _________ 3b. 3 2. m - 6 3a. 2 n + 2n p + 16p 3x - 15 2w 6 x ______ ____ ________ 4a. 4b. 4c. x5 v 2x 3 x 2 + 5x + 6 5. approximately 0.23 m 2 - 10m 7. 3y -6 9. ________ 11. a 3 + 10a 2 + 2 a + 6b 2r + 28 1 _____ __ 15. 17. ______ 19. b 25a 13. 9 37. b b. ___ 236 5 1b. x 2 + __13 - __ 2a. k + 5 2b. b - 7 2x 2c. s + 6 3a. 2y + 1 3b. a - 2 13 20 4a. 3m - 5 + _____ 4b. y + 6 + ____ m+3 y-3 r-4 10y + 20 - 4 25. _______ 3y + 15 7r 3n 2 - 3n 29. _______ 31. 1 n+8 x2 ___________ 2 3p 8q 2 _____ 4(4x 2 + 8x - 1) 1 1 39. ___ 41. ___ 2m 16x x 1 51. __13 53. __ c. 4 47. H 49. ______ 2 2 3x + 9x z 1 57. m ≤ 9 55. _____ 2a + 2 11. x + 1 13. c + 3 15. x - 2 -1 -1 17. a + 2 + ____ 19. n + 4 + ____ n+4 a+2 -2 21. 4n - 5 + _____ 2n + 1 3a. 15f 2h 2 4d - 3 3b. (x - 6)(x + 2)(x + 5) 4a. _____ 2 a+8 5 5. __ h or 12.5 min 4b. ____ a-2 24 3d 1 2 Exercises 1. __2y 3. ____ 5. ____ x-4 a+1 7. 6x 3y 2z 9. (y + 4)(y - 4)(y + 9) x+3 260 1 __ 13a. ___ 11. ____ r b. 6 2 h 15. a - 1 x+2 9. y = __1x y 35 23. -2x 2 + 6x - 15 + ____ x+3 x -10 27. 4k 2 - 4k + 2 + ____ k+1 2 29. 3t + 4 - __2t 31. -4p + 1 + __ 3 33. 4t + 3 35. x - 3 37. 3a - 1 14 39. 3x + 4 + ____ x-2 -2 41. 3x + 1 + _____ 2x - 1 -216 43. 2t 2- 6t + 25 + _____ 3t + 9 10. -15 11. $13,200 12. -4; x = -4 and y = 0 13. -1; x = -1 and y = 3 14. -3; x = -3 and y = -4 15. __74 ; x = __74 and y = 5 16. 2y x √ 15 65. 3 m 67. ____ 15 73. 2k 2 + 5k + 2 2a. -4 2b. -4 2c. 1, 3 3. 22 __29 min 4a. 5; 7 is extraneous. 4b. 1 and 5; no extraneous solutions 4c. 4; 0 is extraneous. y 1 61. 3x - __ + __ x 63. x + 2 2y 12-7 p Check It Out! 1a. 2 1b. 1 1c. - __76 Check It Out! 1a. 2 1b. 3y y x 12-5 2b. 8. y = - __4x x 5 - 5 √ 2 71. 4(x + 1) 69. 6 √ 2a. 1. rational expression 2. rational function 3. rational equation 4. inverse variation 5. discontinuous function 6. Yes; the product xy is constant. 7. No; the product xy is not constant. 14 Exercises 1. 2x - __12 3. 7b - __ + __b8 3 3 __ 5. 2x + 4 + 7. 2x - 3 9. 2y + 5 3t 61. x ≠ 3; x = 3 and y = 0 59. 8 √ 63. x ≠ 0; x = 0 and y = 3 65. x ≠ 0; x = 0 and y = 0 4b + 12 ________ b 2 + 3b - 4 Study Guide: Review -7 5a. x 2 - 2x - 4 + ____ x-2 3 45. -20 47. 2x - 5 + ____ x+1 51. 0.5m + 1 57. C 59. B 43. 1 45a. 64 cm b. 80 cm 3 ____ a-2 31. 5 33. __32 35. -4, 3 37. 6 h 39. 2; 3 is extraneous. 41. No solution; 4 is extraneous. 240 43. ___ ; t - 2; 40 mi/h t 1 1 45a. __ = __ + __1y b. 40 cm 15 24 c. It will increase to 72 cm. 49. F 53. Eddie: 6 h; Luke 3 h; Ryan: 4 h 1 55. y = -2x and y = __ x + 4 are 2 perpendicular. 59. 5 3 25. m + 1 + _____ m-1 6h 2x - 4 Exercises 1. ___ 3. _____ 5. __a6 3 5jk 2 2a Check It Out! 1a. -2p + 1 - __p3 21. __12 h, or 30 min 23. - __43 ; 1 is extraneous. 25. -2 27. 0 29. __45 -7 5b. 2p 2 - 2p + 6 + ____ p+1 12-4 1 35a. 33. -___ 3 (x + 4)(x + 2) 2 simplified; m ≠ 4 31. __8t ; t ≠ 0 27. 4m 2 - 4m 2b 8x + 20 43. A 41. __________ x - 4y 25. 0 27. - __12 , 4 29. already 1 21. ______ 23. 3x - 15 4(m - 2) 3 x-5 1 b. 14 h 35. ____ 37. ___ 39. ____ 2 7+c 3 )( ( ) az + by +cx _________ 55. ; x ≠ 0, y ≠ 0, and xyz z ≠ 0 61. __1 , 4 63. 2; t ≠ ±2 b2 will be the same: ______ . b2 27. ______ 3 (y - 3 ) 19 -m 2 - 6m 700 ________ 29. ___ 31. 33a. ___ 2 r 21z ; x ≠ y and x ≠ -y 53. __________ x+y x-y b x+2 y+2 (y + 4) 47. 4x 2; 8x 2; 8x 3 49. A 51. D 5 2 21. - _____ 23a. ______ b. They 10 + q b +b 1 17. m 19. 3a + 1 21. 36a(3a + 1) 23. 10xy 3z 25. (y + 5)(y + 2) 17. y x Exercises 1. rational equation 11 3. -24 5. - __83 7. __32 9. __35 11. - __ 5 15 4 __ __ 13. 19 15. 3, - 3 17. -2, 3 19. -1, __32 Selected Answers SA29 18. x x-2 π x 20. D: x > 0; R: y > 0 2b 2 + 8b n 2 - n - 42 y x 2 + 2x - 3 12n 3 1 ________ 40. ____ 41. __ 42. ____ m 43. 4x 2 - 16 3 b-3 (b + 8)(b + 7) 44. ____________ 45. 10a 2b 2 2(b + 4)(2b + 7) b2 + 8 46. 10x (x - 3) 47. _____ 2 2b 8p - 2 3x 2 + 2x - 4 48. _________ 49. _________ 2 2 x -2 ,ENGTHCM p - 4p + 2 n -1 10m h 2 + 5h - 1 40 54. __ 55. 2n2 - 3n - 5 53. _________ 3r h-5 5 57. x + 2 58. 3n + 1 56. x - __2x + __ 2 x SA30 3 4x 2 - 12x 15b 2 - 3c 3 36. ____ 37. ____ 35. _______ 3 2 4d 2 b+2 n 2 + 3n + 2 _______ ________ 39. 38. 7m + 2 5b - 1 -10 51. _____ 52. ______ 50. _____ 2 2 7-b 34 18 12 70. - __34 71. __ 72. - __ 17 + ____ 7 x+2 11 -2 ; x ≠ ±3 2 30. ____ x+3 x+3 3 ____ 31. x - 1 ; x ≠ -5 and x ≠ 1 32. ____ ; x-5 2 2b + 2b 4 __ _______ x ≠ -6 and x ≠ 5 33. 34. y 18 69. -4x 2 + 10x 8 + ____ b-2 1 ; k ≠ 0 and r ≠ 0 28. _____ 2k - 3 3 1 __ ____ ; x ≠ -6 and x ≠ k ≠ 29. 2 19. 36 67. 2n + 7 + ____ 68. 3b 2 + 6b + n-5 21. 0 22. 7 23. 0, 1 24. -1, 5 1 25. 5, -5 26. 4, 7 27. __ ; 3r y 7IDTHCM Selected Answers 59. h + 12 60. 3x + 2 61. m - 6 62. 3m + 4 63. x + 2 64. x + 6 -3 65. p - 2 66. 2x - 1 + ____ x+2 73. - __76 ; 0 is extraneous. 74. - __23 ; 1 is extraneous. 75. - __13 , 1 1 79. -2; 4 is 76. -3 77. ±1 78. - __ 12 extraneous. 80. 4, 5 81. -12, 1 82. -19 83. 0; 2 is extraneous. Glossary/Glosario KEYWORD: MA7 Glossary A ENGLISH absolute value (p. 14) The absolute value of x is the distance from zero to x on a number line, denoted ⎪x⎥. ⎧x if x ≥ 0 ⎩ -x if x < 0 SPANISH valor absoluto El valor absoluto de x es la distancia de cero a x en una recta numérica, y se expresa ⎪x⎥. ⎧x si x ≥ 0 ⎩ -x si x < 0 EXAMPLES ⎪3⎥ = 3 ⎪-3⎥ = 3 ⎪x⎥ = ⎨ ⎪x⎥ = ⎨ absolute-value equation (p. 112) An equation that contains absolute-value expressions. ecuación de valor absoluto Ecuación que contiene expresiones de valor absoluto. ⎪x + 4⎥ = 7 absolute-value function (p. 378) A function whose rule contains absolute-value expressions. función de valor absoluto Función cuya regla contiene expresiones de valor absoluto. y = ⎪x + 4⎥ absolute-value inequality (p. 212) An inequality that contains absolute-value expressions. desigualdad de valor absoluto Desigualdad que contiene expresiones de valor absoluto. ⎪x + 4⎥ > 7 acute angle An angle that measures greater than 0° and less than 90°. ángulo agudo Ángulo que mide más de 0° y menos de 90°. acute triangle A triangle with three acute angles. triángulo acutángulo Triángulo con tres ángulos agudos. Addition Property of Equality (p. 79) For real numbers a, b, and c, if a = b, then a + c = b + c. Propiedad de igualdad de la suma Dados los números reales a, b y c, si a = b, entonces a + c = b + c. x-6= 8 +6 +6 −−−− −−− x = 14 Addition Property of Inequality (p. 176) For real numbers a, b, and c, if a < b, then a + c < b + c. Also holds true for >, ≤, ≥, and ≠. Propiedad de desigualdad de la suma Dados los números reales a, b y c, si a < b, entonces a + c < b + c. Es válido también para >, ≤, ≥ y ≠. x-6< 8 +6 +6 −−−− −−− x < 14 additive inverse (p. 15) The opposite of a number. Two numbers are additive inverses if their sum is zero. inverso aditivo El opuesto de un número. Dos números son inversos aditivos si su suma es cero. The additive inverse of 5 is -5. algebraic expression (p. 6) An expression that contains at least one variable. expresión algebraica Expresión que contiene por lo menos una variable. algebraic order of operations See order of operations. orden algebraico de las operaciones Ver orden de las operaciones. The additive inverse of -5 is 5. 2x + 3y 4x Glossary/Glosario G1 ENGLISH SPANISH AND (p. 204) A logical operator representing the intersection of two sets. Y Operador lógico que representa la intersección de dos conjuntos. angle A figure formed by two rays with a common endpoint. ángulo Figura formada por dos rayos con un extremo común. EXAMPLES A = {2, 3, 4, 5} B = {1, 3, 5, 7} The set of values that are in A AND B is A B = {3, 5}. area The number of nonoverlapping unit squares of a given size that will exactly cover the interior of a plane figure. área Cantidad de cuadrados unitarios de un determinado tamaño no superpuestos que cubren exactamente el interior de una figura plana. arithmetic sequence (p. 276) A sequence whose successive terms differ by the same nonzero number d, called the common difference. sucesión aritmética Sucesión cuyos términos sucesivos difieren en el mismo número distinto de cero d, denominado diferencia común. Associative Property of Addition (p. 46) For all numbers a, b, and c, (a + b) + c = a + (b + c). Propiedad asociativa de la suma Dados tres números cualesquiera a, b y c, (a + b) + c = a + (b + c). Associative Property of Multiplication (p. 46) For all numbers a, b, and c, (a · b) · c = a · (b · c). Propiedad asociativa de la multiplicación Dados tres números cualesquiera a, b y c, (a · b) · c = a · (b · c). asymptote (p. 878) A line that a graph gets closer to as the value of a variable becomes extremely large or small. asíntota Línea recta a la cual se aproxima una gráfica a medida que el valor de una variable se hace sumamente grande o pequeño. x Ó The area is 10 square units. 4, 7, 10, +3+3 +3 +3 d=3 (5 + 3) + 7 = 5 + (3 + 7) (5 · 3) · 7 = 5 · (3 · 7) Þ { asymptote Ý ä { average See mean. promedio Ver media. axis of a coordinate plane (p. 54) One of two perpendicular number lines, called the x-axis and the y-axis, used to define the location of a point in a coordinate plane. eje de un plano cartesiano Una de las dos rectas numéricas perpendiculares, denominadas eje x y eje y, utilizadas para definir la ubicación de un punto en un plano cartesiano. axis of symmetry (p. 378, p. 620) A line that divides a plane figure or a graph into two congruent reflected halves. eje de simetría Línea que divide una figura plana o una gráfica en dos mitades reflejadas congruentes. 13, 16, … { y-axis 0 x-axis ÝÃÊvÊÃÞiÌÀÞ Þ { ÞÊNÝN Ó Ý { Ó ä Ó { G2 Glossary/Glosario Ó { B ENGLISH back-to-back stem-and-leaf plot (p. 709) A graph used to organize and compare two sets of data so that the frequencies can be compared. See also stem-andleaf plot. SPANISH diagrama doble de tallo y hojas Gráfica utilizada para organizar y comparar dos conjuntos de datos para poder comparar las frecuencias. Ver también diagrama de tallo y hojas. EXAMPLES Data set A: 9, 12, 14, 16, 23, 27 Data set B: 6, 8, 10, 13, 15, 16, 21 Set A Set B 9 0 68 642 1 0356 73 2 1 Key: ⎪2⎥ 1 means 21 7 ⎪2⎥ means 27 gráfica de barras Gráfica con barras horizontales o verticales para mostrar datos. -Õ} ̽ÃÊ/À>ÛiÊ/i ÌÊ*>iÌà {nää xäää ÓÈää À À ÌÕ -> Ìi « >À à ÇÈä Õ > ÀÌ {äää Îäää Óäää £äää xää /iÊî bar graph (p. 700) A graph that uses vertical or horizontal bars to display data. *>iÌ 3 4 = 3 · 3 · 3 · 3 = 81 3 is the base. base of a power (p. 26) The number in a power that is used as a factor. base de una potencia Número de una potencia que se utiliza como factor. base of an exponential function (p. 796) The value of b in a function of the form f (x) = ab x, where a and b are real numbers with a ≠ 0, b > 0, and b ≠ 1. base de una función exponencial Valor de b en una función del tipo f(x) = ab x, donde a y b son números reales con a ≠ 0, b > 0 y b ≠ 1. biased sample (p. 733) A sample that does not fairly represent the population. muestra no representativa Muestra que no representa adecuadamente una población. To find out about the exercise habits of average Americans, a fitness magazine surveyed its readers about how often they exercise. The population is all Americans and the sample is readers of the fitness magazine. This sample will likely be biased because readers of fitness magazines may exercise more often than other people do. binomial (p. 497) A polynomial with two terms. binomio Polinomio con dos términos. x+y 2a 2 + 3 4m 3n 2 + 6mn 4 boundary line (p. 428) A line that divides a coordinate plane into two half-planes. línea de límite Línea que divide un plano cartesiano en dos semiplanos. In the function f (x) = 5(2) , the base is 2. x Î Þ Õ`>ÀÞÊi Ý Î ä Î Î Glossary/Glosario G3 ENGLISH box-and-whisker plot (p. 718) A method of showing how data are distributed by using the median, quartiles, and minimum and maximum values; also called a box plot. SPANISH gráfica de mediana y rango Método para mostrar la distribución de datos utilizando la mediana, los cuartiles y los valores mínimo y máximo; también llamado gráfica de caja. EXAMPLES &IRSTQUARTILE -INIMUM ä Ó { 4HIRDQUARTILE -EDIAN È n £ä £Ó -AXIMUM £{ C Cartesian coordinate system See coordinate plane. sistema de coordenadas cartesianas Ver plano cartesiano. center of a circle The point inside a circle that is the same distance from every point on the circle. centro de un círculo Punto dentro de un círculo que se encuentra a la misma distancia de todos los puntos del círculo. central angle of a circle An angle whose vertex is the center of a circle. ángulo central de un círculo Ángulo cuyo vértice es el centro de un círculo. circle The set of points in a plane that are a fixed distance from a given point called the center of the circle. círculo Conjunto de puntos en un plano que se encuentran a una distancia fija de un punto determinado denominado centro del círculo. circle graph (p. 702) A way to display data by using a circle divided into non-overlapping sectors. gráfica circular Forma de mostrar datos mediante un círculo dividido en sectores no superpuestos. ,iÃ`iÌÃÊvÊiÃ>]Ê< Èx³ £Î¯ {xqÈ{ Óǯ 1`iÀÊ £n £¯ ££¯ Îä¯ £nqÓ{ Óxq{{ circumference The distance around a circle. circunferencia Distancia alrededor de un círculo. ÀVÕviÀiVi G4 coefficient (p. 48) A number that is multiplied by a variable. coeficiente Número que se multiplica por una variable. In the expression 2x + 3y, 2 is the coefficient of x and 3 is the coefficient of y. combination (p. 761) A selection of a group of objects in which order is not important. The number of combinations of r objects chosen from a group of n objects is denoted nCr. combinación Selección de un grupo de objetos en la cual el orden no es importante. El número de combinaciones de r objetos elegidos de un grupo de n objetos se expresa así: nCr. For objects A, B, C, and D, there are 6 different combinations of 2 objects. AB, AC, AD, BC, BD, CD commission (p. 139) Money paid to a person or company for making a sale, usually a percent of the sale amount. comisión Dinero que se paga a una persona o empresa por realizar una venta; generalmente se trata de un porcentaje del total de la venta. Glossary/Glosario ENGLISH SPANISH EXAMPLES common difference (p. 276) In an arithmetic sequence, the nonzero constant difference of any term and the previous term. diferencia común En una sucesión aritmética, diferencia constante distinta de cero entre cualquier término y el término anterior. In the arithmetic sequence 3, 5, 7, 9, 11, …, the common difference is 2. common factor (p. 545) A factor that is common to all terms of an expression or to two or more expressions. factor común Factor que es común a todos los términos de una expresión o a dos o más expresiones. Expression: 4x 2 + 16x 3 - 8x Common factor: 4x Expressions: 12 and 18 Common factors: 2, 3, and 6 common ratio (p. 790) In a geometric sequence, the constant ratio of any term and the previous term. razón común En una sucesión geométrica, la razón constante entre cualquier término y el término anterior. Commutative Property of Addition (p. 46) For any two numbers a and b, a + b = b + a. Propiedad conmutativa de la suma Dados dos números cualesquiera a y b, a + b = b + a. Commutative Property of Multiplication (p. 46) For any two numbers a and b, a · b = b · a. Propiedad conmutativa de la multiplicación Dados dos números cualesquiera a y b, a · b = b · a. complement of an event (p. 745) The set of all outcomes that are not the event. complemento de un suceso Todos los resultados que no están en el suceso. complementary angles Two angles whose measures have a sum of 90°. ángulos complementarios Dos ángulos cuyas medidas suman 90°. In the geometric sequence 32, 16, 8, 4, 2, . . ., the 1 common ratio is __ . 2 3+4=4+3=7 3 · 4 = 4 · 3 = 12 In the experiment of rolling a number cube, the complement of rolling a 3 is rolling a 1, 2, 4, 5, or 6. ÎÇ xΠcompleting the square (p. 663) A process used to form a perfectsquare trinomial. To complete () 2 the square of x 2 + bx, add __b2 . completar el cuadrado Proceso utilizado para formar un trinomio cuadrado perfecto. Para completar el cuadrado de x 2 + bx, hay que () 2 sumar __b2 . complex fraction (p. 904) A fraction that contains one or more fractions in the numerator, the denominator, or both. fracción compleja Fracción que contiene una o más fracciones en el numerador, en el denominador, o en ambos. composite figure (p. 83) A plane figure made up of triangles, rectangles, trapezoids, circles, and other simple shapes, or a three-dimensional figure made up of prisms, cones, pyramids, cylinders, and other simple threedimensional figures. figura compuesta Figura plana compuesta por triángulos, rectángulos, trapecios, círculos y otras figuras simples, o figura tridimensional compuesta por prismas, conos, pirámides, cilindros y otras figuras tridimensionales simples. x 2 + 6x + ( ) = 9. 6 Add _ 2 2 x 2 + 6x + 9 1 _ 2 _ 2 1+_ 3 Glossary/Glosario G5 ENGLISH SPANISH compound event (p. 761) An event made up of two or more simple events. suceso compuesto Suceso formado por dos o más sucesos simples. compound inequality (p. 204) Two inequalities that are combined into one statement by the word and or or. desigualdad compuesta Dos desigualdades unidas en un enunciado por la palabra y u o. EXAMPLES In the experiment of tossing a coin and rolling a number cube, the event of the coin landing heads and the number cube landing on 3. x ≥ 2 AND x < 7 (also written 2 ≤ x < 7) ä Ó { È n x < 2 OR x > 6 ä compound interest (p. 860) Interest earned or paid on both the principal and previously earned interest. The formula for compound interest is A = P(1 + ) r nt __ n , where A is the final amount, P is the principal, r is the interest rate expressed as a decimal, n is the number of times interest is compounded, and t is the time. G6 interés compuesto Intereses ganados o pagados sobre el capital y los intereses ya devengados. La fórmula de interés compuesto es r A = P(1 + __ n ) , donde A es la nt cantidad final, P es el capital, r es la tasa de interés expresada como un decimal, n es la cantidad de veces que se capitaliza el interés y t es el tiempo. compound statement (p. 203) Two statements that are connected by the word and or or. enunciado compuesto Dos enunciados unidos por la palabra y u o. cone (p. 894) A three-dimensional figure with a circular base and a curved surface that connects the base to a point called the vertex. cono Figura tridimensional con una base circular y una superficie lateral curva que conecta la base con un punto denominado vértice. congruent Having the same size and shape, denoted by . congruente Que tiene el mismo tamaño y la misma forma, expresado por . conjugate of an irrational number (p. 845) The conjugate of is a number in the form a + √b √ a - b. conjugado de un número irracional El conjugado de un número en la . forma a + √ b es a - √b consistent system (p. 420) A system of equations or inequalities that has at least one solution. sistema consistente Sistema de ecuaciones o desigualdades que tiene por lo menos una solución. constant (p. 6) A value that does not change. constante Valor que no cambia. Glossary/Glosario Ó { È n If $100 is put into an account with an interest rate of 5% compounded monthly, then after 2 years, the account will have ( 100 1 + 0.05 12·2 ____ ) = $110.49. 12 The sky is blue and the grass is green. I will drive to school or I will take the bus. + , * −− −− PQ RS - The conjugate of 1 + √ 2 is 1 - √ 2. ⎧x + y = 6 ⎨ ⎩x - y = 4 solution: (5, 1) 3, 0, π SPANISH EXAMPLES constant of variation (p. 336) The constant k in direct and inverse variation equations. constante de variación La constante k en ecuaciones de variación directa e inversa. continuous graph (p. 235) A graph made up of connected lines or curves. gráfica continua Gráfica compuesta por líneas rectas o curvas conectadas. y = 5x constant of variation }iµÕi½ÃÊi>ÀÌÊ,>Ìi Þ i>ÀÌÊÀ>Ìi ENGLISH Ý /i convenience sample (p. 733) A sample based on members of the population that are readily available. muestra de conveniencia Una muestra basada en miembros de la población que están fácilmente disponibles. conversion factor (p. 121) The ratio of two equal quantities, each measured in different units. factor de conversión Razón entre dos cantidades iguales, cada una medida en unidades diferentes. coordinate (p. 54) A number used to identify the location of a point. On a number line, one coordinate is used. On a coordinate plane, two coordinates are used, called the x-coordinate and the y-coordinate. coordenada Número utilizado para identificar la ubicación de un punto. En una recta numérica se utiliza una coordenada. En un plano cartesiano se utilizan dos coordenadas, denominadas coordenada x y coordenada y. A reporter surveys people he personally knows. 12 inches _ 1 foot A { Î Ó £ ä £ Ó Î { x È The coordinate of A is 2. { Þ Ó Ý { ä Ó Ó { The coordinates of B are (-2, 3). coordinate plane (p. 54) A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis. plano cartesiano Plano dividido en cuatro regiones por una línea horizontal denominada eje x y una línea vertical denominada eje y. correlation (p. 266) A measure of the strength and direction of the relationship between two variables or data sets. correlación Medida de la fuerza y dirección de la relación entre dos variables o conjuntos de datos. Þ>Ýà ä Ý>Ýà *ÃÌÛiÊVÀÀi>Ì Þ ÊVÀÀi>Ì Þ Ý i}>ÌÛiÊVÀÀi>Ì Þ Ý Ý corresponding angles of polygons (p. 127) Angles in the same relative position in polygons with an equal number of angles. ángulos correspondientes de los polígonos Ángulos que se ubican en la misma posición relativa en polígonos que tienen el mismo número de ángulos. ∠A and ∠D are corresponding angles. Glossary/Glosario G7 ENGLISH corresponding sides of polygons (p. 127) Sides in the same relative position in polygons with an equal number of sides. cosine (p. 928) In a right triangle, the cosine of angle A is the ratio of the length of the leg adjacent to angle A to the length of the hypotenuse. SPANISH lados correspondientes de los polígonos Lados que se ubican en la misma posición relativa en polígonos que tienen el mismo número de lados. EXAMPLES −− −− AB and DE are corresponding sides. coseno En un triángulo rectángulo, el coseno del ángulo A es la razón entre la longitud del cateto adyacente al ángulo A y la longitud de la hipotenusa. Þ«ÌiÕÃi >`>ViÌ adjacent cos A = _________ hypotenuse _1 = _3 cross products (p. 121) In the c statement __ab = __ , bc and ad are the d cross products. productos cruzados En el c enunciado __ab = __ , bc y ad son d productos cruzados. Cross Product Property (p. 121) For any real numbers a, b, c, and c d, where b ≠ 0 and d ≠ 0, if __ab = __ , d then ad = bc. Propiedad de productos cruzados Dados los números reales a, b, c c y d, donde b ≠ 0 y d ≠ 0, si __ab = __ , d entonces ad = bc. cube A prism with six square faces. cubo Prisma con seis caras cuadradas. cube in numeration (p. 26) The third power of a number. cubo en numeración Tercera potencia de un número. cube root (p. 32) A number, 3 written as √ x , whose cube is x. raíz cúbica Número, expresado 3 como √ x , cuyo cubo es x. cubic equation (p. 681) An equation that can be written in the form a x 3 + bx 2 + cx + d = 0, where a, b, c, and d are real numbers and a ≠ 0. ecuación cúbica Ecuación que se puede expresar como a x 3 + bx 2 + c x + d = 0, donde a, b, c, y d son números reales y a ≠ 0. cubic function (p. 680) A function that can be written in the form f (x) = a x 3 + bx 2 + c x + d, where a, b, c, and d are real numbers and a ≠ 0. función cúbica Función que se puede expresar como f (x) = a x 3 + b x 2 + c x + d, donde a, b, c, y d son números reales y a ≠ 0. cubic polynomial (p. 497) A polynomial of degree 3. polinomio cúbico Polinomio de grado 3. x3 + 4 x2 - 6x + 2 cumulative frequency (p. 711) The frequency of all data values that are less than or equal to a given value. frecuencia acumulativa Frecuencia de todos los valores de los datos que son menores que o iguales a un valor dado. For the data set 2, 2, 3, 5, 5, 6, 7, 7, 8, 8, 8, 9, the cumulative frequency table is shown below. 2 6 Cross products: 2 · 3 = 6 and 1 · 6 = 6 _ 10 , then 4x = 60, If 4 = _ x 6 so x = 15. 8 is the cube of 2. 3 √ 64 = 4, because 4 3 = 64; 4 is the cube root of 64. 4 x 3 + x 2 - 3x - 1 = 0 f ( x) = x 3 + 2 x 2 - 6 x + 8 Data 2 3 5 6 7 8 9 G8 Glossary/Glosario Frequency 2 1 2 1 2 3 1 Cumulative Frequency 2 3 5 6 8 11 12 ENGLISH cylinder (p. 894) A threedimensional figure with two parallel congruent circular bases and a curved surface that connects the bases. SPANISH EXAMPLES cilindro Figura tridimensional con dos bases circulares congruentes paralelas y una superficie lateral curva que conecta las bases. D data Information gathered from a survey or experiment. datos Información reunida en una encuesta o experimento. degree measure of an angle A unit of angle measure; one 1 degree is ___ of a circle. 360 medida en grados de un ángulo Unidad de medida de los ángulos; 1 un grado es ___ de un círculo. 360 degree of a monomial (p. 496) The sum of the exponents of the variables in the monomial. grado de un monomio Suma de los exponentes de las variables del monomio. degree of a polynomial (p. 496) The degree of the term of the polynomial with the greatest degree. grado de un polinomio Grado del término del polinomio con el grado máximo. dependent events (p. 750) Events for which the occurrence or nonoccurrence of one event affects the probability of the other event. sucesos dependientes Dos sucesos son dependientes si el hecho de que uno de ellos ocurra o no afecta la probabilidad del otro suceso. dependent system (p. 421) A system of equations that has infinitely many solutions. sistema dependiente Sistema de ecuaciones que tiene infinitamente muchas soluciones. dependent variable (p. 250) The output of a function; a variable whose value depends on the value of the input, or independent variable. variable dependiente Salida de una función; variable cuyo valor depende del valor de la entrada, o variable independiente. diameter A segment that has endpoints on the circle and that passes through the center of the circle; also the length of that segment. diámetro Segmento que atraviesa el centro de un círculo y cuyos extremos están sobre la circunferencia; longitud de dicho segmento. difference of two cubes (p. 584) A polynomial of the form a 3 - b 3, which may be written as the product (a - b)(a 2 + ab + b 2). diferencia de dos cubos Polinomio del tipo a 3 - b 3, que se puede expresar como el producto (a - b)(a 2 + ab + b 2). x 3 - 8 = (x - 2)(x 2 + 2x + 4) difference of two squares (p. 523) A polynomial of the form a 2 - b 2, which may be written as the product (a + b)(a - b). diferencia de dos cuadrados Polinomio del tipo a 2 - b 2, que se puede expresar como el producto (a + b)(a - b). x 2 - 4 = (x + 2)(x - 2) 4x 2y 5z 3 Degree: 2 + 5 + 3 = 10 5 = 5x 0 Degree: 0 3x 2y 2 + 4xy 5 - 12x 3y 2 Degree 4 Degree 6 Degree 5 Degree 6 From a bag containing 3 red marbles and 2 blue marbles, draw a red marble, and then draw a blue marble without replacing the first marble. ⎧x + y = 2 ⎨ ⎩ 2x + 2y = 4 For y = 2x + 1, y is the dependent variable. input: x output: y Glossary/Glosario G9 ENGLISH SPANISH EXAMPLES dimensional analysis (p. 121) A process that uses rates to convert measurements from one unit to another. análisis dimensional Un proceso que utiliza tasas para convertir medidas de unidad a otra. direct variation (p. 336) A linear relationship between two variables, x and y, that can be written in the form y = kx, where k is a nonzero constant. variación directa Relación lineal entre dos variables, x e y, que puede expresarse en la forma y = kx, donde k es una constante distinta de cero. 1 qt _ = 6 qt 12 pt · 2 pt { Þ Ó Ý { Ó Ó { { y = 2x discontinuous function (p. 878) A function whose graph has one or more jumps, breaks, or holes. función discontinua Función cuya gráfica tiene uno o más saltos, interrupciones u hoyos. Þ { Ý ä { descuento Cantidad por la que se reduce un precio original. discrete graph (p. 235) A graph made up of unconnected points. gráfica discreta Gráfica compuesta de puntos no conectados. Theme Park Attendance People discount (p. 145) An amount by which an original price is reduced. { Years discriminant (p. 672) The discriminant of the quadratic equation ax 2 + bx + c = 0 is b 2 - 4ac. discriminante El discriminante de la ecuación cuadrática ax 2 + bx + c = 0 es b 2 - 4ac. Distance Formula (p. 331) In a coordinate plane, the distance from (x 1, y 1) to (x 2, y 2) is Fórmula de distancia En un plano cartesiano, la distancia desde (x 1, y 1) hasta (x 2, y 2) es d = √ (x 2 - x 1)2 + (y 2 - y 1) 2 . d = √ (x 2 - x 1)2 + (y 2 - y 1) 2 . The discriminant of 2x 2 - 5x - 3 = 0 is (-5) 2 - 4(2)(-3) or 49. Þ Ó]Êx® { £]Ê£® { Ó Ó Ý ä Ó { The distance from (2, 5) to (-1, 1) is (-1 - 2)2 + (1 - 5)2 d = √ = (-3) 2 + (-4) 2 √ = √ 9 + 16 = √ 25 = 5. Distributive Property (p. 47) For all real numbers a, b, and c, a(b + c) = ab + ac, and (b + c)a = ba + ca. Propiedad distributiva Dados los números reales a, b y c, a(b + c) = ab + ac, y (b + c)a = ba + ca. Division Property of Equality (p. 86) For real numbers a, b, and c, where c ≠ 0, if a = b, then __ac = __bc . Propiedad de igualdad de la división Dados los números reales a, b y c, donde c ≠ 0, si a = b, entonces __ac = __bc . G10 Glossary/Glosario 3(4 + 5) = 3 · 4 + 3 · 5 (4 + 5)3 = 4 · 3 + 5 · 3 4x = 12 4x = _ 12 _ 4 4 x=3 ENGLISH SPANISH Division Property of Inequality (p. 182, p. 183) If both sides of an inequality are divided by the same positive quantity, the new inequality will have the same solution set. If both sides of an inequality are divided by the same negative quantity, the new inequality will have the same solution set if the inequality symbol is reversed. Propiedad de desigualdad de la división Cuando ambos lados de una desigualdad se dividen entre el mismo número positivo, la nueva desigualdad tiene el mismo conjunto solución. Cuando ambos lados de una desigualdad se dividen entra el mismo número negativo, la nueva desigualdad tiene el mismo conjunto solución si se invierte el símbolo de desigualdad. domain (p. 240) The set of all first coordinates (or x-values) of a relation or function. dominio Conjunto de todos los valores de la primera coordenada (o valores de x ) de una función o relación. EXAMPLES 4x ≥ 12 12 4x ≥ _ _ 4 4 x≥3 -4x ≥ 12 -4x ≤ 12 -4 -4 x ≤ -3 _ _ The domain of the function {(-5, 3), (-3, -2), (-1, -1), (1, 0)} is {-5, -3, -1, 1}. E element Each member in a set or matrix. See also entry. elemento Cada miembro en un conjunto o matriz. Ver también entrada. elimination method (p. 411) A method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations of the system. eliminación Método utilizado para resolver sistemas de ecuaciones por el cual se elimina una variable sumando o restando dos ecuaciones del sistema. empty set (p. 102) A set with no elements. conjunto vacío Conjunto sin elementos. The solution set of ⎪x⎥ < 0 is the empty set, { }, or . entry (p. 770) Each value in a matrix; also called an element. entrada Cada valor de una matriz, también denominado elemento. 3 is the entry in the first row and second column of ⎡2 3⎤ A=⎢ , denoted a 12. ⎣0 1⎦ equally likely outcomes (p. 744) Outcomes are equally likely if they have the same probability of occurring. If an experiment has n equally likely outcomes, then the 1 probability of each outcome is __ n. resultados igualmente probables Los resultados son igualmente probables si tienen la misma probabilidad de ocurrir. Si un experimento tiene n resultados igualmente probables, entonces la probabilidad de cada 1 resultado es __ n. If a fair coin is tossed, then P(heads) = P(tails) = 1 . 2 So the outcome “heads” and the outcome “tails” are equally likely. equation (p. 77) A mathematical statement that two expressions are equivalent. ecuación Enunciado matemático que indica que dos expresiones son equivalentes. x+4=7 2+3=6-1 (x - 1)2 + (y + 2)2 = 4 equilateral triangle A triangle with three congruent sides. triángulo equilátero Triángulo con tres lados congruentes. equivalent ratios (p. 120) Ratios that name the same comparison. razones equivalentes Razones que expresan la misma comparación. _ _1 and _2 are equivalent ratios. 2 4 Glossary/Glosario G11 ENGLISH SPANISH EXAMPLES evaluate (p. 7) To find the value of an algebraic expression by substituting a number for each variable and simplifying by using the order of operations. evaluar Calcular el valor de una expresión algebraica sustituyendo cada variable por un número y simplificando mediante el orden de las operaciones. Evaluate 2x + 7 for x = 3. 2x + 7 2(3) + 7 6+7 13 event (p. 737) An outcome or set of outcomes of an experiment. suceso Resultado o conjunto de resultados en un experimento. In the experiment of rolling a number cube, the event “an odd number” consists of the outcomes 1, 3, and 5. excluded values (p. 878) Values of x for which a function or expression is not defined. valores excluidos Valores de x para los cuales no está definida una función o expresión. The excluded values of (x + 2) __ (x - 1)(x + 4) are x = 1 and x = -4, which would make the denominator equal to 0. experiment (p. 737) An operation, process, or activity in which outcomes can be used to estimate probability. experimento Una operación, proceso o actividad en la que se usan los resultados para estimar una probabilidad. experimental probability (p. 738) The ratio of the number of times an event occurs to the number of trials, or times, that an activity is performed. probabilidad experimental Razón entre la cantidad de veces que ocurre un suceso y la cantidad de pruebas, o veces, que se realiza una actividad. exponent (p. 26) The number that indicates how many times the base in a power is used as a factor. exponente Número que indica la cantidad de veces que la base de una potencia se utiliza como factor. exponential decay (p. 807) An exponential function of the form f (x) = ab x in which 0 < b < 1. If r is the rate of decay, then the function can be written t y = a (1 - r) , where a is the initial amount and t is the time. decremento exponencial Función exponencial del tipo f (x) = ab x en la cual 0 < b < 1. Si r es la tasa decremental, entonces la función se puede expresar como t y = a (1 - r) , donde a es la cantidad inicial y t es el tiempo. Tossing a coin 10 times and noting the number of heads Kendra attempted 27 free throws and made 16 of them. The experimental probability that she will make her next free throw is P(free throw) = 16 ≈ 0.59. number made __ =_ 27 number attempted 3 4 = 3 · 3 · 3 · 3 = 81 4 is the exponent. () 1 f (x) = 3 _ 2 x Þ Ý exponential expression An algebraic expression in which the variable is in an exponent with a fixed number as the base. expresión exponencial Expresión algebraica en la que la variable está en un exponente y que tiene un número fijo como base. exponential function (p. 796) A function of the form f (x) = ab x, where a and b are real numbers with a ≠ 0, b > 0, and b ≠ 1. función exponencial Función del tipo f (x) = ab x, donde a y b son números reales con a ≠ 0, b > 0 y b ≠ 1. 2 x+1 f (x) = 3 · 4 x Þ Ý G12 Glossary/Glosario ENGLISH SPANISH EXAMPLES exponential growth (p. 805) An exponential function of the form f (x) = a b x in which b > 1. If r is the rate of growth, then the function can be written t y = a(1 + r) , where a is the initial amount and t is the time. crecimiento exponencial Función exponencial del tipo f (x) = ab x en la que b > 1. Si r es la tasa de crecimiento, entonces la función se t puede expresar como y = a(1 + r) , donde a es la cantidad inicial y t es el tiempo. expression (p. 6) A mathematical phrase that contains operations, numbers, and/or variables. expresión Frase matemática que contiene operaciones, números y/o variables. extraneous solution (p. 848) A solution of a derived equation that is not a solution of the original equation. solución extraña Solución de una ecuación derivada que no es una solución de la ecuación original. To solve √ x = -2, square both sides; x = 4. 4 = -2 is false; so 4 Check √ is an extraneous solution. factor (p. 544) A number or expression that is multiplied by another number or expression to get a product. See also factoring. factor Número o expresión que se multiplica por otro número o expresión para obtener un producto. Ver también factoreo. 12 = 3 · 4 3 and 4 are factors of 12. factorial (p. 762) If n is a positive integer, then n factorial, written n!, is n · (n - 1) · (n - 2) · … · 2 · 1. The factorial of 0 is defined to be 1. factorial Si n es un entero positivo, entonces el factorial de n, expresado como n!, es n · (n - 1) · (n - 2) · … · 2 · 1. Por definición, el factorial de 0 será 1. factoring (p. 544) The process of writing a number or algebraic expression as a product. factorización Proceso por el que se expresa un número o expresión algebraica como un producto. fair (p. 744) When all outcomes of an experiment are equally likely. justo Cuando todos los resultados When tossing a fair coin, heads de un experimento son igualmente and tails are equally likely. Each has a probability of __12 . probables. family of functions (p. 369) A set of functions whose graphs have basic characteristics in common. Functions in the same family are transformations of their parent function. familia de funciones Conjunto de funciones cuyas gráficas tienen características básicas en común. Las funciones de la misma familia son transformaciones de su función madre. f (x) = 2 x Þ Ó Ý Ó ä Ó 6x + 1 F x 2 - 1 = (x - 1)(x + 1) (x - 1) and (x + 1) are factors of x 2 - 1. 7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 x 2 - 4x - 21 = (x - 7)(x + 3) ÊÞÊÊÝÓÊÊ£ n Þ ÊÞÊÊÎÝÓ ÊÞÊÊÝÊÊÓ®Ó È { ÊÞÊÊÝÓ Ó Ý { Ó ä Ó { Glossary/Glosario G13 ENGLISH first differences (p. 610) The differences between y-values of a function for evenly spaced x-values. SPANISH EXAMPLES primeras diferencias Diferencias entre los valores de y de una función para valores de x espaciados uniformemente. Constant change in x-values +1 +1 +1 +1 x 0 1 2 3 4 y = x2 0 1 4 9 16 +1 +3 +5 +7 First differences first quartile (p. 718) The median of the lower half of a data set, denoted Q 1. Also called lower quartile. primer cuartil Mediana de la mitad inferior de un conjunto de datos, expresada como Q 1. También se llama cuartil inferior. FOIL (p. 513) A mnemonic (memory) device for a method of multiplying two binomials: Multiply the First terms. Multiply the Outer terms. Multiply the Inner terms. Multiply the Last terms. FOIL Regla mnemotécnica para recordar el método de multiplicación de dos binomios: Multiplicar los términos Primeros F L (First). Multiplicar los términos Externos (x + 2)(x - 3) = x 2 - 3x + 2x - 6 = x2 - x - 6 (Outer). I Multiplicar los términos Internos O (Inner). Multiplicar los términos Últimos (Last). formula (p. 107) A literal equation that states a rule for a relationship among quantities. fórmula Ecuación literal que establece una regla para una relación entre cantidades. fractional exponent See rational exponent. exponente fraccionario Ver exponente racional. frequency (p. 710) The number of times the value appears in the data set. frecuencia Cantidad de veces que aparece el valor en un conjunto de datos. frequency table (p. 710) A table that lists the number of times, or frequency, that each data value occurs. tabla de frecuencia Tabla que enumera la cantidad de veces que ocurre cada valor de datos, o la frecuencia. function (p. 241) A relation in which every domain value is paired with exactly one range value. función Relación en la que a cada valor de dominio corresponde exactamente un valor de rango. Lower half 18, 23, 28, First quartile A = πr 2 In the data set 5, 6, 6, 7, 8, 9, the data value 6 has a frequency of 2. Data set: 1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 6, 6, 6 Frequency table: Data 1 2 3 4 5 6 Frequency 2 2 1 1 3 4 È x Ó £ G14 Glossary/Glosario Upper half 29, 36, 42 { £ ä ENGLISH SPANISH EXAMPLES function notation (p. 250) If x is the independent variable and y is the dependent variable, then the function notation for y is f (x), read “f of x,” where f names the function. notación de función Si x es la variable independiente e y es la variable dependiente, entonces la notación de función para y es f (x), que se lee “f de x,” donde f nombra la función. function rule (p. 250) An algebraic expression that defines a function. regla de función Expresión algebraica que define una función. Fundamental Counting Principle (p. 760) If one event has m possible outcomes and a second event has n possible outcomes after the first event has occurred, then there are mn total possible outcomes for the two events. Principio fundamental de conteo Si un suceso tiene m resultados posibles y otro suceso tiene n resultados posibles después de ocurrido el primer suceso, entonces hay mn resultados posibles en total para los dos sucesos. equation: y = 2x function notation: f(x) = 2x f (x) = 2x 2 + 3x - 7 function rule If there are 4 colors of shirts, 3 colors of pants, and 2 colors of shoes, then there are 4 · 3 · 2 = 24 possible outfits. G geometric sequence (p. 790) A sequence in which the ratio of successive terms is a constant r, called the common ratio, where r ≠ 0 and r ≠ 1. sucesión geométrica Sucesión en la que la razón de los términos sucesivos es una constante r, denominada razón común, donde r ≠ 0 y r ≠ 1. graph of a function (p. 256) The set of points in a coordinate plane with coordinates (x, y), where x is in the domain of the function f and y = f(x). gráfica de una función Conjunto de los puntos de un plano cartesiano con coordenadas (x, y), donde x está en el dominio de la función f e y = f (x). 1, 2, 4, 8, 16, … ·2 ·2 ·2 ·2 { r=2 Þ Þ Ó Ý { Ó ä Ó Ý n { ä { Ó { { n { n Þ graph of a system of linear inequalities (p. 435) The region in a coordinate plane consisting of points whose coordinates are solutions to all of the inequalities in the system. gráfica de un sistema de desigualdades lineales Región de un plano cartesiano que consta de puntos cuyas coordenadas son soluciones de todas las desigualdades del sistema. graph of an inequality in one variable (p. 171) The set of points on a number line that are solutions of the inequality. gráfica de una desigualdad en una variable Conjunto de los puntos de una recta numérica que representan soluciones de la desigualdad. graph of an inequality in two variables (p. 428) The set of points in a coordinate plane whose coordinates (x, y) are solutions of the inequality. gráfica de una desigualdad en dos variables Conjunto de los puntos de un plano cartesiano cuyas coordenadas (x, y) son soluciones de la desigualdad. Ó]£®ÊÃÊÊÌ iÊ ÛiÀ>««}Ê Ã >`i`ÊÀi}Ã] ÃÊÌÊÃÊ>ÊÃÕÌ° Ó ä Ó Ý Ó Ó x≥2 { Î Ó £ ä £ Ó Î { x È y≤x+1 Î Þ Ý Î ä Î Î Glossary/Glosario G15 ENGLISH SPANISH EXAMPLES graph of an ordered pair (p. 54) For the ordered pair (x, y), the point in a coordinate plane that is a horizontal distance of x units from the origin and a vertical distance of y units from the origin. gráfica de un par ordenado Dado el par ordenado (x, y), punto en un plano cartesiano que está a una distancia horizontal de x unidades desde el origen y a una distancia vertical de y unidades desde el origen. greatest common factor (monomials) (GCF) (p. 575) The product of the greatest integer and the greatest power of each variable that divide evenly into each monomial. máximo común divisor (monomios) (MCD) Producto del entero mayor y la potencia mayor de cada variable que divide exactamente cada monomio. greatest common factor (numbers) (GCF) The largest common factor of two or more given numbers. máximo común divisor (números) (MCD) El mayor de los factores The GCF of 27 and 45 is 9. comunes compartidos por dos o más números dados. grouping symbols (p. 40) Symbols such as parentheses ( ), brackets [ ], and braces { } that separate part of an expression. A fraction bar, absolute-value symbols, and radical symbols may also be used as grouping symbols. símbolos de agrupación Símbolos tales como paréntesis ( ), corchetes [ ] y llaves { } que separan parte de una expresión. La barra de fracciones, los símbolos de valor absoluto y los símbolos de radical también se pueden utilizar como símbolos de agrupación. { Þ Ó Ý { Ó ä Ó { Ó { - S(2, -4) The GCF of 4x 3y and 6x 2y is 2x 2y. ⎧ ⎫ 6 + ⎨3 - ⎡⎣(4 - 3) + 2⎤⎦ + 1⎬ - 5 ⎧⎩ ⎫ ⎩ ⎭ 6 + ⎨3 - ⎡⎣1 + 2⎤⎦ + 1⎬ - 5 6 + {3 - 3 + 1} - 5 6+1-5 2 ⎭ H half-life (p. 807) The half-life of a substance is the time it takes for one-half of the substance to decay into another substance. vida media La vida media de una sustancia es el tiempo que tarda la mitad de la sustancia en desintegrarse y transformarse en otra sustancia. half-plane (p. 428) The part of the coordinate plane on one side of a line, which may include the line. semiplano La parte del plano cartesiano de un lado de una línea, que puede incluir la línea. Carbon-14 has a half-life of 5730 years, so 5 g of an initial amount of 10 g will remain after 5730 years. Î Ý Î ä Î Heron’s Formula (p. 834) A triangle with side lengths a, b, and c has area A = √ s(s - a)(s - b)(s - c) , where s is one-half the perimeter, or s = __12 (a + b + c). G16 Glossary/Glosario fórmula de Herón Un triángulo con longitudes de lado a, b y c tiene un área A = √ s(s - a)(s - b)(s - c) , donde s es la mitad del perímetro ó s = __12 (a + b + c). Þ Î histogram (p. 710) A bar graph used to display data grouped in intervals. SPANISH histograma Gráfica de barras utilizada para mostrar datos agrupados en intervalos de clases. EXAMPLES -Ì>ÀÌ}Ê->>Àià ÀiµÕiVÞ ENGLISH {ä Îä Óä £ä ä n n n n ->>ÀÞÊÀ>}iÊÌ ÕÃ>`Êf® horizontal line (p. 316) A line described by the equation y = b, where b is the y-intercept. línea horizontal Línea descrita por la ecuación y = b, donde b es la intersección con el eje y. y=4 Þ Ó Ý { Ó ä hypotenuse The side opposite the right angle in a right triangle. hipotenusa Lado opuesto al ángulo recto de un triángulo rectángulo. Ó { Þ«ÌiÕÃi I identity (p. 101) An equation that is true for all values of the variables. identidad Ecuación verdadera para todos los valores de las variables. inclusive events (p. 758) Events that have one or more outcomes in common. sucesos inclusivos Sucesos que tienen uno o más resultados en común. inconsistent system (p. 420) A system of equations or inequalities that has no solution. sistema inconsistente Sistema de ecuaciones o desigualdades que no tiene solución. independent events (p. 750) Events for which the occurrence or nonoccurrence of one event does not affect the probability of the other event. sucesos independientes Dos sucesos son independientes si el hecho de que se produzca o no uno de ellos no afecta la probabilidad del otro suceso. independent system (p. 421) A system of equations that has exactly one solution. sistema independiente Sistema de ecuaciones que tiene sólo una solución. independent variable (p. 250) The input of a function; a variable whose value determines the value of the output, or dependent variable. variable independiente Entrada de una función; variable cuyo valor determina el valor de la salida, o variable dependiente. 3=3 2(x - 1) = 2x - 2 In the experiment of rolling a number cube, rolling an even number and rolling a number less than 3 are inclusive events because both contain the outcome 2. ⎧x + y = 0 ⎨ ⎩x + y = 1 From a bag containing 3 red marbles and 2 blue marbles, draw a red marble, replace it, and then draw a blue marble. ⎧x + y = 7 ⎨ ⎩x - y = 1 Solution: (4, 3) For y = 2x + 1, x is the independent variable. Glossary/Glosario G17 ENGLISH SPANISH n √ x, EXAMPLES n √ x, index (p. 488) In the radical which represents the nth root of x, n is the index. In the radical √ x, the index is understood to be 2. índice En el radical que representa la enésima raíz de x, n es el índice. En el radical √x , se da por sentado que el índice es 2. indirect measurement (p. 128) A method of measurement that uses formulas, similar figures, and/or proportions. medición indirecta Método de medición en el que se usan fórmulas, figuras semejantes y/o proporciones. inequality (p. 170) A statement that compares two expressions by using one of the following signs: <, >, ≤, ≥, or ≠. desigualdad Enunciado que compara dos expresiones utilizando uno de los siguientes signos: <, >, ≤, ≥, o ≠. input (p. 55) A value that is substituted for the independent variable in a relation or function. entrada Valor que sustituye a la variable independiente en una relación o función. input-output table A table that displays input values of a function or expression together with the corresponding outputs. tabla de entrada y salida Tabla que muestra los valores de entrada de una función o expresión junto con las correspondientes salidas. integer (p. 34) A member of the set of whole numbers and their opposites. entero Miembro del conjunto de números cabales y sus opuestos. intercept See x-intercept and y-intercept. intersección Ver intersección con el eje x e intersección con el eje y. interest (p. 139) The amount of money charged for borrowing money or the amount of money earned when saving or investing money. See also compound interest, simple interest. interés Cantidad de dinero que se cobra por prestar dinero o cantidad de dinero que se gana cuando se ahorra o invierte dinero. Ver también interés compuesto, interés simple. interquartile range (IQR) (p. 718) The difference of the third (upper) and first (lower) quartiles in a data set, representing the middle half of the data. rango entre cuartiles Diferencia entre el tercer cuartil (superior) y el primer cuartil (inferior) de un conjunto de datos, que representa la mitad central de los datos. intersection (p. 205) The intersection of two sets is the set of all elements that are common to both sets, denoted by . intersección de conjuntos La A = {1, 2, 3, 4} intersección de dos conjuntos es el B = {1, 3, 5, 7, 9} conjunto de todos los elementos A B = {1, 3} que son comunes a ambos conjuntos, expresado por . inverse operations Operations that undo each other. operaciones inversas Operaciones que se anulan entre sí. G18 Glossary/Glosario 3 The radical √ 8 has an index of 3. x≥2 { Î Ó £ ä £ Ó Î { x È For the function f (x) = x + 5, the input 3 produces an output of 8. Input x 1 2 3 Output y 4 7 10 13 4 …, -3, -2, -1, 0, 1, 2, 3, … Lower half Upper half 18, 23, 28, 29, 36, 42 First quartile Third quartile Interquartile range: 36 - 23 = 13 Addition and subtraction of the same quantity are inverse operations: 5 + 3 = 8, 8 - 3 = 5 Multiplication and division by the same quantity are inverse operations: 2 · 3 = 6, 6 ÷ 3 = 2 ENGLISH SPANISH EXAMPLES inverse variation (p. 871) A relationship between two variables, x and y, that can be written in the form y = __kx , where k is a nonzero constant and x ≠ 0. variación inversa Relación entre dos variables, x e y, que puede expresarse en la forma y = __kx , donde k es una constante distinta de cero y x ≠ 0. 8 y=_ x irrational number (p. 34) A real number that cannot be expressed as the ratio of two integers. número irracional Número real que no se puede expresar como una razón de enteros. √ 2 , π, e isolate the variable (p. 77) To isolate a variable in an equation, use inverse operations on both sides until the variable appears by itself on one side of the equation and does not appear on the other side. despejar la variable Para despejar la variable de una ecuación, utiliza operaciones inversas en ambos lados hasta que la variable aparezca sola en uno de los lados de la ecuación y no aparezca en el otro lado. isosceles triangle A triangle with at least two congruent sides. triángulo isósceles Triángulo que tiene al menos dos lados congruentes. 10 = 6 - 2x -6 -6 −− −−−−−− 4= -2x 4 = -2x -2 -2 -2 = x _ _ L leading coefficient (p. 497) The coefficient of the first term of a polynomial in standard form. coeficiente principal Coeficiente del primer término de un polinomio en forma estándar. least common denominator (LCD) (p. 907) The least common multiple of the denominators of two or more given fractions or rational expressions. mínimo común denominador (MCD) Mínimo común múltiplo de los denominadores de dos o más fracciones dadas o expresionnes racionales. least common multiple (monomials) (LCM) (p. 906) The product of the smallest positive number and the lowest power of each variable that divide evenly into each monomial. mínimo común múltiplo (monomios) (MCM) El producto del número positivo más pequeño y la menor The LCM of 6 x 2 and 4 x is 12 x 2. potencia de cada variable que divide exactamente cada monomio. least common multiple (numbers) (LCM) The smallest whole number, other than zero, that is a multiple of two or more given numbers. mínimo común múltiplo (números) (MCM) El menor de los números cabales, distinto de cero, que es múltiplo de dos o más números dados. leg of a right triangle One of the two sides of a right triangle that form the right angle. cateto de un triángulo rectángulo Uno de los dos lados de un triángulo rectángulo que forman el ángulo recto. 3x 2 + 7x - 2 Leading coefficient: 3 5 is 12. 3 and _ The LCD of _ 4 6 The LCM of 10 and 18 is 90. i} i} Glossary/Glosario G19 ENGLISH SPANISH EXAMPLES like radicals (p. 835) Radical terms having the same radicand and index. radicales semejantes Términos radicales que tienen el mismo radicando e índice. 2x and √ 2x 3 √ like terms (p. 47) Terms with the same variables raised to the same exponents. términos semejantes Términos con las mismas variables elevadas a los mismos exponentes. 3a 3b 2 and 7a 3b 2 line A straight path that has no thickness and extends forever. línea Un trazo recto que no tiene grosor y se extiende infinitamente. line graph (p. 701) A graph that uses line segments to show how data changes. gráfica lineal Gráfica que se vale de segmentos de recta para mostrar cambios en los datos. Ű -VÀi >À½ÃÊ6`iÊ>iÊ-VÀià £Óää nää {ää ä £ Ó Î { x >iÊÕLiÀ linear equation in one variable An equation that can be written in the form ax = b where a and b are constants and a ≠ 0. ecuación lineal en una variable Ecuación que puede expresarse en la forma ax = b donde a y b son constantes y a ≠ 0. x+1=7 linear equation in two variables (p. 302) An equation that can be written in the form Ax + By = C where A, B, and C are constants and A and B are not both 0. ecuación lineal en dos variables Ecuación que puede expresarse en la forma Ax + By = C donde A, B y C son constantes y A y B no son ambas 0. 2x + 3y = 6 linear function (p. 300) A function that can be written in the form y = mx + b, where x is the independent variable and m and b are real numbers. Its graph is a line. función lineal Función que puede expresarse en la forma y = mx + b, donde x es la variable independiente y m y b son números reales. Su gráfica es una línea. y=x-1 { Þ Ó Ý { Ó ä Ó { { linear inequality in one variable An inequality that can be written in one of the following forms: ax < b, ax > b, ax ≤ b, ax ≥ b, or ax ≠ b, where a and b are constants and a ≠ 0. desigualdad lineal en una variable Desigualdad que puede expresarse de una de las siguientes formas: ax < b, ax > b, ax ≤ b, ax ≥ b o ax ≠ b, donde a y b son constantes y a ≠ 0. 3x - 5 ≤ 2(x + 4) linear inequality in two variables (p. 428) An inequality that can be written in one of the following forms: Ax + By < C, Ax + By > C, Ax + By ≤ C, Ax + By ≥ C, or Ax + By ≠ C, where A, B, and C are constants and A and B are not both 0. desigualdad lineal en dos variables Desigualdad que puede expresarse de una de las siguientes formas: Ax + By < C, Ax + By > C, Ax + By ≤ C, Ax + By ≥ C o Ax + By ≠ C, donde A, B y C son constantes y A y B no son ambas 0. 2x + 3y > 6 G20 Glossary/Glosario È ENGLISH SPANISH EXAMPLES d = rt 1h b + b A=_ ( 1 2) 2 literal equation (p. 108) An equation that contains two or more variables. ecuación literal Ecuación que contiene dos o más variables. lower quartile See first quartile. cuartil inferior Ver primer cuartil. M mapping diagram (p. 240) A diagram that shows the relationship of elements in the domain to elements in the range of a relation or function. diagrama de correspondencia Diagrama que muestra la relación entre los elementos del dominio y los elementos del rango de una función. markup (p. 145) The amount by which a wholesale cost is increased. margen de ganancia Cantidad que se agrega a un costo mayorista. matrix (p. 770) A rectangular array of numbers. matriz Arreglo rectangular de números. >««}Ê>}À> > Ó ⎢-2 ⎣ 7 máximo de una función Valor de y del punto más alto en la gráfica de la función. mean (p. 716) The sum of all the values in a data set divided by the number of data values. Also called the average. media Suma de todos los valores de un conjunto de datos dividida entre el número de valores de datos. También llamada promedio. measure of an angle Angles are measured in degrees. A degree 1 is ___ of a complete circle. 360 medida de un ángulo Los ángulos se miden en grados. Un grado es 1 ___ de un círculo completo. 360 measure of central tendency (p. 716) A measure that describes the center of a data set. medida de tendencia dominante Medida que describe el centro de un conjunto de datos. median (p. 716) For an ordered data set with an odd number of values, the median is the middle value. For an ordered data set with an even number of values, the median is the average of the two middle values. mediana Dado un conjunto de datos ordenado con un número impar de valores, la mediana es el valor medio. Dado un conjunto de datos con un número par de valores, la mediana es el promedio de los dos valores medios. midpoint (p. 330) The point that divides a segment into two congruent segments. punto medio Punto que divide un segmento en dos segmentos congruentes. ⎡ 1 maximum of a function (p. 380, p. 612) The y-value of the highest point on the graph of the function. ,>}i 0 3⎤ 2 -5 -6 3⎦ ä]ÊÓ® The maximum of the function is 2. Data set: 4, 6, 7, 8, 10 4 + 6 + 7 + 8 + 10 Mean: __ 5 35 = 7 =_ 5 ÓÈ°n mean, median, or mode 8, 9, 9, 12, 15 Median: 9 4, 6, 7, 10, 10, 12 7 + 10 Median: _ = 8.5 2 −− Point B is the midpoint of AC. Glossary/Glosario G21 ENGLISH minimum of a function (p. 380, p. 612) The y-value of the lowest point on the graph of the function. SPANISH EXAMPLES mínimo de una función Valor de y del punto más bajo en la gráfica de la función. ä]ÊÓ® The minimum of the function is -2. mode (p. 716) The value or values that occur most frequently in a data set; if all values occur with the same frequency, the data set is said to have no mode. moda El valor o los valores que se presentan con mayor frecuencia en un conjunto de datos. Si todos los valores se presentan con la misma frecuencia, se dice que el conjunto de datos no tiene moda. monomial (p. 496) A number or a product of numbers and variables with whole-number exponents, or a polynomial with one term. monomio Número o producto de números y variables con exponentes de números cabales, o polinomio con un término. Multiplication Property of Equality (p. 86) If a, b, and c are real numbers and a = b, then ac = bc. Propiedad de igualdad de la multiplicación Si a, b y c son números reales y a = b, entonces ac = bc. Multiplication Property of Inequality (p. 182, p. 183) If both sides of an inequality are multiplied by the same positive quantity, the new inequality will have the same solution set. If both sides of an inequality are multiplied by the same negative quantity, the new inequality will have the same solution set if the inequality symbol is reversed. Propiedad de desigualdad de la multiplicación Si ambos lados de una desigualdad se multiplican por el mismo número positivo, la nueva desigualdad tendrá el mismo conjunto solución. Si ambos lados de una desigualdad se multiplican por el mismo número negativo, la nueva desigualdad tendrá el mismo conjunto solución si se invierte el símbolo de desigualdad. multiplicative inverse (p. 21) The reciprocal of the number. inverso multiplicativo Recíproco de un número. mutually exclusive events (p. 758) Two events are mutually exclusive if they cannot both occur in the same trial of an experiment. sucesos mutuamente excluyentes Dos sucesos son mutuamente excluyentes si ambos no pueden ocurrir en la misma prueba de un experimento. Data set: 3, 6, 8, 8, 10 Mode: 8 Data set: 2, 5, 5, 7, 7 Modes: 5 and 7 Data set: 2, 3, 6, 9, 11 No mode 3x 2 y 4 1x =7 _ 3 1 x = (3)(7) (3) _ 3 x = 21 ( ) 1x >7 _ 3 1 _ (3) x > (3)(7) 3 x > 21 ( ) -x ≤ 2 (-1)(-x) ≥ (-1)(2) x ≥ -2 The multiplicative inverse of 5 is __15 . In the experiment of rolling a number cube, rolling a 3 and rolling an even number are mutually exclusive events. N natural number (p. 34) A counting number. número natural Número que se utiliza para contar. negative correlation (p. 267) Two data sets have a negative correlation if one set of data values increases as the other set decreases. correlación negativa Dos conjuntos de datos tienen una correlación negativa si un conjunto de valores de datos aumenta a medida que el otro conjunto disminuye. G22 Glossary/Glosario 1, 2, 3, 4, 5, 6, … Þ Ý ENGLISH SPANISH negative exponent (p. 460) For any nonzero real number x and 1 any integer n, x -n = __ . xn exponente negativo Para cualquier número real distinto de cero x y 1 cualquier entero n, x -n = __ . xn negative number A number that is less than zero. Negative numbers lie to the left of zero on a number line. número negativo Número menor que cero. Los números negativos se ubican a la izquierda del cero en una recta numérica. negative square root (p. 32) The opposite of the principal square root of a number a, written as - √a . raíz cuadrada negativa Opuesto de la raíz cuadrada principal de un número a, que se expresa como - √a . net (p. 894) A diagram of the faces of a three-dimensional figure arranged in such a way that the diagram can be folded to form the three-dimensional figure. plantilla Diagrama de las caras de una figura tridimensional que se puede plegar para formar la figura tridimensional. no correlation (p. 267) Two data sets have no correlation if there is no relationship between the sets of values. sin correlación Dos conjuntos de datos no tienen correlación si no existe una relación entre los conjuntos de valores. n th root (p. 488) The nth root n of1 a number a, written as √ a or __ n a , is a number that is equal to a when it is raised to the nth power. enésima raíz La enésima raíz de un 1 __ n número a, que se escribe √ a o a n, es un número igual a a cuando se eleva a la enésima potencia. number line (p. 14) A line used to represent the real numbers. recta numérica Línea utilizada para representar los números reales. numerical expression (p. 6) An expression that contains only numbers and/or operations. expresión numérica Expresión que contiene únicamente números y operaciones. EXAMPLES 1 ; 3 -2 = _ 1 x -2 = _ x2 32 -2 is a negative number. { Î Ó £ ä £ Ó Î { The negative square root of 9 is - √ 9 = -3. 10 m 10 m 6m 6m Þ Ý 5 √ 32 = 2, because 2 5 = 32. { Î Ó £ ä £ Ó Î { x È 2 · 3 + (4 - 6) O obtuse angle An angle that measures greater than 90° and less than 180°. ángulo obtuso Ángulo que mide más de 90° y menos de 180°. obtuse triangle A triangle with one obtuse angle. triángulo obtusángulo Triángulo con un ángulo obtuso. odds (p. 746) A comparison of favorable and unfavorable outcomes. The odds in favor of an event are the ratio of the number of favorable outcomes to the number of unfavorable outcomes. The odds against an event are the ratio of the number of unfavorable outcomes to the number of favorable outcomes. probabilidades a favor y en contra Comparación de los resultados favorables y desfavorables. Las probabilidades a favor de un suceso son la razón entre la cantidad de resultados favorables y la cantidad de resultados desfavorables. Las probabilidades en contra de un suceso son la razón entre la cantidad de resultados desfavorables y la cantidad de resultados favorables. The odds in favor of rolling a 3 on a number cube are 1 : 5. The odds against rolling a 3 on a number cube are 5 : 1. Glossary/Glosario G23 ENGLISH SPANISH EXAMPLES opposite (p. 15) The opposite of a number a, denoted -a, is the number that is the same distance from zero as a, on the opposite side of the number line. The sum of opposites is 0. opuesto El opuesto de un número a, expresado -a, es el número que se encuentra a la misma distancia de cero que a, del lado opuesto de la recta numérica. La suma de los opuestos es 0. opposite reciprocal (p. 364) The opposite of the reciprocal of a number. The opposite reciprocal of any nonzero number a is - __a1 . recíproco opuesto Opuesto del recíproco de un número. El recíproco opuesto de a es - __a1 . OR (p. 204) A logical operator representing the union of two sets. O Operador lógico que representa la unión de dos conjuntos. order of operations (p. 40) A process for evaluating expressions: First, perform operations in parentheses or other grouping symbols. Second, simplify powers and roots. Third, perform all multiplication and division from left to right. Fourth, perform all addition and subtraction from left to right. orden de las operaciones Regla para evaluar las expresiones: Primero, realizar las operaciones entre paréntesis u otros símbolos de agrupación. Segundo, simplificar las potencias y las raíces. Tercero, realizar todas las multiplicaciones y divisiones de izquierda a derecha. Cuarto, realizar todas las sumas y restas de izquierda a derecha. ordered pair (p. 54) A pair of numbers (x, y) that can be used to locate a point on a coordinate plane. The first number x indicates the distance to the left or right of the origin, and the second number y indicates the distance above or below the origin. par ordenado Par de números (x, y) que se pueden utilizar para ubicar un punto en un plano cartesiano. El primer número, x, indica la distancia a la izquierda o derecha del origen y el segundo número, y, indica la distancia hacia arriba o hacia abajo del origen. origin (p. 54) The intersection of the x- and y-axes in a coordinate plane. The coordinates of the origin are (0, 0). origen Intersección de los ejes x e y en un plano cartesiano. Las coordenadas de origen son (0, 0). outcome (p. 737) A possible result of a probability experiment. resultado Resultado posible de un experimento de probabilidad. outlier (p. 716) A data value that is far removed from the rest of the data. valor extremo Valor de datos que está muy alejado del resto de los datos. output (p. 55) The result of substituting a value for a variable in a function. salida Resultado de la sustitución de una variable por un valor en una función. G24 Glossary/Glosario xÊÕÌà xÊÕÌÃ È x { Î Ó £ ä £ Ó Î { x È 5 and -5 are opposites. 3. 2 is - _ The opposite reciprocal of _ 3 2 A = {2, 3, 4, 5} B = {1, 3, 5, 7} The set of values that are in A OR B is A B = {1, 2, 3, 4, 5, 7}. 2 + 3 2 - (7 + 5) ÷ 4 · 3 2 + 3 2 - 12 ÷ 4 · 3 Add inside parentheses. 2 + 9 - 12 ÷ 4 · 3 Simplify the power. 2+9-3·3 Divide. 2+9-9 Multiply. 11 - 9 Add. 2 Subtract. { Þ Ó Ý { Ó ä Ó { The ordered pair (-2, 3) can be used to locate B. À} ä In the experiment of rolling a number cube, the possible outcomes are 1, 2, 3, 4, 5, and 6. -OSTOFDATA -EAN /UTLIER For the function f (x) = x 2 + 1, the input 3 produces an output of 10. ENGLISH SPANISH EXAMPLES parabola (p. 611) The shape of the graph of a quadratic function. parábola Forma de la gráfica de una función cuadrática. parallel lines (p. 361) Lines in the same plane that do not intersect. líneas paralelas Líneas en el mismo plano que no se cruzan. parallelogram A quadrilateral with two pairs of parallel sides. paralelogramo Cuadrilátero con dos pares de lados paralelos. parent function (p. 369) The simplest function with the defining characteristics of the family. Functions in the same family are transformations of their parent function. función madre La función más básica que tiene las características distintivas de una familia. Las funciones de la misma familia son transformaciones de su función madre. Pascal’s triangle (p. 590) A triangular arrangement of numbers in which every row starts and ends with 1 and each other number is the sum of the two numbers above it. triángulo de Pascal Arreglo triangular de números en el cual cada fila comienza y termina con 1 y los demás números son la suma de los dos valores que están arriba de cada uno. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 percent (p. 133) A ratio that compares a number to 100. porcentaje Razón que compara un número con 100. 17 = 17% _ 100 percent change (p. 144) An increase or decrease given as a percent of the original amount. See also percent decrease, percent increase. porcentaje de cambio Incremento o disminución dada como un porcentaje de la cantidad original. Ver también porcentaje de disminución, porcentaje de incremento. percent decrease (p. 144) A decrease given as a percent of the original amount. porcentaje de disminución Disminución dada como un porcentaje de la cantidad original. If an item that costs $8.00 is marked down to $6.00, the amount of the decrease is $2.00, so the percent decrease is 2.00 ____ = 0.25 = 25%. 8.00 percent increase (p. 144) An increase given as a percent of the original amount. porcentaje de incremento Incremento dado como un porcentaje de la cantidad original. If an item’s wholesale cost of $8.00 is marked up to $12.00, the amount of the increase is $4.00, so the percent increase 4.00 is ____ = 0.5 = 50%. 8.00 perfect square (p. 32) A number whose positive square root is a whole number. cuadrado perfecto Número cuya raíz cuadrada positiva es un número cabal. P À à f (x) = x 2 is the parent function for g (x) = x 2 + 4 and h (x) = (5x + 2)2 - 3. 36 is a perfect square 36 = 6. because √ Glossary/Glosario G25 ENGLISH SPANISH EXAMPLES perfect-square trinomial (p. 521) A trinomial whose factored form is the square of a binomial. A perfect-square trinomial has the 2 form a 2 - 2ab + b 2 = (a - b) or 2 a 2 + 2ab + b 2 = (a + b) . trinomio cuadrado perfecto Trinomio cuya forma factorizada es el cuadrado de un binomio. Un trinomio cuadrado perfecto tiene 2 la forma a 2 - 2ab + b 2 = (a - b) 2 o a 2 + 2ab + b 2 = (a + b) . x 2 + 6x + 9 is a perfectsquare trinomial, because x 2 + 6x + 9 = (x + 3) 2. perimeter (p. 52) The sum of the side lengths of a closed plane figure. perímetro Suma de las longitudes de los lados de una figura plana cerrada. 18 ft 6 ft Perimeter = 18 + 6 + 18 + 6 = 48 ft permutation (p. 761) An arrangement of a group of objects in which order is important. permutación Arreglo de un grupo de objetos en el cual el orden es importante. perpendicular Intersecting to form 90° angles. perpendicular Que se cruza para formar ángulos de 90°. perpendicular lines (p. 363) Lines that intersect at 90° angles. líneas perpendiculares Líneas que se cruzan en ángulos de 90°. plane A flat surface that has no thickness and extends forever. plano Una superficie plana que no tiene grosor y se extiende infinitamente. point A location that has no size. punto Ubicación exacta que no tiene ningún tamaño. point-slope form (p. 351) The point-slope form of a linear equation is y - y 1 = m(x - x 1), where m is the slope and (x 1, y 1) is a point on the line. forma de punto y pendiente La forma de punto y pendiente de una ecuación lineal es y - y 1 = m(x - x 1), donde m es la pendiente y (x 1, y 1) es un punto en la línea. polygon (p. 52) A closed plane figure formed by three or more segments such that each segment intersects exactly two other segments only at their endpoints and no two segments with a common endpoint are collinear. polígono Figura plana cerrada formada por tres o más segmentos tal que cada segmento se cruza únicamente con otros dos segmentos sólo en sus extremos y ningún segmento con un extremo común a otro es colineal con éste. polynomial (p. 496) A monomial or a sum or difference of monomials. polinomio Monomio o suma o diferencia de monomios. G26 Glossary/Glosario For objects A, B, C, and D, there are 12 different permutations of 2 objects. AB, AC, AD, BC, BD, CD BA, CA, DA, CB, DB, DC * point P y - 3 = 2(x - 3) 2x 2 + 3xy - 7y 2 ENGLISH SPANISH EXAMPLES x+1 x + 2 x 2 + 3x + 5 -(x 2 + 2x) −−−−−−− x+5 -(x + 2) −−−−−− 3 x 2+ 3x + 5 3 __ =x+1+_ x+2 x +2 polynomial long division (p. 914) A method of dividing one polynomial by another. división larga polinomial Método por el que se divide un polinomio entre otro. population (p. 732) The entire group of objects or individuals considered for a survey. población Grupo completo de objetos o individuos que se desea estudiar. positive correlation (p. 267) Two data sets have a positive correlation if both sets of data values increase. correlación positiva Dos conjuntos de datos tienen correlación positiva si los valores de ambos conjuntos de datos aumentan. positive number A number greater than zero. número positivo Número mayor que cero. positive square root (p. 32) The positive square root of a number, indicated by the radical sign. raíz cuadrada positiva Raíz cuadrada positiva de un número, expresada por el signo de radical. power (p. 26) An expression written with a base and an exponent or the value of such an expression. potencia Expresión escrita con una base y un exponente o el valor de dicha expresión. Power of a Power Property (p. 476) If a is any nonzero real number and m and n are integers, n then (a m) = a mn. Propiedad de la potencia de una potencia Dado un número real a distinto de cero y los números n enteros m y n, entonces (a m) = a mn. (6 7)4 = 6 7·4 Power of a Product Property (p. 477) If a and b are any nonzero real numbers and n is any integer, n then (ab) = a nb n. Propiedad de la potencia de un producto Dados los números reales a y b distintos de cero y un número n entero n, entonces (ab) = a nb n. (2 · 4)3 = 2 3 · 4 3 Power of a Quotient Property (p. 483, p. 484) If a and b are any nonzero real numbers and n is an Propiedad de la potencia de un cociente Dados los números reales a y b distintos de cero y un número integer, then a n __ (b) n a = __ . bn entero n, entonces a n __ (b) n a = __ . bn prediction (p. 739) An estimate or guess about something that has not yet happened. predicción Estimación o suposición sobre algo que todavía no ha sucedido. prime factorization (p. 544) A representation of a number or a polynomial as a product of primes. factorización prima Representación de un número o de un polinomio como producto de números primos. In a survey about the study habits of high school students, the population is all high school students. Þ Ý 2 is a positive number. { Î Ó £ ä £ Ó Î { The positive square root of 36 = 6. 36 is √ 2 3 = 8, so 8 is the third power of 2. = 6 28 = 8 · 64 = 512 (_35 ) = _53 · _53 · _53 · _53 4 ·3·3·3 = 3__ 5·5·5·5 34 =_ 54 The prime factorization of 60 is 2 · 2 · 3 · 5. Glossary/Glosario G27 ENGLISH SPANISH EXAMPLES prime number (p. 544) A whole number greater than 1 that has exactly two positive factors, itself and 1. número primo Número cabal mayor 5 is prime because its only que 1 que es divisible únicamente positive factors are 5 and 1. entre sí mismo y entre 1. principal (p. 139) An amount of money borrowed or invested. capital Cantidad de dinero que se pide prestado o se invierte. prism (p. 894) A polyhedron formed by two parallel congruent polygonal bases connected by faces that are parallelograms. prisma Poliedro formado por dos bases poligonales congruentes y paralelas conectadas por caras laterales que son paralelogramos. probability (p. 737) A number from 0 to 1 (or 0% to 100%) that is the measure of how likely an event is to occur. probabilidad Número entre 0 y 1 (o entre 0% y 100%) que describe cuán probable es que ocurra un suceso. Product of Powers Property (p. 474) If a is any nonzero real number and m and n are integers, then a m · a n = a m+n. Propiedad del producto de potencias Dado un número real a distinto de cero y los números enteros m y n, entonces a m · a n = a m+n. Product Property of Square Roots (p. 830) For a ≥ 0 and ab = √ a · √ b. b ≥ 0, √ Propiedad del producto de raíces cuadradas Dados a ≥ 0 y b ≥ 0, √ ab = √ a · √ b. proportion (p. 120) A statement c that two ratios are equal; __ab = __ . d proporción Ecuación que establece c . que dos razones son iguales; __ab = __ d pyramid (p. 894) A polyhedron formed by a polygonal base and triangular lateral faces that meet at a common vertex. pirámide Poliedro formado por una base poligonal y caras laterales triangulares que se encuentran en un vértice común. Pythagorean Theorem If a right triangle has legs of lengths a and b and a hypotenuse of length c, then a 2 + b 2 = c 2. Teorema de Pitágoras Dado un triángulo rectángulo con catetos de longitudes a y b y una hipotenusa de longitud c, entonces a 2 + b 2 = c 2. A bag contains 3 red marbles and 4 blue marbles. The probability of randomly choosing a red marble is __37 . 6 7 · 6 4 = 6 7+4 = 6 11 √ 9 · 25 = √ 9 · √ 25 = 3 · 5 = 15 4 2 =_ _ 3 6 £ÎÊV xÊV £ÓÊV 5 2 + 12 2 = 13 2 25 + 144 = 169 Pythagorean triple (p. 539) A set of three positive integers a, b, and c such that a 2 + b 2 = c 2. Tripleta de Pitágoras Conjunto de tres enteros positivos a, b y c tal que a 2 + b 2 = c 2. The numbers 3, 4, and 5 form a Pythagorean triple because 3 2 + 4 2 = 5 2. Q quadrant (p. 54) One of the four regions into which the x- and y-axes divide the coordinate plane. cuadrante Una de las cuatro regiones en las que los ejes x e y dividen el plano cartesiano. +Õ>`À>ÌÊ +Õ>`À>ÌÊ ä +Õ>`À>ÌÊ +Õ>`À>ÌÊ6 G28 Glossary/Glosario ENGLISH SPANISH quadratic equation (p. 642) An equation that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. ecuación cuadrática Ecuación que se puede expresar como ax 2 + bx + c = 0, donde a, b y c son números reales y a ≠ 0. Quadratic Formula (p. 670) fórmula cuadrática La fórmula -b ± √ b 2 - 4ac -b ± √ b 2 - 4ac EXAMPLES x 2 + 3x - 4 = 0 x2 - 9 = 0 The solutions of 2x 2 - 5x - 3 = 0 are given by (-5)2 - 4(2)(-3) -(-5) ± √ x = ___ 2(2) √ 5 ± 25 + 24 5±7 = __ = _ 4 4 1 x = 3 or x = - _ 2 The formula x = ____________ , 2a which gives solutions, or roots, of equations in the form ax 2 + bx + c = 0, where a ≠ 0. x = ____________ , que da 2a soluciones, o raíces, para las ecuaciones del tipo ax 2 + bx + c = 0, donde a ≠ 0. quadratic function (p. 610) A function that can be written in the form f (x) = ax 2 + bx + c, where a, b, and c are real numbers and a ≠ 0. función cuadrática Función que se puede expresar como f (x) = ax 2 + bx + c, donde a, b y c son números reales y a ≠ 0. quadratic polynomial (p. 497) A polynomial of degree 2. polinomio cuadrático Polinomio de grado 2. quartile The median of the upper or lower half of a data set. See also first quartile, third quartile. cuartil La mediana de la mitad superior o inferior de un conjunto de datos. Ver también primer cuartil, tercer cuartil. Quotient of Powers Property (p. 481) If a is a nonzero real number and m and n are integers, am then ___ = a m-n. an Propiedad del cociente de potencias Dado un número real a distinto de cero y los números am enteros m y n, entonces ___ = a m-n. an Quotient Property of Square Roots (p. 830) For a ≥ 0 and Propiedad del cociente de raíces cuadradas Dados a ≥ 0 y √a a = ___ . b > 0, __ b √b √a a = ___ . b > 0, __ b f (x) = x 2 - 6x + 8 x 2 - 6x + 8 &IRSTQUARTILE -INIMUM ä Ó { 4HIRDQUARTILE -EDIAN È n £ä -AXIMUM £Ó £{ 6 7 = 6 7-4 = 6 3 _ 64 √ 9 9 =_ 3 _ =_ 25 √ 25 5 √b R radical equation (p. 846) An equation that contains a variable within a radical. ecuación radical Ecuación que contiene una variable dentro de un radical. radical expression (p. 829) An expression that contains a radical sign. expresión radical Expresión que contiene un signo de radical. radical symbol (p. 32) The symbol √ used to denote a root. The symbol is used alone to indicate a square root or with n an index, √, to indicate the nth root. símbolo de radical Símbolo √ que se utiliza para expresar una raíz. Puede utilizarse solo para indicar una raíz cuadrada, o con un n índice, √, para indicar la enésima raíz. √ x+3+4=7 √ x+3+4 √ 36 = 6 3 √ 27 = 3 Glossary/Glosario G29 ENGLISH SPANISH EXAMPLES radicand (p. 829) The expression under a radical sign. radicando Número o expresión debajo del signo de radical. radius A segment whose endpoints are the center of a circle and a point on the circle; the distance from the center of a circle to any point on the circle. radio Segmento cuyos extremos son el centro de un círculo y un punto de la circunferencia; distancia desde el centro de un círculo hasta cualquier punto de la circunferencia. random sample (p. 727) A sample selected from a population so that each member of the population has an equal chance of being selected. muestra aleatoria Muestra seleccionada de una población tal que cada miembro de ésta tenga igual probabilidad de ser seleccionada. Mr. Hansen chose a random sample of the class by writing each student’s name on a slip of paper, mixing up the slips, and drawing five slips without looking. range of a data set (p. 716) The difference of the greatest and least values in the data set. rango de un conjunto de datos La diferencia del mayor y menor valor en un conjunto de datos. The data set {3, 3, 5, 7, 8, 10, 11, 11, 12} has a range of 12 - 3 = 9. range of a function or relation (p. 240) The set of all second coordinates (or y-values) of a function or relation. rango de una función o relación Conjunto de todos los valores de la segunda coordenada (o valores de y) de una función o relación. The range of the function {(-5, 3), (-3, -2), (-1, -1), (1, 0)} is {-2, -1, 0, 3}. rate (p. 120) A ratio that compares two quantities measured in different units. tasa Razón que compara dos cantidades medidas en diferentes unidades. rate of change (p. 314) A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. tasa de cambio Razón que compara la cantidad de cambio de la variable dependiente con la cantidad de cambio de la variable independiente. Expression: √ x+3 Radicand: x + 3 ,>`Õà 55 miles = 55 mi/h _ 1 hour The cost of mailing a letter increased from 22 cents in 1985 to 25 cents in 1988. During this period, the rate of change was change in cost 25 - 22 ___________ = _________ = __3 change in year 1988 - 1985 3 = 1 cent per year. ratio (p. 120) A comparison of two quantities by division. razón Comparación de dos cantidades mediante una división. rational equation (p. 920) An equation that contains one or more rational expressions. ecuación racional Ecuación que contiene una o más expresiones racionales. rational exponent (p. 489) An exponent that can be expressed m as __ n such that if m and n are exponente racional Exponente que m se puede expresar como __ n tal que si m y n son números enteros, m _ m n n m ) . integers, then b n = √b = ( √b rational expression (p. 886) An algebraic expression whose numerator and denominator are polynomials and whose denominator has a degree ≥ 1. G30 Glossary/Glosario m _ 1 or 1 : 2 _ 2 x+2 __ =6 2 x + 3x - 1 1 _ 6 64 6 = √ 64 m n n m entonces b n = √b = ( √ b) . expresión racional Expresión algebraica cuyo numerador y denominador son polinomios y cuyo denominador tiene un grado ≥ 1. x+2 __ x 2 + 3x - 1 ENGLISH SPANISH EXAMPLES rational function (p. 878) A function whose rule can be written as a rational expression. función racional Función cuya regla se puede expresar como una expresión racional. x+2 f(x) = __ 2 x + 3x - 1 rational number (p. 34) A number that can be written in the form __ab , where a and b are integers and b ≠ 0. número racional Número que se puede expresar como __ab , donde a y b son números enteros y b ≠ 0. − 2, 0 3, 1.75, 0.3, - _ 3 rationalizing the denominator (p. 842) A method of rewriting a fraction by multiplying by another fraction that is equivalent to 1 in order to remove radical terms from the denominator. racionalizar el denominador Método que consiste en escribir nuevamente una fracción multiplicándola por otra fracción equivalente a 1 a fin de eliminar los términos radicales del denominador. √ √ 2 2 1 ·_ _ =_ 2 √2 √ 2 ray A part of a line that starts at an endpoint and extends forever in one direction. rayo Parte de una recta que comienza en un extremo y se extiende infinitamente en una dirección. real number (p. 34) A rational or irrational number. Every point on the number line represents a real number. número real Número racional o irracional. Cada punto de la recta numérica representa un número real. ,i>Ê ÕLiÀà ,>Ì>Ê ÕLiÀÃÊύ® ÚÚÚ ÊÓÇÊÊÊ Ü ä°ÊÎÊ Ìi}iÀÃÊϖ® { Î >ÌÕÀ>Ê ÕLiÀÃÊϊ® £ i Ó recíproco Dado el número real a ≠ 0, el recíproco de a es __a1 . El producto de los recíprocos es 1. rectangle A quadrilateral with four right angles. rectángulo Cuadrilátero con cuatro ángulos rectos. rectangular prism (p. 894) A prism whose bases are rectangles. prisma rectangular Prisma cuyas bases son rectángulos. rectangular pyramid (p. 894) A pyramid whose base is a rectangle. pirámide rectangular Pirámide cuya base es un rectángulo. reflection (p. 371) A transformation that reflects, or “flips,” a graph or figure across a line, called the line of reflection. reflexión Transformación en la que una gráfica o figura se refleja o se invierte sobre una línea, denominada la línea de reflexión. regular polygon A polygon that is both equilateral and equiangular. polígono regular Polígono equilátero de ángulos iguales. ÊÊȖ££Ê е еÊ е ÊȖÓÊ Ê ä Î ÚÚÊxÊÊÊ {°x reciprocal (p. 21) For a real number a ≠ 0, the reciprocal of a is __a1 . The product of reciprocals is 1. е ÊȖ£ÇÊ еÊ Ó 7 iÊ ÕLiÀÃÊϓ® £ ÀÀ>Ì>Ê ÕLiÀà ÊÊÊ ÊÚÚÚ Ê£ä ££ û Number Reciprocal 2 1 __ 1 1 -1 -1 0 No reciprocal 2 Ī Ī Ī Glossary/Glosario G31 ENGLISH SPANISH relation (p. 240) A set of ordered pairs. relación Conjunto de pares ordenados. repeating decimal (p. 34) A rational number in decimal form that has a nonzero block of one or more digits that repeat continuously. decimal periódico Número racional en forma decimal que tiene un bloque de uno o más dígitos que se repite continuamente. replacement set (p. 8) A set of numbers that can be substituted for a variable. conjunto de reemplazo Conjunto de números que pueden sustituir una variable. rhombus A quadrilateral with four congruent sides. rombo Cuadrilátero con cuatro lados congruentes. right angle An angle that measures 90°. ángulo recto Ángulo que mide 90°. rise (p. 315) The difference in the y-values of two points on a line. distancia vertical Diferencia entre los valores de y de dos puntos de una línea. rotation (p. 370) A transformation that rotates or turns a figure about a point called the center of rotation. rotación Transformación que rota o gira una figura sobre un punto llamado centro de rotación. run (p. 315) The difference in the x-values of two points on a line. distancia horizontal Diferencia entre los valores de x de dos puntos de una línea. EXAMPLES ⎧ ⎫ ⎨(0, 5), (0, 4), (2, 3), (4, 0)⎬ ⎩ ⎭ − − −− − 1.3, 0.6, 2.14, 6.773 For the points (3, -1) and (6, 5), the rise is 5 - (-1) = 6. Ī Ī Ī Ī For the points (3, -1) and (6, 5), the run is 6 - 3 = 3. S sales tax (p. 140) A percent of the cost of an item that is charged by governments to raise money. impuesto sobre la venta Porcentaje del costo de un artículo que cobran los gobiernos para recaudar dinero. sample (p. 732) A part of the population. muestra Una parte de la población. In a survey about the study habits of high school students, a sample is a survey of 100 students. sample space (p. 737) The set of all possible outcomes of a probability experiment. espacio muestral Conjunto de todos los resultados posibles de un experimento de probabilidad. In the experiment of rolling a number cube, the sample space is {1, 2, 3, 4, 5, 6}. scalar (p. 771) A number that is multiplied by a matrix. escalar Número que se multiplica por una matriz. 3 [ ] [ ] 3 -6 1 -2 = 2 3 6 9 scalar scale (p. 122) The ratio between two corresponding measurements. G32 Glossary/Glosario escala Razón entre dos medidas correspondientes. 1 cm : 5 mi ENGLISH scale drawing (p. 122) A drawing that uses a scale to represent an object as smaller or larger than the actual object. SPANISH dibujo a escala Dibujo que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto original. EXAMPLES A blueprint is an example of a scale drawing. scale factor (p. 129) The multiplier used on each dimension to change one figure into a similar figure. factor de escala El multiplicador utilizado en cada dimensión para transformar una figura en una figura semejante. 6 in. 4 in. 2 in. 3 in. 3 = 1.5 Scale factor: _ 2 scale model (p. 122) A threedimensional model that uses a scale to represent an object as smaller or larger than the actual object. modelo a escala Modelo tridimensional que utiliza una escala para representar un objeto como más pequeño o más grande que el objeto real. scalene triangle A triangle with no congruent sides. triángulo escaleno Triángulo sin lados congruentes. scatter plot (p. 266) A graph with points plotted to show a possible relationship between two sets of data. diagrama de dispersión Gráfica con puntos que se usa para demostrar una relación posible entre dos conjuntos de datos. Þ n È { Ó Ý ä scientific notation (p. 467) A method of writing very large or very small numbers, by using powers of 10, in the form m × 10 n, where 1 ≤ m < 10 and n is an integer. notación científica Método que consiste en escribir números muy grandes o muy pequeños utilizando potencias de 10 del tipo m × 10 n, donde 1 ≤ m < 10 y n es un número entero. second differences (p. 610) Differences between first differences of a function. segundas diferencias Diferencias entre las primeras diferencias de una función. Ó { È 12,560,000,000,000 = 1.256 × 10 13 0.0000075 = 7.5 × 10 -6 Constant change in x-values +1 +1 +1 +1 x 0 1 2 3 4 y = x2 0 1 4 9 16 First differences +1 +3 +5 +7 Second differences sequence (p. 276) A list of numbers that often form a pattern. sucesión Lista de números que generalmente forman un patrón. set-builder notation (p. 170) A notation for a set that uses a rule to describe the properties of the elements of the set. notación de conjuntos Notación para un conjunto que se vale de una regla para describir las propiedades de los elementos del conjunto. n +2 +2 +2 1, 2, 4, 8, 16, … {x | x > 3} is read “The set of all x such that x is greater than 3.” Glossary/Glosario G33 ENGLISH SPANISH similar (p. 127) Two figures are similar if they have the same shape but not necessarily the same size. semejantes Dos figuras con la misma forma pero no necesariamente del mismo tamaño. similarity statement (p. 127) A statement that indicates that two polygons are similar by listing the vertices in the order of correspondence. enunciado de semejanza Enunciado que indica que dos polígonos son semejantes enumerando los vértices en orden de correspondencia. EXAMPLES È x x°{ { £Ó £ä £ä°n n quadrilateral ABCD ∼ quadrilateral EFGH simple event (p. 761) An event consisting of only one outcome. suceso simple Suceso que tiene sólo un resultado. simple interest (p. 139) A fixed percent of the principal. For principal P, interest rate r, and time t in years, the simple interest is I = Prt. interés simple Porcentaje fijo del capital. Dado el capital P, la tasa de interés r y el tiempo t expresado en años, el interés simple es I = Prt. simplest form of a square root expression (p. 829) A square root expression is in simplest form if it meets the following criteria: 1. No perfect squares are in the radicand. 2. No fractions are in the radicand. 3. No square roots appear in the denominator of a fraction. forma simplificada de una expresión de raíz cuadrada Una expresión de raíz cuadrada está en forma simplificada si reúne los siguientes requisitos: 1. No hay cuadrados perfectos en el radicando. 2. No hay fracciones en el radicando. 3. No aparecen raíces cuadradas en el denominador de una fracción. See also rationalizing the denominator. Ver también racionalizar el denominador. simplest form of a rational expression (p. 887) A rational expression is in simplest form if the numerator and denominator have no common factors. forma simplificada de una expresión racional Una expresión racional está en forma simplificada cuando el numerador y el denominador no tienen factores comunes. In the experiment of rolling a number cube, the event consisting of the outcome 3 is a simple event. If $100 is put into an account with a simple interest rate of 5%, then after 2 years, the account will have earned I = 100 · 0.05 · 2 = $10 in interest. Not Simplest Form Simplest Form √ 180 6 √ 5 √ 216a 2b 2 6ab √ 6 √ 7 _ √ 2 √ 14 _ 2 x - 1)(x + 1) x 2 - 1 = (__ _ 2 x +x-2 (x - 1)(x + 2) x +1 =_ x+2 Simplest form G34 Glossary/Glosario ENGLISH SPANISH simplest form of an exponential expression (p. 474) An exponential expression is in simplest form if it meets the following criteria: 1. There are no negative exponents. 2. The same base does not appear more than once in a product or quotient. 3. No powers, products, or quotients are raised to powers. 4. Numerical coefficients in a quotient do not have any common factor other than 1. forma simplificada de una expresión exponencial Una expresión exponencial está en forma simplificada si reúne los siguientes requisitos: 1. No hay exponentes negativos. 2. La misma base no aparece más de una vez en un producto o cociente. 3. No se elevan a potencias productos, cocientes ni potencias. 4. Los coeficientes numéricos en un cociente no tienen ningún factor común que no sea 1. simplify (p. 40) To perform all indicated operations. simplificar Realizar todas las operaciones indicadas. simulation (p. 736) A model of an experiment, often one that would be too difficult or timeconsuming to actually perform. simulación Modelo de un experimento; generalmente se recurre a la simulación cuando realizar dicho experimento sería demasiado difícil o llevaría mucho tiempo. sine (p. 928) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the hypotenuse. seno En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud de la hipotenusa. EXAMPLES Not Simplest Form Simplest Form 78 · 74 7 12 (x 2)-4 · x 5 1 _ x3 a 5b 9 _ (ab)4 ab 5 13 - 20 + 8 -7 + 8 1 Þ«ÌiÕÃi ««ÃÌi opposite sin A = __ hypotenuse slope (p. 315) A measure of the steepness of a line. If (x 1, y 1) and (x 2, y 2) are any two points on the line, the slope of the line, known as m, is represented by the y2 - y1 equation m = _____ x2 - x1 . pendiente Medida de la inclinación de una línea. Dados dos puntos (x 1, y 1) y (x 2, y 2) en una línea, la pendiente de la línea, denominada m, se representa con la ecuación y2 - y1 m = _____ x2 - x1 . { Þ Ó Ó]ÊÓ® Ó]Ê£® { ä Ó Ý { Ó { _ y - y1 3 -1 - 2 = _ m = x2 - x = _ 4 -2 - 2 1 2 slope-intercept form (p. 345) The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. forma de pendiente-intersección La forma de pendiente-intersección de una ecuación lineal es y = mx + b, donde m es la pendiente y b es la intersección con el eje y. y = -2x + 4 The slope is -2. The y-intercept is 4. solution of a linear equation in two variables An ordered pair or ordered pairs that make the equation true. solución de una ecuación lineal en dos variables Un par ordenado o pares ordenados que hacen que la ecuación sea verdadera. (4, 2) is a solution of x + y = 6. Glossary/Glosario G35 ENGLISH SPANISH EXAMPLES solution of a linear inequality in two variables (p. 428) An ordered pair or ordered pairs that make the inequality true. solución de una desigualdad lineal en dos variables Un par ordenado o pares ordenados que hacen que la desigualdad sea verdadera. (3, 1) is a solution of x + y < 6. solution of a system of linear equations (p. 397) Any ordered pair that satisfies all the equations in a system. solución de un sistema de ecuaciones lineales Cualquier par ordenado que resuelva todas las ecuaciones de un sistema. ⎧x + y = -1 ⎨ ⎩ -x + y = -3 solution of a system of linear inequalities (p. 435) Any ordered pair that satisfies all the inequalities in a system. solución de un sistema de desigualdades lineales Cualquier par ordenado que resuelva todas las desigualdades de un sistema. Solution: (1, -2) ⎧y ≤ x + 1 ⎨ ⎩ y < -x + 4 Þ Ó]£®ÊÃÊÊÌ iÊ ÛiÀ>««}Ê Ã >`i`ÊÀi}Ã] ÃÊÌÊÃÊ>ÊÃÕÌ° Ó ä Ó Ó Ó solution of an equation in one variable (p. 77) A value or values that make the equation true. solución de una ecuación en una variable Valor o valores que hacen que la ecuación sea verdadera. Equation: x + 2 = 6 Solution: x = 4 solution of an inequality in one variable (p. 170) A value or values that make the inequality true. solución de una desigualdad en una variable Valor o valores que hacen que la desigualdad sea verdadera. Inequality: x + 2 < 6 Solution: x < 4 solution set (p. 77) The set of values that make a statement true. conjunto solución Conjunto de valores que hacen verdadero un enunciado. Inequality: x + 3 ≥ 5 Solution set: {x | x ≥ 2} { Î Ó £ square A quadrilateral with four congruent sides and four right angles. cuadrado Cuadrilátero con cuatro lados congruentes y cuatro ángulos rectos. square in numeration (p. 26) The second power of a number. cuadrado en numeración La segunda potencia de un número. square root (p. 32) A number that is multiplied by itself to form a product is called a square root of that product. raíz cuadrada El número que se multiplica por sí mismo para formar un producto se denomina la raíz cuadrada de ese producto. square-root function (p. 822) A function whose rule contains a variable under a square-root sign. función de raíz cuadrada Función cuya regla contiene una variable bajo un signo de raíz cuadrada. standard form of a linear equation (p. 302) Ax + By = C, where A, B, and C are real numbers and A and B are not both 0. forma estándar de una ecuación lineal Ax + By = C, donde A, B y C son números reales y A y B no son ambos cero. G36 Glossary/Glosario ä £ Ó Î { x 16 is the square of 4. √ 16 = 4, because 4 2 = 4 · 4 = 16. -5 y = √3x 2x + 3y = 6 È Ý ENGLISH SPANISH EXAMPLES standard form of a polynomial (p. 497) A polynomial in one variable is written in standard form when the terms are in order from greatest degree to least degree. forma estándar de un polinomio Un polinomio de una variable se expresa en forma estándar cuando los términos se ordenan de mayor a menor grado. standard form of a quadratic equation (p. 642) ax 2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. forma estándar de una ecuación cuadrática ax 2 + bx + c = 0, donde a, b y c son números reales y a ≠ 0. stem-and-leaf plot (p. 709) A graph used to organize and display data by dividing each data value into two parts, a stem and a leaf. diagrama de tallo y hojas Gráfica utilizada para organizar y mostrar datos dividiendo cada valor de datos en dos partes, un tallo y una hoja. stratified random sample (p. 732) A sample in which a population is divided into distinct groups and members are selected at random from each group. muestra aleatoria estratificada Muestra en la que la población está dividida en grupos diferenciados y los miembros de cada grupo se seleccionan al azar. substitution method (p. 404) A method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation(s). sustitución Método utilizado para resolver sistemas de ecuaciones resolviendo una ecuación para una variable y sustituyendo la expresión resultante en las demás ecuaciones. Subtraction Property of Equality (p. 79) If a, b, and c are real numbers and a = b, then a - c = b - c. Propiedad de igualdad de la resta Si a, b y c son números reales y a = b, entonces a - c = b - c. x+6= 8 -6 -6 −−−− −− x = 2 Subtraction Property of Inequality (p. 176) For real numbers a, b, and c, if a < b, then a - c < b - c. Also holds true for >, ≤, ≥, and ≠. Propiedad de desigualdad de la resta Dados los números reales a, b y c, si a < b, entonces a - c < b - c. Es válido también para >, ≤, ≥ y ≠. x+6< 8 -6 -6 −−−− −− x < 2 supplementary angles Two angles whose measures have a sum of 180°. ángulos suplementarios Dos ángulos cuyas medidas suman 180°. surface area (p. 520) The total area of all faces and curved surfaces of a three-dimensional figure. área total Área total de todas las caras y superficies curvas de una figura tridimensional. 4x 5 - 2 x 4 + x 2 - x + 1 2x 2 + 3x - 1 = 0 -Ìi Î { x i>Ûià ÓÊÎÊ{Ê{ÊÇÊ äÊ£ÊxÊÇÊÇÊÇÊn £ÊÓÊÓÊÎ iÞ\ÊÎ]ÓÊi>ÃÊÎ°Ó Ms. Carter chose a stratified random sample of her school’s student population by randomly selecting 30 students from each grade level. Îäc £xäc £ÓÊV ÈÊV nÊV Surface area = 2(8)(12) + 2(8)(6) + 2(12)(6) = 432 cm 2 Glossary/Glosario G37 ENGLISH SPANISH EXAMPLES system of linear equations (p. 397) A system of equations in which all of the equations are linear. sistema de ecuaciones lineales Sistema de ecuaciones en el que todas las ecuaciones son lineales. ⎧2x + 3y = -1 ⎨ ⎩ x - 3y = 4 system of linear inequalities (p. 435) A system of inequalities in which all of the inequalities are linear. sistema de desigualdades lineales Sistema de desigualdades en el que todas las desigualdades son lineales. ⎧2x + 3y > -1 ⎨ ⎩ x - 3y ≤ 4 systematic random sample (p. 732) A sample based on selecting one member of the population at random and then selecting other members by using a pattern. muestra sistemática Muestra en la que se elige a un miembro de la población al azar y luego se elige a otros miembros mediante un patrón. Mr. Martin chose a systematic random sample of customers visiting a store by selecting one customer at random and then selecting every tenth customer after that. T tangent (p. 928) In a right triangle, the ratio of the length of the leg opposite ∠A to the length of the leg adjacent to ∠A. tangente En un triángulo rectángulo, razón entre la longitud del cateto opuesto a ∠A y la longitud del cateto adyacente a ∠A. ««ÃÌi >`>ViÌ tan A = opposite _ adjacent 3x 2 + 6x - 8 term of an expression (p. 47) The parts of the expression that are added or subtracted. término de una expresión Parte de una expresión que debe sumarse o restarse. term of a sequence (p. 276) An element or number in the sequence. término de una sucesión Elemento o número de una sucesión. 5 is the third term in the sequence 1, 3, 5, 7, … terminating decimal (p. 34) A decimal that ends, or terminates. decimal finito Decimal con un número determinados de posiciones decimales. 1.5, 2.75, 4.0 theoretical probability (p. 744) The ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. probabilidad teórica Razón entre el número de resultados igualmente probables de un suceso y el número total de resultados posibles. In the experiment of rolling a number cube, the theoretical probability of rolling an odd number is __36 = __12 . third quartile (p. 718) The median of the upper half of a data set. Also called upper quartile. tercer cuartil La mediana de la mitad superior de un conjunto de datos. También se llama cuartil superior. Lower half 18, 23, 28, tip (p. 140) An amount of money added to a bill for service; usually a percent of the bill. propina Cantidad que se agrega a una factura por servicios; generalmente, un porcentaje de la factura. transformation (p. 369) A change in the position, size, or shape of a figure or graph. transformación Cambio en la posición, tamaño o forma de una figura o gráfica. Term Term Term B B Preimage A C ABC G38 Glossary/Glosario Upper half 29, 36, 42 Third quartile Image A ABC C ENGLISH SPANISH translation (p. 369) A transformation that shifts or slides every point of a figure or graph the same distance in the same direction. traslación Transformación en la que todos los puntos de una figura o gráfica se mueven la misma distancia en la misma dirección. trapezoid A quadrilateral with exactly one pair of parallel sides. trapecio Cuadrilátero con sólo un par de lados paralelos. tree diagram (p. 760) A branching diagram that shows all possible combinations or outcomes of an experiment. EXAMPLES Ī Ī Ī diagrama de árbol Diagrama con ramificaciones que muestra todas las combinaciones o resultados posibles de un experimento. Ī ( 4 The tree diagram shows the possible outcomes when tossing a coin and rolling a number cube. línea de tendencia Línea en un diagrama de dispersión que sirve para mostrar la correlación entre conjuntos de datos más claramente. Õ`À>ÃiÀ £Óää iÞÊÀ>Ãi`Êf® trend line (p. 269) A line on a scatter plot that helps show the correlation between data sets more clearly. £äää nää Èää {ää Óää ä xä £ää £xä Óää ,ÃÊÃ`Ê In the experiment of rolling a number cube, each roll is one trial. trial (p. 737) Each repetition or observation of an experiment. prueba Una sola repetición u observación de un experimento. triangle A three-sided polygon. triángulo Polígono de tres lados. triangular prism (p. 894) A prism whose bases are triangles. prisma triangular Prisma cuyas bases son triángulos. triangular pyramid (p. 894) A pyramid whose base is a triangle. pirámide triangular Pirámide cuya base es un triángulo. trigonometric ratio (p. 928) Ratio of the lengths of two sides of a right triangle. razón trigonométrica Razón entre dos lados de un triángulo rectángulo. "ASES V L > a , cos A = _ b , tan A = _ a sin A = _ c c b trinomial (p. 497) A polynomial with three terms. trinomio Polinomio con tres términos. 4x 2 + 3xy - 5y 2 Glossary/Glosario G39 ENGLISH SPANISH EXAMPLES union (p. 206) The union of two sets is the set of all elements that are in either set, denoted by . unión La unión de dos conjuntos es el conjunto de todos los elementos que se encuentran en ambos conjuntos, expresado por . A = {1, 2, 3, 4} B = {1, 3, 5, 7, 9} A B = {1, 2, 3, 4, 5, 7, 9} unit rate (p. 120) A rate in which the second quantity in the comparison is one unit. tasa unitaria Tasa en la que la segunda cantidad de la comparación es una unidad. 30 mi = 30 mi/h _ 1h unlike radicals (p. 835) Radicals with a different quantity under the radical. radicales distintos Radicales con cantidades diferentes debajo del signo de radical. unlike terms Terms with different variables or the same variables raised to different powers. términos distintos Términos con variables diferentes o las mismas variables elevadas a potencias diferentes. upper quartile See third quartile. cuartil superior Ver tercer cuartil. U 2 √ 2 and 2 √3 4xy 2 and 6x 2y V value of a function (p. 251) The result of replacing the independent variable with a number and simplifying. valor de una función Resultado de reemplazar la variable independiente por un número y luego simplificar. The value of the function f (x) = x + 1 for x = 3 is 4. value of a variable (p. 7) A number used to replace a variable to make an equation true. valor de una variable Número utilizado para reemplazar una variable y hacer que una ecuación sea verdadera. In the equation x + 1 = 4, the value of x is 3. value of an expression (p. 7) The result of replacing the variables in an expression with numbers and simplifying. valor de una expresión Resultado de reemplazar las variables de una expresión por un número y luego simplificar. The value of the expression x + 1 for x = 3 is 4. variable (p. 6) A symbol used to represent a quantity that can change. variable Símbolo utilizado para representar una cantidad que puede cambiar. In the expression 2x + 3, x is the variable. Venn diagram A diagram used to show relationships between sets. diagrama de Venn Diagrama utilizado para mostrar la relación entre conjuntos. Brand A Brand B Both Neither: 15 vertex of a parabola (p. 612) The highest or lowest point on the parabola. vértice de una parábola Punto más alto o más bajo de una parábola. ä]ÊÓ® The vertex is (0, -2). G40 Glossary/Glosario ENGLISH vertex of an absolute-value graph (p. 378) The point on the axis of symmetry of the graph. SPANISH EXAMPLES vértice de una gráfica de valor absoluto Punto en el eje de simetría de la gráfica. { Þ ÞÊNÝN Ó 6iÀÌiÝ { vertical angles The nonadjacent angles formed by two intersecting lines. ángulos opuestos por el vértice Ángulos no adyacentes formados por dos líneas que se cruzan. Ý ä Ó Ó { £ Î { Ó ∠1 and ∠3 are vertical angles. ∠2 and ∠4 are vertical angles. vertical line (p. 316) A line whose equation is x = a, where a is the x-intercept. línea vertical Línea cuya ecuación es x = a, donde a es la intersección con el eje x. Þ x { Î Ó £ ÝÊÊÓ x {ÎÓ £ £ vertical-line test (p. 247) A test used to determine whether a relation is a function. If any vertical line crosses the graph of a relation more than once, the relation is not a function. prueba de la línea vertical Prueba utilizada para determinar si una relación es una función. Si una línea vertical corta la gráfica de una relación más de una vez, la relación no es una función. volume (p. 520) The number of nonoverlapping unit cubes of a given size that will exactly fill the interior of a three-dimensional figure. volumen Cantidad de cubos unitarios no superpuestos de un determinado tamaño que llenan exactamente el interior de una figura tridimensional. voluntary response sample (p. 733) A sample in which members choose to be in the sample. muestra de respuesta voluntaria Una muestra en la que los miembros eligen participar. £ Ó Î { x x Ý Þ x x Ý x Function Not a function {ÊvÌ ÎÊvÌ £ÓÊvÌ Volume = (3)(4)(12) = 144 ft 3 A store provides survey cards for customers who wish to fill them out. W whole number (p. 34) A member of the set of natural numbers and zero. número cabal Miembro del conjunto de los números naturales y cero. 0, 1, 2, 3, 4, 5, … X x-axis (p. 54) The horizontal axis in a coordinate plane. eje x Eje horizontal en un plano cartesiano. x-axis 0 Glossary/Glosario G41 ENGLISH SPANISH EXAMPLES x-coordinate (p. 54) The first number in an ordered pair, which indicates the horizontal distance of a point from the origin on the coordinate plane. coordenada x Primer número de un par ordenado, que indica la distancia horizontal de un punto desde el origen en un plano cartesiano. 4 x-intercept (p. 307) The x-coordinate(s) of the point(s) where a graph intersects the x-axis. intersección con el eje x Coordenada(s) x de uno o más puntos donde una gráfica corta el eje x. y 2 x -4 0 -2 x-coordinate 2 4 -2 P (-2, -3) -4 4 -2 y (2, 0) x 4 0 -2 The x-intercept is 2. Y y-axis y-axis (p. 54) The vertical axis in a coordinate plane. eje y Eje vertical en un plano cartesiano. 0 y-coordinate (p. 54) The second number in an ordered pair, which indicates the vertical distance of a point from the origin on the coordinate plane. coordenada y Segundo número de un par ordenado, que indica la distancia vertical de un punto desde el origen en un plano cartesiano. y-intercept (p. 307) The y-coordinate(s) of the point(s) where a graph intersects the y-axis. intersección con el eje y Coordenada(s) y de uno o más puntos donde una gráfica corta el eje y. y 4 2 x -4 -2 0 2 4 y-coordinate P -2 (-2, -3) -4 4 y (0, 2) -2 0 2 x -2 The y-intercept is 2. Z zero exponent (p. 460) For any nonzero real number x, x 0 = 1. exponente cero Dado un número real distinto de cero x, x 0 = 1. zero of a function (p. 619) For the function f, any number x such that f (x) = 0. cero de una función Dada la función f, todo número x tal que f (x) = 0. 50 = 1 Ó £ x { ÎÓ £ £ Î]Êä® Ó £ Ó Î { x £]Êä® Î { x The zeros are -3 and 1. Zero Product Property (p. 650) For real numbers p and q, if pq = 0, then p = 0 or q = 0. G42 Glossary/Glosario Propiedad del producto cero Dados los números reales p y q, si pq = 0, entonces p = 0 o q = 0. If (x - 1)(x + 2) = 0, then x - 1 = 0 or x + 2 = 0, so x = 1 or x = -2. Index A Aaron, Hank, 42 Absolute value definition of, 14 equations, 112–114 functions, 378–381, AT5 inequalities, 212–215 Addition of matrices, 770 of polynomials, 502–503, 504–506 properties of, 46 of radical expressions, 835–837 of rational expressions, 905–908 of real numbers, 14–17 solving equations by, 77–79 solving inequalities by, 176–179 Addition Property of Equality, 79, 86 Addition Property of Inequality, 176 Additive identity, 15 Additive inverse, 15 Advertising, 721 Agriculture, 677 Air Force Academy, 439 Air Force One, 254 Albers, Josef, 30 Algebraic expressions, 6, 7, 8, 10, 40–42, 72, 248 Algebra Lab, see also Technology Lab Compound Events, 758–759 Explore Changes in Population, 150–151 Explore Constant Changes, 322–323 Explore Properties of Exponents, 472–473 Explore the Axis of Symmetry, 618 Model Completing the Square, 662 Model Equations with Variables on Both Sides, 99 Model Factoring, 550, 558–559 Model Growth and Decay, 804 Model Inverse Variation, 870 Model One-Step Equations, 76 Model Polynomial Addition and Subtraction, 502–503 Model Polynomial Division, 912 Model Polynomial Multiplication, 510–511 Model Systems of Linear Equations, 403 Model Variable Relationships, 248 Simulations, 736 Truth Tables and Compound Statements, 203 Vertical-Line Test, 247 Algebra tiles, 76, 99, 502–503, 510–511, 550, 558–559, 912 All of the Above, 862–863 Altitude sickness, 356 Amusement parks, 833 Anglerfish, 81 Angles central, 703 corresponding, 127 exterior, AT21 remote interior, AT21 Animals, 186 Animals Link, 186 Annulus, 557 Answers, choosing combinations of, 936–937 Any Question Type read the problem for understanding, 450–451 spatial reasoning, 780–781 translate words to math, 600–601 use a diagram, 536–537 Applications Advertising, 721 Agriculture, 677 Amusement Parks, 833 Animals, 186 Aquatics, 644 Archaeology, 124, 810 Archery, 624 Architecture, 622, 834 Art, 30, 565 Astronomy, 10, 333, 340, 468, 469, 475, 479, 486, 722, 826 Athletics, 281, 423, 646 Automobiles, 713 Aviation, 86, 334, 409, 928 Basketball, 757 Biology, 16, 29, 89, 97, 105, 121, 122, 124, 137, 208, 216, 310, 311, 463, 464, 469, 479, 489, 491, 646, 712, 889, 891, AT4 Business, 18, 58, 98, 139, 142, 193, 197, 347, 349, 372, 416, 422, 433, 437, 438, 506, 721, 722, 817, 883, AT12 Camping, 185 Career, 658 Carpentry, 534 Chemistry, 123, 138, 204, 209, 416, 470, 486, 713 City Planning, 582 Communication, 115, 193, 470, 877 Construction, 109, 115, 320, 566, 890, 929 Consumer Application, 102, 184, 244, 314, 347 Consumer Economics, 88, 96, 103, 179, 271, 321, 356, 407, 408, 414, 415, 430, 482, 817 Contests, 801 Data Collection, 238, 320, 637 Decorating, 125 Design, 31 Diving, 24, 929 Earth Science, 88 Ecology, 271 Economics, 17, 88, 96, 103, 179, 271, 321, 407, 408, 414, 415, 482, 817 Education, 24, 186, 200 Electricity, 624, 630, 875 Employment, 148 Engineering, 114, 179, 624, 630, 875 Entertainment, 23, 30, 37, 111, 125, 137, 194, 201, 216, 280, 304, 373, 402, 424, 469, 526, 660, 714, 756, 903, AT8, AT12 Environment, 340 Environmental Science, 124, 310, 328 Farming, 439 Finance, 80, 88, 109, 124, 139, 143, 409, 555, 806, 819, 884 Fitness, 79, 349, 401, 838, 910 Fund-raising, 269 Games, 653 Gardening, 33, 124, 192, 573 Gemology, 881 Geography, 19, 332, 478, 486 Geology, 80, 81, 424, 826 Geometry, 10, 30, 36, 43, 44, 45, 50, 57, 81, 88, 96, 104, 109, 126, 185, 194, 199, 201, 244, 306, 334, 362, 363, 365, 366, 367, 409, 416, 433, 478, 479, 487, 491, 492, 493, 499, 500, 507, 508, 509, 517, 518, 519, 525, 529, 534, 549, 556, 557, 565, 572, 573, 583, 591, 646, 654, 655, 659, 660, 661, 666, 667, 669, 749, 757, 825, 836, 837, 838, 839, 843, 844, 845, 849, 850, 851, 891, 892, 918, 919, AT4, AT21 Health, 179, 180, 244, 326, 469 Hiking, 773 History, 97, 356, 731, 765 Hobbies, 130, 374, 433, 667, 675, AT17 Home Economics, 876 Landscaping, 401 Law Enforcement, 825 Logic, 731, AT21 Manufacturing, 115, 124, 461 Marine Biology, 616 Math History, 255, 416, 526, 590, 834 Measurement, 128, 305, 355, 470, 582, 794 Mechanics, 875, 877 Medicine, 463, 501 Index IN1 Meteorology, 10, 17, 110, 208, 209, 356, 851 Military, 439 Money, 408 Music, 208, 216, 218, 516, 548, 873 Navigation, 929 Number Sense, 548, 549 Number Theory, 280, 654, 660 Nutrition, 87, 88, 136, 148, 215, 244 Oceanography, 121, 260 Personal Finance, 311, 551, 812 Pet Care, 819, AT8 Photography, 507, 517 Physical Science, 306, 479, 793, AT16 Physics, 498, 555, 573, 589, 636, 638, 654, 660, 673, 800, 844, 851, 874 Population, 126 Probability, 901, 902 Problem-Solving, 28, 94, 177–178, 259, 354, 399, 523–524, 579–580, 627–628, 665–666, 753, 815–816, 921–922 Quality Control, 739 Real Estate, 131 Recreation, 22, 87, 116, 200, 238, 244, 280, 321, 333, 334–335, 408, 439, 832, 908 Recycling, 8 Remodeling, 565 Safety, 214 School, 81, 199, 374, 400, 433, 755 Science, 124, 356, 478, 552, 808 Shipping, 279 Solar Energy, 918 Space Shuttle, 115 Sports, 42, 44, 50, 104, 107, 110, 121, 125, 172, 178, 180, 209, 238, 310, 335, 348, 381, 424, 485, 518, 529, 616, 645, 652, 654, 676, 712, 718, 721, 740, 741, 792, 793, 800, 831 Statistics, 82, 88, 469, 799 Technology, 29, 142, 471, 534, 546, 766, 801 Temperature, 116, 215, 217 Transportation, 96, 124, 179, 209, 254, 261, 271, 304, 305, 341, 500, 851 Travel, 18, 36, 90, 104, 185, 278, 279, 308, 319, 334, 630, 875, 909, 910, 924, AT8 Wages, 305, 339 Waterfalls, 645 Weather, 23, 712, 713, 721 Winter Sports, 876 Applied Sciences major, 402 Approximating solutions, 91 Aquatics, 644 Archaeology, 124, 810 Archery, 624 Architecture, 622, 834 Area of composite figures, 83 in the coordinate plane, 313 of a rectangle, 83 of a square, 83 IN2 Index surface, 493, 520 of a trapezoid, 335, 654 of a triangle, 83 Are You Ready?, 3, 73, 167, 231, 297, 393, 457, 541, 607, 695, 787, 867 Arguments, writing convincing, 395 Arithmetic sequences, 276–278 Art, 30, 565 Art Link, 565 Assessment Chapter Test, 66, 158, 224, 288, 386, 448, 534, 598, 688, 778, 860, 934 College Entrance Exam Practice ACT, 159, 289, 449, 599 SAT, 67, 387, 535 SAT Mathematics Subject Tests, 689, 779, 861, 935 SAT Student-Produced Responses, 225 Cumulative Assessment, 70–71, 162–163, 228–229, 292–293, 390–391, 452–453, 538–539, 602–603, 692–693, 782–783, 864–865, 938–939 Multi-Step Test Prep, 38, 60, 118, 152, 188, 218, 264, 282, 342, 376, 426, 442, 494, 528, 576, 592, 640, 678, 734, 768, 820, 854, 896, 926 Multi-Step Test Prep questions are also found in every exercise set. Some examples: 10, 18, 24, 30, 36 Ready to Go On?, 39, 61, 153, 189, 219, 265, 283, 343, 377, 427, 443, 495, 529, 577, 593, 641, 679, 735, 769, 821, 855, 897, 927 Standardized Test Prep, 70–71, 162–163, 228–229, 292–293, 390–391, 452–453, 538–539, 602–603, 692–693, 782–783, 864–865, 938–939 Study Guide: Preview, 4, 74, 168, 232, 298, 394, 458, 542, 608, 696, 788, 868 Study Guide: Review, 62–65, 154–157, 220–223, 284–287, 382–385, 444–447, 530–533, 594–597, 684–687, 774–777, 856–859, 930–933 Test Prep Test Prep questions are found in every exercise set. Some examples: 11, 19, 25, 31, 37 Test Tackler Any Question Type Read the Problem for Understanding, 450–451 Spatial Reasoning, 780–781 Translate Words to Math, 600–601 Use a Diagram, 536–537 Extended Response Explain Your Reasoning, 690–691 Understand the Scores, 290–291 Gridded Response Fill in Answer Grids Correctly, 68–69 Multiple Choice Choose Combinations of Answers, 936–937 Eliminate Answer Choices, 160–161 None of the Above or All of the Above, 862–863 Recognize Distracters, 388–389 Short Response Understand Short Response Scores, 226–227 Associative Properties of Addition and Multiplication, 46 Astronomy, 10, 333, 340, 468, 469, 475, 479, 486, 722, 826 Astronomy Link, 10, 340 Asymptotes definition of, 878 graphing rational functions using, 880 identifying, 879 Athletics, 281, 423, 646 Atoms, 470 Automobiles, 713 Automobiles Link, 713 Average, 716–719 Aviation, 86, 334, 409, 928 Axes, 54 Axis of symmetry in absolute-value functions, 378 in a parabola, 618, 620 definition of, 618 exploring, 618 finding by using the formula, 621 by using zeros, 620 B Back-to-back stem-and-leaf plot, 709 Bald eagles, 891 Bamboo, 311 Bar graphs, 698–699, 700 Bases of numbers, 26 Basketball, 757 Biased samples, 733 Binomial(s) cubic, 532 definition of, 497 division of polynomials by, 914–915 opposite, 554, 888 special products of, 521–525 Biology, 16, 29, 89, 97, 105, 121, 122, 124, 137, 208, 216, 310, 311, 463, 464, 469, 479, 489, 491, 646, 712, 889, 891, AT4 Biology Link, 105, 311, 464, 646, 891 Biology major, 106 Biostatistics major, 767 Blood loss, 464 Blood volume, 121 Biology Link, 105, 311, 464, 646, 891 Biology major, 106 Biostatistics major, 767 Blood loss, 464 Blood volume, 121 Boiling point, 356 Box-and-whisker plot, 718–719, 725 Boyle’s law, 874 Braces, 40 Brackets, 40 Business, 18, 58, 98, 139, 142, 193, 197, 347, 349, 372, 416, 422, 433, 437, 438, 506, 721, 722, 817, 883, AT12 C Calculator, see Graphing calculator Camping, 185 Career Path Applied Sciences major, 402 Biology major, 106 Biostatistics major, 767 Culinary Arts major, 202 Data Mining major, 358 Environmental Sciences major, 567 Carpentry, 534 Cartesian plane, 58 Caution!, 27, 48, 86, 139, 184, 301, 308, 314, 315, 406, 421, 437, 461, 477, 552, 571, 586, 610, 613, 621, 636, 744, 792, 799, 831, 887, 906 Central angles of circles, 703 Central tendency, measure of, 716 Changes constant, 322–323 percent, 144 rate of constant and variable, AT14–AT15 decrease, 803 definition of, 314, AT14 identifying linear and nonlinear functions from, AT15–AT16 increase, 803 slope and, 314–317 Changing dimensions, 53, 129 Chapter Test, 66, 158, 224, 288, 386, 448, 534, 598, 688, 778, 860, 934, see also Assessment Charts, reading and interpreting, 697 Cheetahs, 105 Chemistry, 123, 138, 204, 209, 416, 470, 486, 713 Chemistry Link, 209, 470 Choosing combinations of answers, 936–937 factoring methods, 586–588 models graphing data for, 813–814 using patterns for, 814 Circle graphs, 698–699, 702 City Planning, 582 Coefficients definition of, 48 leading, of polynomials, 497 opposite, 411 Coincident lines, 421 College Entrance Exam Practice, see also Assessment ACT, 159, 289, 449, 599 SAT, 67, 387, 535 SAT Mathematics Subject Tests, 689, 779, 861, 935 SAT Student-Produced Responses, 225 Combinations of answers, choosing, 936–937 definition of, 761 and permutations, 760–763 Combining like radicals, 835 Combining like terms, 48 Commission, 139 Common difference, 276–277 Common ratio, 790, 792 Communicating math choose, 567, 675 compare, 11, 147, 172, 184, 201, 238, 357, 425, 485, 492, 516, 668, 669, 811, 824, 850, 919 construct, 730 create, 244, 245, 305, 714 define, 547 describe, 19, 31, 50, 56, 79, 82, 89, 135, 137, 148, 192, 195, 207, 215, 217, 238, 242, 245, 261, 272, 305, 320, 327, 328, 333, 347, 349, 364, 372, 374, 375, 424, 433, 506, 546, 556, 582, 623, 625, 636, 638, 644, 653, 659, 667, 675, 705, 707, 721, 808, 818, 826, 844, 875, 884, 918, AT4, AT6, AT10, AT11, AT12 determine, 25, 36, 423, 425, 792, 876, AT4, AT12, AT17 explain, 8, 17, 19, 22, 24, 30, 37, 42, 43, 44, 45, 50, 57, 80, 82, 88, 95, 104, 105, 109, 110, 114, 116, 123, 125, 131, 135, 137, 141, 142, 174, 175, 179, 180, 185, 186, 199, 201, 202, 209, 210, 217, 236, 242, 244, 246, 252, 254, 260, 261, 271, 272, 279, 280, 306, 321, 328, 334, 340, 341, 366, 367, 372, 399, 400, 402, 409, 415, 416, 417, 424, 425, 431, 434, 439, 440, 464, 469, 470, 471, 477, 479, 486, 493, 498, 500, 501, 508, 518, 526, 527, 547, 548, 554, 556, 557, 563, 565, 573, 574, 583, 590, 591, 629, 630, 631, 637, 638, 646, 653, 659, 661, 668, 676, 677, 706, 707, 708, 711, 713, 714, 719, 728, 729, 730, 739, 748, 749, 755, 756, 763, 765, 766, 794, 795, 808, 809, 818, 825, 826, 833, 838, 842, 852, 876, 884, 892, 901, 903, 908, 910, 918, 923, 925, AT4 express, 28 find, 106, 549, 566, 567, 574, 637, 918 give (an) example(s), 29, 56, 125, 187, 236, 246, 254, 500, 519, 527, 588, 638, 708, 719, 727, 739, 754, 827, 837, 838, 892 identify, 35, 349, 381, 506, 809, 884, 891 list, 439, 876 make, 713, 729, 853 name, 129, 175, 252, 309, 433, 704, 874 Reading and Writing Math, 5, 75, 169, 233, 299, 395, 459, 543, 609, 697, 789, 869, see also Reading Math; Reading Strategies; Study Strategies; Writing Math; Writing Strategies show, 80, 81, 82, 88, 89, 97, 104, 124, 126, 142, 148, 179, 180, 187, 367, 492, 556, 566, 573, 590, 616, 766, 832 tell, 49, 87, 90, 103, 116, 216, 238, 329, 381, 431, 468, 616, 625, 636, 666, 741, 747, 800, 837, 852, 882 write, 8, 9, 98, 105, 136, 312, 401, 518, 573, 584, 591, 617, 794, 809, 833, 885, 889, 911 Write About It Write About It questions are found in every exercise set. Some examples: 9, 19, 24, 29, 31 Communication, 115, 193, 470, 877 Commutative Properties of Addition and Multiplication, 46 Compatible numbers, 46 Complement of an event, 745 Completing the square, 663–666, 674 Complex fractions, 904 Composite figures, areas of, 83 Compound events, 758–759, 761 Compound inequalities, 204–207, 212 Compound interest, 806 Compound statements, 203 Conditional statements, AT21 Cones, 520, 894 Congruent segments, 330 Conjecture, making a, 322, 323, 368, 632, 648, 662, 828, 870, 893, AT18 Conjugates, rationalizing denominators using, 845 Connecting Algebra to Data Analysis, 275, 360, 698–699, 732–733 to Geometry, 52–53, 83, 211, 313, 520, 803, 894, 895 to Number Theory, 418–419, 585 Index IN3 Consistent systems, 420–421 Constant definition of, 6 in trinomial factoring, 560–562 of variation, 336, 871 Constant changes, exploring, 322–323 Construction, 109, 115, 320, 566, 890, 929 Consumer Application, 102, 184, 244, 314, 347 Consumer Economics, 88, 96, 103, 179, 271, 321, 356, 407, 408, 414, 415, 430, 482, 817 Contact lenses, 180 Contests, 801 Continuous graphs, 235 Contrapositives, AT21 Convenience sample, 733 Conversion factors, 121, 609 Converting between probabilities and odds, 746 Convincing arguments/ explanations, writing, 395 Coordinate plane area in, 313 distance in, 331–332 locating points in, 54 Correlation, 266 Corresponding angles, 127 Corresponding sides, 127 Cosine, 928 Counterexamples, AT18–AT19 Crash test dummies, 500 Create a table to evaluate expressions, 12–13 Critical Thinking Critical Thinking questions are found in every exercise set. Some examples: 11, 18, 23, 24, 30 Cross products in proportions, 121 solving rational equations by using, 920 Cross Products Property, 121 Cube roots, 32 Cubes, difference of, 584 Cubic binomials, 532 Cubic equations, 681–683 Cubic functions, 680–683 Cubic polynomials, 497 Culinary Arts major, 202 Cumulative Assessment, 70–71,162–163, 228–229, 292–293, 390–391, 452–453, 538–539, 602–603, 692–693, 782–783, 864–865, 938–939 Cumulative frequency, 711 Cylinders properties of, 894 surface area of, 520 volume of, 520, 661 IN4 Index D Data displaying, 700–704 distributions, 716–719 graphing, to choose a model, 813–814 organizing, 700–704 Data Analysis, Connecting Algebra to, 275, 360, 698–699, 732–733 Data Collection applications, 238, 320, 637 Data Mining major, 358 Death Valley National Park, 18 Decay, exponential, 805–808 Decorating, 125 Deductive reasoning, AT19 Degrees of monomials, 496 of polynomials, 496 of power functions, AT3 Denominators like, 905, 906 rationalizing, 842, 845 unlike, 905, 906 Density Property of Real Numbers, 37 Dependent events, 750–754 Dependent systems, 421 Dependent variables, 250, 251, 252, 253, 254 Descartes, Rene, 58 Design, 31 Devon Island, 10 Diagrams ladder, 544 mapping, 240 reading and interpreting, 697 tree, 760 using, 536–537 Difference(s) of cubes, 584 first, 610 second, 610 of two squares, 523, 580 Dimensional analysis, 121 Dimensions changing, 53, 129 of a matrix, 770 Direct variation, 336–339 Discontinuous functions, 878 Discount, 145 Discrete graphs, 235 Discriminant, 672 Quadratic Formula and, 670–675 Displaying data, 700–704 Distance Formula, 331–332 Distracters, recognizing, 388–389 Distributions, data, 716–719 Distributive Property, 47, 551 Diving, 24, 929 Diving Link, 24 Division long, 914–915 of polynomials, 912, 913–917 of radical expressions, 840–842 of rational expressions, 898–901 of real numbers, 20–22 of signed numbers, 20 solving equations by, 84–87 solving inequalities by, 182–184 by zero, 21–22 Division properties of exponents, 481–485 Division Property of Equality, 86 Division Property of Inequality, 182, 183 Domain(s), 240, 241, 242, 243, 244, 245, 246, 252, 253, 254, 255, 256, 257, 259, 260, 263, 264, 265, 839, 877 of linear functions, 303 of quadratic functions, 613 reasonable, 252, 253, 254, 255, 259, 265, 287, 288, 303, 308, 616, 873, 876, 881, 883, 884 of square-root functions, 823–824 Double-bar graphs, 701 Double-line graphs, 702 Dow Jones Industrial Average (DJIA), 17 Draw a diagram, PS2 Drum Corps International, 548 E Eagles, 891 Earned run average (ERA), 110 Earth Science, 88 Ecology, 271 Ecology Link, 271 Economics, 17, 88, 96, 103, 179, 271, 321, 407, 408, 414, 415, 430, 482, 817 Education, 24, 186, 200 Electricity, 624, 630, 875 Electricity Link, 844 Elimination, solving systems of linear equations by, 411–415 Ellipsis, 276 Employment, 148, see also Career Path Empty set, 102 End behavior, AT4 Engineering, 114, 179, 624, 630, 875 Engineering Link, 624 Entertainment, 23, 30, 37, 111, 125, 137, 194, 201, 216, 280, 304, 373, 402, 424, 469, 526, 660, 714, 756, 903, AT8, AT12 Entry of a matrix, 770 Environment, 340 Environmental Science, 124, 310, 328, 567 Environmental Sciences major, 567 Equality Power Property of, 846 properties of, 79, 86 Equally likely, 744 Equations absolute value, 112–114 cubic, 681–683 definition of, 77 finding slope from, 326 linear definition of, 302 point-slope form of, 351–355 slope-intercept form of, 344–347, 350 solving, by using a spreadsheet, 396 standard form of, 302 systems of, 420–423 literal, 108 model one-step, 76 with variables on both sides, 99 point-slope form, 351–355 quadratic definition of, 642 discriminant of, 672 related function of, 642 roots of, 648–649 solving by completing the square, 664, 674 by factoring, 650–653, 674 by graphing, 642–644, 674 by using square roots, 656–659, 674 by using the Quadratic Formula, 670–675 standard form of, 642 radical, 846–850 rational, 920–923, 926 slope-intercept form, 344–347, 350 solutions of, 77 solving by addition, 77–79 by division, 84–87 by elimination, 411–415 by graphing, 91, AT9 by multiplication, 84–87 multi-step, 92–95 by subtraction, 77–79 two-step, 92–95 by using cross products, 920 with variables on both sides, 100–103 standard form, 302, 642 systems of classification of, 421–422 identifying solutions of, 397 with infinitely many solutions, 421 modeling, 403 with no solution, 420–421 solving by elimination, 411–415 by graphing, 397–399 by substitution, 404–407 for trend lines, 360 Equivalent ratios, 120 Equivalents, common, 133 Error Analysis, 18, 58, 80, 88, 125, 142, 174, 186, 201, 245, 254, 261, 272, 328, 357, 416, 425, 434, 439, 464, 479, 492, 501, 508, 526, 557, 573, 583, 590, 630, 654, 660, 668, 713, 730, 748, 810, 818, 838, 852, 884, 903, 910, 918 Escape velocity, 826 Estimating cube roots, 33 with percents, 140 solutions using the Quadratic Formula, 671 square roots, 33 Estimation, 30, 50, 110, 136, 148, 180, 187, 254, 262, 272, 311, 320, 334, 366, 410, 440, 479, 519, 565, 590, 616, 660, 721, 741, 811, 826, 852, 876, 918 Evaluating exponential functions, 796–797 expressions, 7, 12–13 factored and polynomial forms, 563 functions, 251 Events compound, 758–759, 761 definition of, 737 dependent, 750–754 inclusive, 758 independent, 750–754 mutually exclusive, 758 simple, 761 Exam preparation, 869 Excluded values, 868, 878 Experimental probability, 737–739 Experiments, 737 Explanations convincing, 395 for your reasoning in extended responses, 690–691 Exploring axis of symmetry of a parabola, 618 constant changes, 322–323 properties of exponents, 472–473 roots, zeros, and x-intercepts, 648–649 Exponent(s) definition of, 26 division properties of, 481–485 integer, 460–462 and powers of ten, 466–467 in scientific notation, 467–468 using patterns to investigate, 460 multiplication properties of, 474–477 negative, 460, 466 powers and, 26–28 properties of, 472–473 rational, 488–490 reading, 27 zero, 460 Exponential decay, 805–808 Exponential expressions, simplifying, 474 Exponential functions, 796–799 definition of, 796 evaluating, 796–797 general form of, 815 graphs of, 797–799, AT9 Exponential growth, 805–808 Exponential models, 813–816 Expressions algebraic, 6, 7, 8, 10, 40–42, 72, 248 create a table to evaluate, 12–13 exponential, simplifying, 474 numerical, 6 radical, see Radical expressions rational, see Rational expressions simplifying, 46–49 square-root, 829, 841 variables and, 6–8 Extended Response, 71, 163, 229, 246, 290–291, 293, 306, 391, 450, 451, 453, 509, 539, 549, 603, 677, 690–691, 693, 783, 865, 911, 939 Explain Your Reasoning, 690–691 Understand the Scores, 290–291 Extension Absolute-Value Functions, 378–381 Cubic Functions and Equations, 680–683 Matrices, 770–773 Trigonometric Ratios, 928–929 Exterior angles, AT21 Extra Practice, S4–S39 Extraneous solutions, 848–849, 922 F Factor(s) common, 545 definition of, 544 greatest common, 545–546 prime factorization, 544 Factorial, 762 Factoring polynomials choosing a method, 586–588 common binomial factors in, 553 factored form, 551 factoring ax 2 + bx + c, 568–571 factoring perfect-square trinomials, 578–579, 585 factoring the difference of two squares, 580–581, 585 factoring x 2 + bx + c, 560–563 by graphing, 575 by greatest common factor, 551–554, 587 by grouping, 553–554, 588 by guess and check, 560, 568 knowing when an expression is fully factored, 586 Index IN5 modeling, 550, 558–559 by multiple methods, 587–588 with opposite binomials, 554 recognizing special products, 585 in simplifying rational expressions, 886–888 solving quadratic equations by, 650–653, 674 steps in, 586 trinomial constant as product of binomial constants, 560–562 unfactorable polynomials, 587 Factorization, prime, 544 Factor tree, 544 Fair experiments, 744 Families of functions definition of, 369 of linear functions, 368, 882 of quadratic functions, 632, 882 of rational functions, 882 of square-root functions, 882 Farming, 439 Fibonacci sequence, 280 Figures composite, 83 reading and interpreting, 697 similar, 127 solid, 894 Final exam preparation, 869 Finance, 80, 88, 109, 124, 139, 143, 409, 555, 806, 819, 884 Find a pattern, PS6 Finding a term of a geometric sequence, 790–792 Finding a term of an arithmetic sequence, 276–278 First coordinates, 240 First differences, 610 First quartile, 718 Fitness, 79, 349, 401, 838, 910 Flying fish, AT4 FOIL method, 513, 560, 841 Formula(s), 107 area of composite figures, 83 in the coordinate plane, 313 of a rectangle, 83 of a square, 83 surface, 493, 520 of a trapezoid, 335, 654 of a triangle, 83 axis of symmetry of a parabola, 621 combinations, 763 compound interest, 806 distance, 331–332 distance from a light source, 492 experimental probability, 738 exponential decay, 807 exponential growth, 805 half-life, 807 Heron’s, 834 midpoint, 330–331 period of a pendulum, 660 IN6 Index permutations, 762 probability of dependent events, 753 probability of independent events, 751 Quadratic, 670 recursive, AT13 remembering, 789 simple interest, 139 slope, 324 solving for a variable, 108 speed of light, 494 surface area of a sphere, 493 term of a geometric sequence, 790 term of an arithmetic sequence, 276–277 theoretical probability, 744 volume of a cylinder, 661 Foundation plan, 895 Fraction(s) complex, 904 as exponents, 488–490 finding roots of, 33 negative, 352 Frequency, 709–711 cumulative, 711 Frequency table, 710 Function(s) absolute-value, 378–381, AT5 cubic, 680–683 definition of, 241 degree of, 496, AT3 discontinuous, 878 end behavior of, AT4 evaluating, 251 exponential, 796–799, 815, AT9 families of, 369, 632, 882 general forms of, 815 graphing, 256–260 greatest-integer, AT7 identifying, AT10–AT12 introduction to, 54–56 linear definition of, 300, AT15 family of, 368, 882 general form of, 815 graphing, 302 identifying, 300–303 reflections of, 371 rotations of, 370 transformations of, 369–372 vertical translations of, 369 parent definition of, 369 linear, 369, 882 quadratic, 632, 633, 882 rational, 882 square-root, 882 piecewise, AT5–AT7 power, AT2–AT4 quadratic characteristics of, 619–623 comparing graphs of, 635 definition of, 610 domain of 613 in families of functions, 632, 882 finding zeros of, 619 general form of, 815 graphing, 626–629, AT9 using a table of values, 611 identifying, 610–613 range of, 613 transformations of, 633–636 radical, 828 rational, 878–882, 893 relations and, 240–242 square-root, 822–824, 882 step, AT6 writing, 249–252 zeros of, 619 Function notation, 250 Function rules, 250, 263 Function table, 55–56, 263 Fundamental Counting Principle, 760 Fund-raising, 269 G Galileo Galilei, 255 Gallium, 209 Games, 653 Gardening, 33, 124, 192, 573 GCF (greatest common factor), 545-546 factoring by, 551–554 Gemology, 881 General forms of functions, 815 Geodes, 424 Geography, 19, 332, 478, 486 Geology, 80, 81, 424, 826 Geology Link, 81, 424, 826 Geometric models of powers, 26 of special products, 521, 523 Geometric probability, 910 Geometric sequences, 790–792 Geometry, see also Applications angles central, 703 corresponding, 127 exterior, AT21 remote interior, AT21 annulus, 557 area of composite figures, 83 in the coordinate plane, 313 of a rectangle, 83 of a square, 83 surface, 493, 520 of a trapezoid, 335, 654 of a triangle, 83 changing dimensions, 53, 129 circle, diameter of, 334 cones properties of, 894 surface area of, 520 volume of, 520 Connecting Algebra to, 52–53, 83, 211, 313, 520, 894 coordinate plane, 54 area in the, 313 distance in the, 331–332 corresponding sides, 127 cosine, 928 cylinders, 520, 894 dimensions, changing, 53, 129 foundation plan, 895 geometric models of powers, 26 of special products, 521, 523 geometric probability, 910 half-plane, 428 Heron’s formula, 834 indirect measurement, 128 nets, 894 perimeter, 52–53 planes Cartesian, 58 coordinate, 54 polygons, 52–53 prisms, 520, 894 pyramids, 520, 894 Pythagorean Theorem, 331, 661, 831 rectangles, area of, 83 reflections, 371 rotations, 370 scale, 122 scale drawing, 122 scale factor, 129 scale model, 122 sectors, 702 similar figures, 127 sine, 928 slopes of lines, 361–364 surface-area-to-volume ratio, 889 tangent, 928 transformations of absolute-value functions, 379–381 of linear functions, 369–372 of quadratic functions, 633–636 translations, 369, 823 Triangle Inequality, 211 trigonometric ratios, 928–929 volume, 520 Get Organized, see Graphic organizers go.hrw.com, see Online Resources Graph(s), see also Graphing bar, 698–699, 700 circle, 698–699, 702 comparing, of quadratic functions, 635 connecting to function rules and tables, 263 continuous, 235 discrete, 235 double-bar, 701 double-line, 702 finding slope from, 325 finding zeros of quadratic functions from, 619 identifying linear functions by, 300 line, 701–704 misleading, 726–727 reading and interpreting, 697 turning points on, 680 Graphic Organizers Graphic Organizers are found in every lesson. Some examples: 8, 17, 22, 28, 35 Graphics, reading and interpreting, 697 Graphing, see also Graph(s) absolute-value functions, 379–381 data to choose a model, 813–814 exponential functions, 799, AT9 to factor polynomials, 575 functions, 256–260 greatest-integer functions, AT7 inequalities, 170–172 linear functions, 302 linear inequalities, 429 midpoints and endpoints, 330–331 piecewise functions, AT5–AT7 point-slope form, 352–353 power functions, AT2–AT3 quadratic functions, 611, 626–629, AT9 radical functions, 828 rational functions, 880, 893 relationships, 234–236 slope-intercept form, 344–347, 350 solving equations by, 91 solving quadratic equations by, 642–644, 674 solving systems of linear equations by, 397–399 square-root functions, translations of, 823 step functions, AT6 Graphing Calculator, 12, 16, 91, 263, 274, 368, 370, 371, 398, 401, 441, 575, 617, 625, 632, 634, 635, 636, 637, 639, 643, 644, 645, 646, 647, 648, 649, 664, 668, 671, 674, 724–725, 743, 763, 799, 801, 811, 825, 826, 853, 885, 918, AT9 Greatest common factor (GCF), 545–546 factoring by, 551–554 Greatest-integer function, AT7 Gridded Response, 45, 68–69, 71, 98, 105, 132, 149, 163, 195, 229, 255, 293, 310, 329, 341, 350, 367, 391, 401, 451, 453, 465, 539, 584, 601, 603, 625, 639, 693, 783, 827, 865, 877, 892, 939 Fill in Answer Grids Correctly, 68–69 Griffith-Joyner, Florence, 125 Grouping, factoring by, 553–554, 588 Grouping symbols, 40, 41 Growth exponential, 805–808 modeling, 804 Guess and test, PS4 H Half-life, 807 Half-plane, 428 Hamm, Paul, 44 Handball team, 518 Health, 179, 180, 244, 326, 469 Health Link, 180 Helpful Hint, 8, 16, 21, 40, 41, 46, 47, 92, 94, 100, 112, 121, 128, 129, 134, 140, 144, 145, 176, 196, 198, 204, 207, 212, 235, 242, 250, 256, 257, 266, 309, 346, 352, 363, 364, 397, 398, 404, 405, 412, 413, 422, 429, 481, 484, 488, 490, 514, 515, 545, 554, 562, 579, 587, 611, 619, 627, 635, 643, 651, 652, 657, 664, 671, 673, 710, 738, 762, 763, 791, 796, 805, 807, 815, 822, 824, 830, 835, 840, 842, 849, 872, 914, 915, 922, 928, AT10 Heron of Alexandria, 834 Heron’s formula, 834 Hiking, 773 Histograms, 710 frequency and, 709–711 History, 97, 356, 731, 765 History Link, 97, 731, 765 Hobbies, 130, 374, 433, 667, 675, AT17 Hobbies Link, 374 Home Economics, 876 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 23, 29, 35 Horizontal lines, 363, 879 Hot-air balloons, 20, 22 Hot Tip!, 67, 69, 71, 159, 161, 163, 225, 227, 229, 289, 291, 293, 387, 389, 391, 449, 451, 453, 535, 537, 539, 599, 601, 603, 689, 691, 693, 779, 781, 783, 861, 863, 865, 935, 937, 939 Hurricanes, 851 I Identifying asymptotes, 879 Identities equations as, 101 Identity additive, 15 multiplicative, 21 Inclusive events, 758 Inconsistent systems, 420–421 Independent events, 750–754 Index IN7 Independent systems, 421 Independent variables, 250–254 Index, in roots, 32, 488 Indirect measurement, 128 Inductive reasoning, AT18–AT19 Inequalities absolute-value, 212–215 compound, 204–207, 212 definition of, 170 graphing, 170–172, 429 linear definition of, 428 graphing, 429 solutions of, 428 solving, 428–431 systems of, 435–437, 441 properties of, 176, 182, 183 solutions of, 170 solving by addition, 176–179 compound, 204–207 by division, 182–184 by multiplication, 182–184 multi-step, 190–192 by subtraction, 176–179 two-step, 190–192 with variables on both sides, 196–199 triangle, 211 writing, 170–172 Input, 55, 249 Input-output table, 55–56 Integer exponents, 460–462 and powers of ten, 466–467 in scientific notation, 467–468 using patterns to investigate, 460 Integers, 33 Intercepts, 307–309, 353, 626–628 Interest compound, 806 simple, 139 Interpreting graphics, 697 scatter plots and trend lines, 274, 360 Interquartile range (IQR), 718 Intersection, 205 Inverse operations, 77, 84, 92, 100, 107, 176 Inverse Property of Addition, 15 Inverse Property of Multiplication, 21 Inverses additive, 15 multiplicative, 21 squaring and square roots, 32 Inverse variation, 871–874, 896 IQR (interquartile range), 718 Irrational numbers, 34 Ishtar Gate, 526 Isolating variables, 77 IN8 Index K Kangaroos, 646 Key words, 234 King, Martin Luther, Jr., 97 Kites, 200 Know-It Note Know-It Notes are found throughout this book. Some examples: 15, 20, 21, 40, 46 Koopa (turtle), 142 L Ladder diagram, 544 Landscaping, 401 Landscaping Link, 401 Law Enforcement, 825 LCD (least common denominator), 907, 920–921 LCM (least common multiple), 906–907 Leading coefficients of polynomials, 497 Leaning Tower of Pisa, 630 Least common denominator (LCD), 907, 920–921 Least common multiple (LCM), 906–907 Leonardo da Vinci, 624 Light, speed of, 494 Light-year, 469 Like denominators, 905, 906 Likely, equally, 744 Like radicals, 835 Like terms, 47 Lincoln, Abraham, 97 Line(s) coincident, 421 horizontal, 363, 879 median-fit, 275 parallel, 361–364 perpendicular, 361–364 slope of a, 315–317 trend finding equations for, 360 interpreting, 274, 360 scatter plots and, 266–269, 360 vertical, 361, 879 Linear equations definition of, 302 point-slope form of, 351–355 slope-intercept form of, 344–347, 350 solving, by using a spreadsheet, 396 standard form of, 302 systems of, 420–423 Linear functions definition of, 300, AT15 family of, 368, 882 general form of, 815 graphing, 302 identifying, 300–303 reflections of, 371 rotations of, 370 transformations of, 369–372 vertical translations of, 369 Linear inequalities definition of, 428 graphing, 429 solutions of, 428 solving, 428–431 systems of, 435–437, 441 Linear models, 360, 813–816 Linear systems, 420–423 Line graphs, 701–704 Link Animals, 186 Art, 565 Astronomy, 10, 340 Automobiles, 713 Biology, 105, 311, 464, 646, 891 Chemistry, 209, 470 Diving, 24 Ecology, 271 Electricity, 844 Engineering, 624 Geology, 81, 424, 826 Health, 180 History, 97, 731, 765 Hobbies, 374 Landscaping, 401 Math History, 58, 416, 526, 590, 834 Meteorology, 851 Military, 439 Music, 548 Number Theory, 280 Recreation, 200 School, 755 Science, 356 Solar Energy, 918 Sports, 44, 125, 238, 518 Statistics, 88 Technology, 142, 801 Transportation, 254, 500 Travel, 319, 630, 910 Winter Sports, 876 Lists of ordered pairs identifying exponential functions by, 797, 814 identifying linear functions by, 301, 814 identifying quadratic functions by, 610, 814 Literal equations, 108 Logic, 731, AT21 Long division, polynomial 914–915 Lookout Mountain Incline Railway, 319 Lower quartile, see First quartile Luminosity, 492 M Magnification, 926 Make a Conjecture, 322, 323, 632, 648, 662, 828, 870, 893, AT18 Make a model, PS3 Make an organized list, PS11 Make a table, PS7 Manufacturing, 115, 124, 461 Mapping diagrams, 240 Maps, 122 Marine Biology, 616 Markup, 145 Mars lander, 340 Math History, 255, 416, 526, 590, 834 Math History Link, 58, 416, 526, 590, 834 Math symbols, 233 Matrices, 770–773 Maximum values of absolute-value functions, 380 of parabolas, 612 of power functions, AT3 Mean, 716, 717 Measurement applications, 128, 305, 355, 470, 582, 794 indirect, 128 Measures of central tendency, 716–719 Mechanics, 875, 877 Median, 716, 717 Median-fit line, 275 Medicine, 463, 501 Mental Math, 46, 47, 140, 161, 585 Meteorology, 10, 17, 110, 208, 209, 356, 851 Meteorology Link, 851 Middleton Place Gardens, 401 Midpoint formula, 330–331 Military, 439 Military Link, 439 Minimum values of absolute-value functions, 380 of parabolas, 612 of power functions, AT3 Misleading graphs and statistics, 726–727 Mode, 716, 717 Model(s) choosing, 813–814 exponential, 813–816 geometric of powers, 26 of special products, 521, 523 linear, 360, 813–816 quadratic, 813–816 rectangle, for multiplying polynomials, 514 Modeling addition and subtraction of real numbers, 14 completing the square, 662 equations with variables on both sides, 99 factoring, 550, 558–559 growth and decay, 804 inverse variation, 870 one-step equations, 76 polynomial addition and subtraction, 502–503 polynomial division, 912 polynomial multiplication, 510–511 systems of linear equations, 403 variable relationships, 248 Money, 408 Monomials, 496 Moore’s law, 801 Multiple Choice, 70–71, 160–161, 162–163, 292–293, 390–391, 451, 452–453, 536, 537, 538–539, 601, 602–603, 692–693, 781, 782–783, 862–863, 864–865, 936–937, 938–939 Choose Combinations of Answers, 936–937 Eliminate Answer Choices, 160–161 None of the Above or All of the Above, 862–863 Recognize Distracters, 388–389 Multiple representations, 15, 21, 26, 27, 46, 47, 76, 79, 86, 99, 101, 171, 176, 182, 183, 198, 204, 205, 206, 240, 248, 263, 299, 324, 344, 361, 363, 403, 460, 466, 474, 476, 477, 481, 483, 484, 502, 503, 510, 511, 521, 523, 550, 558, 559, 561, 578, 580, 612, 620, 621, 633, 648, 649, 650, 656, 662, 663, 762, 763, 822, 830, 846, 871, 879, 913 Multiplication of polynomials, 512–516 by powers of ten, 467 properties of, 46 of radical expressions, 840–842 of rational expressions, 898–901 of real numbers, 20–22 scalar, of matrices, 771 of signed numbers, 20 solving equations by, 84–87 solving inequalities by, 182–184 of square-root expressions containing two terms, 841 by zero, 21–22 Multiplication properties of exponents, 474–477 Multiplication Property of Equality, 86 Multiplication Property of Inequality, 182, 183 Multiplicative identity, 21 Multiplicative inverse, 21 Multi-Step, 11, 57, 59, 115, 132, 137, 143, 148, 149, 216, 243, 244, 279, 334, 357, 400, 409, 415, 424, 432, 439, 465, 480, 494, 518, 526, 548, 549, 583, 617, 629, 637, 654, 660, 667, 668, 676, 794, 801, 811, 817, 826, 827, 833, 839, 852, 876, 883, 910, 918, 924 Multi-step equations, solving, 92–95 Multi-step inequalities, solving, 190–192 Multi-Step Test Prep, 38, 60, 118, 152, 188, 218, 264, 282, 342, 376, 426, 442, 494, 528, 576, 592, 640, 678, 734, 768, 820, 854, 896, 926 Multi-Step Test Prep questions are also found in every exercise set. Some examples: 10, 18, 24, 30, 36 Music, 208, 216, 218, 516, 548, 873 Music Link, 548 Mutually exclusive events, 758 N Natural numbers, 33 Navigation, 929 Negative correlation, 267 Negative exponents, 460 Negative Power of a Quotient Property, 484 Negative slope, 316 Nets, 894 Nightingale, Florence, 731 No correlation, 267 None of the Above, 862–863 Nonlinear equations, graphing to solve, AT9 Nonlinear functions, AT15–AT16 Notation, scientific, 467–468 Null set, 102 Index IN9 Numbers compatible, 46 irrational, 34 natural, 33 prime, 544 random, 743 rational, 33 real, see Real numbers signed, 20 whole, 33 Number Sense, 548, 549 Number Theory, 280, 654, 660 Connecting Algebra to, 418–419, 585 Number Theory Link, 280 Numerical expressions, 6 Nutrition, 87, 88, 136, 148, 215, 244 O Oceanography, 121, 260 Ocelots, 271 Odds, 746 Okeechobee, Lake, 332 Online Resources Career Resources Online, 106, 202, 402, 567, 767 Chapter Project Online, 2, 72, 166, 230, 296, 392, 456, 540, 606, 694, 786, 866 Homework Help Online Homework Help Online is available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 23, 29, 35 Lab Resources Online, 12, 76, 99, 150, 263, 274, 396, 403, 502, 575, 632, 648 Parent Resources Online Parent Resources Online are available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 23, 29, 35 State Test Practice Online, 70, 162, 228, 292, 390, 452, 538, 602, 692, 782, 864, 938 Operations inverse, 77, 84, 92, 100, 107, 176 order of, 40–42 Opposite binomials, 554, 888 factoring with, 554 Opposite coefficients, 411 Opposites, 15 Orangutans, 186 Ordered pairs definition of, 54 identifying exponential functions by using, 610, 814 IN10 Index identifying linear functions by using, 301, 814 identifying quadratic functions by using, 797, 814 showing relations by, 240 Order of operations, 40–42 Organizing data, 700–704 Origin, 54 Outcome, 737 Outliers, 716–717, 725 Output, 55, 249 P Paella, 336 Parabola(s) axis of symmetry of a, 620–621 definition of, 611 exploring, 618 identifying the direction of a, 612 vertex of a, 612, 621 vertical translations of a, 635 width of a, 633 Parallel lines, slopes of, 361–364 Parent functions definition of, 369 linear, 369, 882 quadratic, 632, 633, 882 rational, 882 square-root, 882 Parentheses, 40 Parent Resources Online Parent Resources Online are available for every lesson. Refer to the go.hrw.com box at the beginning of each exercise set. Some examples: 9, 17, 23, 29, 35 Pascal, Blaise, 590 Pascal’s Triangle, 590 Patterns in choosing a model, 814 in finding properties of exponents, 472–473 identifying, AT10–AT11 in investigating integer exponents, 460 in investigating powers of ten, 466 recursive, AT10–AT11 Pearl, Nancy, 374 Pendulum, period of a, 660 Percent(s), 133–135 applications of, 139–141 Percent change, 144–146 Percent proportion, 133 Perfect squares, 32, 33 Perfect-square trinomials, 521, 578 Perimeter, 52–53 Period, of a pendulum, 844 Permutations, 760–763 Perpendicular lines, slopes of, 361–364 Personal Finance, 311, 551, 812 Pet Care, 819, AT8 pH, 486 Photography, 507, 517 Physical Science, 306, 479, 793, AT16 Physics, 498, 555, 573, 589, 636, 638, 654, 660, 673, 800, 844, 851, 874 Piecewise functions, AT5–AT7 Pimlico Race Course, 238 Plane(s) Cartesian, 58 coordinate, 54 Point-slope form of linear equations, 351–355 Polygons, 52–53 Polynomial(s), see also Factoring polynomials addition of, 504–506 cubic, 497, 532 degrees of, 496–497 division of, 913–917 leading coefficients of, 497 long division, 914–915 multiplication of, 512–516 quadratic, 497 standard form of, 497, 916 subtraction of, 502–506 unfactorable, 587 Population, 126 Population density, 486 Positive correlation, 267 Positive Power of a Quotient Property, 483 Positive slope, 316 Power(s) definition of, 26 exponents and, 26–28 geometric models of, 26 Negative, of a Quotient Property, 484 Positive, of a Quotient Property, 483 of a Power Property, 476 of a Product Property, 477 of ten, 466–467 Power functions, AT2–AT4 Power of a Power Property, 476 Power of a Product Property, 477 Power Property of Equality, 846 Powers of ten, 466–467 Powers Property Power of, 476 Product of, 474–475 Quotient of, 481, 898 Prediction, 739 Preparing for your final exam, 869 Prime factorization, 544 Prime numbers, 544 Principal, 139 Principal square root, 32 Prisms, 520, 894 Probability applications, 901, 902 converting between odds and, 746 definition of, 737 of dependent events, 753 experimental, 737–739 geometric, 910 of inclusive events, 758 of independent events, 751 of mutually exclusive events, 758 theoretical, 744–747 Problem-Solving Applications, 28, 94, 177–178, 259, 354, 399, 523–524, 579–580, 627–628, 665–666, 753, 815–816, 921–922 Problem-Solving Handbook, PS2–PS11 Draw a diagram, PS2 Find a pattern, PS6 Guess and test, PS4 Make a model, PS3 Make an organized list, PS11 Make a table, PS7 Solve a simpler problem, PS8 Use logical reasoning, PS9 Use a Venn diagram, PS10 Work backward, PS5 Product of Powers Property, 474–475 Product Property Power of a, 476 of Square Roots, 830 Zero, 886 Product Rule for Inverse Variation, 873 Properties of addition, 46 of equality, 79, 86 of inequality, 176, 182, 183 of multiplication, 46 of zero, 21 Proportions applications of, 127–129 cross products in, 121 definition of, 120 percent, 133 rates, ratios and, 120–123 Pyramids, 520, 894 Pythagorean Theorem, 331, 661, 831 Q Qin Jiushao, 416 Quadrants, 54 Quadratic equations definition of, 642 discriminant of, 672 related function of, 642 roots of, 648–649 solving by completing the square, 664, 674 by factoring, 650–653, 674 by graphing, 642–644, 674 by using square roots, 656–659, 674 by using the Quadratic Formula, 670–675 standard form of, 642 Quadratic Formula discriminant and, 670–675 in estimating solutions, 671–672 solving quadratic equations by, 670–675 Quadratic functions characteristics of, 619–623 comparing graphs of, 635 definition of, 610 domain of, 613 in families of functions, 632, 882 finding zeros of, 619 general form of, 815 graphing, 626–629, AT9 using a table of values, 611 identifying, 610–613 range of, 613 transformations of, 633–636 Quadratic models, 813–816 Quadratic parent functions, 632, 633, 882 Quadratic polynomials, 497 Quality Control, 739 Quotient of Powers Property, 481, 898 Quotient Property Negative Power of a, 484 Positive Power of a, 483 of Square Roots, 830 R Radical equations, 846–850 Radical expressions, 829–832 addition of, 835–837 definition of, 829 division of, 840–842 multiplication of, 840–842 Product Property of, 830–831 Quotient Property of, 830–831 square-root expressions, 829–831, 841 subtraction of, 835–837 Radical functions, graphing, 828 Radicals, like, 835 Radical symbol, 32, 488, 490 Radicand, 829 Rainbows, 494 Random numbers, 743 Random samples, 727, 732 Range, 82, 240, 241, 242, 243, 244, 245, 246, 252, 253, 254, 255, 260, 264, 265, 877 of linear functions, 303 of quadratic functions, 613 reasonable, 252, 253, 254, 255, 259, 265, 287, 288, 303, 308, 372, 617, 873, 876, 881, 883 Range (of a data set), 716, 717 Rate of change constant and variable, AT14–AT15 decrease, 803 definition of, 314, AT14 identifying linear and nonlinear functions from, AT15–AT16 increase, 803 slope and, 314–317 Rates, 120–123 Ratio(s) equivalent, 120 rates, proportions and, 120–123 surface-area-to-volume, 889 trigonometric, 928–929 Rational equations, 920–923, 926 Rational exponents, 488–490 Rational expressions addition of, 905–908 definition of, 886 division of, 898–901 multiplication of, 898–901 simplifying, 886–889 subtraction of, 905–908 Rational functions, 878–882 definition of, 878 excluded values in, 878 family of, 882 graphing, 880–881, 893 identifying asymptotes, 878–880 Rationalizing denominators, 841, 845 Rational numbers, 33 Reading graphics, 697 the problem, 459 Reading and Writing Math, 5, 75, 169, 233, 299, 395, 459, 543, 609, 697, 789, 869, see also Reading Strategies; Study Strategies; Writing Strategies Reading Math, 34, 54, 120, 122, 127, 172, 251, 276, 324, 422, 460, 467, 468, 581, 656, 702, 718, 745, 746, 806, 874 Reading Strategies Read a Lesson for Understanding, 543 Read and Interpret Graphics, 697 Read and Interpret Math Symbols, 233 Read and Understand the Problem, 459 Use Your Book for Success, 5 Index IN11 Ready to Go On?, 39, 61, 119, 153, 189, 219, 265, 283, 343, 377, 427, 443, 495, 529, 577, 593, 641, 679, 735, 769, 821, 855, 897, 927, see also Assessment Real Estate, 131 Real numbers addition of, 14–17 classification of, 33 definition of, 14 Density Property of, 37 division of, 20–22 integers, 33, 34 irrational, 34 multiplication of, 20–22 natural numbers, 33, 34 rational, 33, 34 square roots and, 32–35 subtraction of, 14–17 whole numbers, 33, 34 Real-World Connections, 164–165, 294–295, 454–455, 604–605, 784–785, 940–941 Reasonable answer, 79, 80, 81, 82, 88, 89, 95, 97, 104, 121, 123, 124, 126, 128, 130, 134, 137, 142, 144, 145, 148, 149, 161, 172, 178, 184, 235, 259, 399, 400, 430, 437, 636, 643, 666, 674, 796, 805, 807, 822 Reasonable domain, 252, 253, 254, 255, 259, 265, 287, 288, 303, 308, 372, 617, 873, 876, 881, 883 Reasonableness, 79, 80, 81, 82, 88, 89, 95, 97, 104, 123, 124, 126, 128, 130, 134, 137, 142, 144, 145, 148, 149, 161, 254, 255, 259, 265, 269, 643, 822 Reasonable range, 252, 253, 254, 255, 259, 265, 287, 288, 303, 308, 372, 617, 873, 876, 881, 883 Reasoning deductive, AT19 explaining your, in extended responses, 690–691 inductive, AT18–AT19 spatial, 780–781 Reciprocals, 21 Recognizing distracters, 388–389 Recreation, 22, 87, 116, 200, 238, 244, 280, 321, 333, 334–335, 408, 439, 832, 908 Recreation Link, 200 Rectangle model for multiplying polynomials, 514 Rectangles, area of, 83 Recursive formula, AT13 Recursive patterns, AT10–AT11 Recycling, 8 Reflections, 371 Relations, functions and, 240–242 Relationships graphing, 234–236 variable, model, 248 IN12 Index Remember!, 42, 108, 113, 190, 205, 214, 302, 303, 362, 369, 412, 435, 460, 475, 488, 496, 497, 504, 512, 524, 544, 560, 569, 627, 633, 658, 665, 670, 682, 700, 761, 814, 836, 841, 871, 873, 881, 886, 889, 898, 899, 901, 907, 916 Remembering formulas, 789 Remodeling, 565 Remote interior angles, AT21 Repeating decimals, 33 Replacement set, 8 Representations multiple, 15, 21, 26, 27, 46, 47, 76, 79, 86, 99, 101, 171, 176, 182, 183, 198, 204, 205, 206, 240 248, 263, 299, 324, 344, 361, 363, 403, 460, 466, 474, 476, 477, 481, 483, 484, 502, 503, 510, 511, 521, 523, 550, 558, 559, 561, 578, 580, 612, 620, 621, 633, 648, 649, 650, 656, 662, 663, 762, 763, 822, 830, 846, 871, 879, 913 of solid figures, 894 Rise, 315 Root(s), see also Radical expressions cube, 32 exploring, 648–649 of fractions, 33 principal square, 32 of quadratic equations, 648–649 square, 32 symbol for, 32 Rotations, 370 Run, 315 S Safety, 214 Sales tax, 140 Samples, random, 727, 732 Sample space, 737 Sandia Peak Tramway, 308 Scalar, 771 Scale, 122 Scale drawing, 122 Scale factor, 129 Scale model, 122 Scatter plots interpreting, 274 trend lines and, 266–269, 360 School, 81, 199, 374, 400, 433, 755 School Link, 755 Science, 124, 356, 478, 552, 808 Science Link, 356 Scientific notation, 467–468 Seabiscuit, 238 Sea horses, 121 Second coordinates, 240 Second differences, 610 Sectors, 702 Selected Answers, SA2–SA30 Sequences arithmetic, 276–278 definition of, 276 geometric, 790–792 recursive, AT10–AT11 Set-builder notation, 170 Sheppard, Alfred, 124 Shipping, 279 Short Response, 25, 71, 82, 90, 143, 163, 187, 202, 226–227, 229, 239, 273, 293, 335, 350, 391, 417, 440, 451, 453, 465, 501, 536, 537, 539, 566, 591, 600, 601, 603, 617, 631, 669, 693, 708, 723, 742, 766, 781, 783, 811, 865, 885, 904, 939 Understand Short Response Scores, 226–227 Signed numbers, multiplication and division of, 20 Silicon chips, 801 Similar figures, 127 Simple events, 761 Simple interest, 139 Simple random sample, 732 Simplest form of a square-root expression, 829 Simplifying exponential expressions, 474 expressions, 46–49 rational expressions, 886–889 Simulations, 736 Sine, 928 Slope(s) comparing, 317 defined, 315 finding, 325–326 formula, 324–327 negative, 316 of parallel lines, 361–364 of perpendicular lines, 361–364 positive, 316 rate of change and, 314–317 of trend lines, 360 undefined, 316 zero, 316 Slope formula, 324–327 Slope-intercept form of linear equations, 344–347, 350 Snowshoes, 876 Solar cars, 713 Solar Energy, 918 Solar Energy Link, 918 Solar-powered aircraft, 918 Solid figures, representing, 894 Solutions of absolute-value equations, 112–114 of absolute-value inequalities, 212–215 approximating, 91 of equations, 77 estimating using the Quadratic Formula, 671–672 extraneous, 848–849, 922 of inequalities, 170 of linear equations by using a spreadsheet, 396 of linear inequalities, 428 of rational equations, 920–921 of systems of linear equations, 397 of systems of linear inequalities, 435, 441 Solution set, 77, 102 Solve a simpler problem, PS8 Space Shuttle, 115 Spatial reasoning, 780–781 Special products of binomials, 521–525 factoring, 578–581 geometric models of, 521, 523 Special systems, solving, 420–423 Speed of light, 494 Speed squares, 534 Spheres surface area, 493 volume, 493, AT4 Sports, 42, 44, 50, 104, 107, 110, 121, 125, 172, 178, 180, 209, 238, 310, 335, 348, 381, 424, 485, 518, 529, 616, 645, 652, 654, 676, 712, 718, 721, 740, 741, 792, 793, 800, 831 Sports Link, 44, 125, 238, 518 Spreadsheet, 13, 150–151, 396 Square(s) difference of two, 523, 580 perfect, 32 Square, area of a, 83 Square, completing the, 663–666, 674 Square root(s) definition of, 32 estimating, 33 of fractions, 33 irrational, 34 principal, 32 Product Property of, 830 Quotient Property of, 830 real numbers and, 32–35 solving quadratic equations by using, 656–659, 674 Square-root expressions, see also Radical expressions multiplication of, 841 simplest form of, 829 Square-root functions, 822–824 domain of, 823 family of, 882 graphs of translations of, 823 Square-Root Property, 656 Standard form of linear equations, 302 of polynomials, 497, 916 of quadratic equations, 642 Standardized Test Prep, 70–71, 162–163, 228–229, 292–293, 390–391, 452–453, 538–539, 602–603, 692–693, 782–783, 864–865, 938–939, see also Assessment Statistics applications, 82, 88, 469, 799 measures of central tendency, 716–719 misleading, 726–727 Statistics Link, 88 Stem-and-leaf plot, 709, 715 Step functions, AT6 Stonehenge II, 124 Stratified random sample, 732 Student to Student, 47, 76, 171, 242, 308, 406, 476, 571, 674, 751, 816, 888 Study Guide: Preview, 4, 74, 168, 232, 298, 394, 458, 542, 608, 696, 788, 868, see also Assessment Study Guide: Review, 62–65, 154–157, 220–223, 284–287, 382–385, 444–447, 530–533, 594–597, 684–687, 774–777, 856–859, 930–933, see also Assessment Study Strategies Learn Vocabulary, 609 Prepare for Your Final Exam, 869 Remember Formulas, 789 Use Multiple Representations, 299 Use Your Notes Effectively, 169 Use Your Own Words, 75 Substitution, solving systems of linear equations by, 404–407 Subtraction of matrices, 771 of polynomials, 502–503, 504–506 of radical expressions, 835–837 of rational expressions, 905–908 of real numbers, 14–17 solving equations by, 77–79 solving inequalities by, 176–179 Subtraction Property of Equality, 79, 86 Subtraction Property of Inequality, 176 Surface area, 493, 520 Surface-area-to-volume ratio, 889 Sydney Harbour Bridge, 114 Symmetric Property of Equality, 187 Symmetry, axis of in absolute-value functions, 378 in a parabola, 618, 620 definition of, 618 exploring, 618 finding by using the formula, 621 by using zeros, 620 System of linear inequalities, 435–437 System(s) of linear equations classification of, 421–422 identifying solutions of, 397 with infinitely many solutions, 421 modeling, 403 with no solution, 420–421 solving by elimination, 411–415 by graphing, 397–399 by substitution, 404–407 Systematic random sample, 732 T Tables connecting to function rules and graphs, 263 evaluate expressions using, 12–13 finding slope from, 325 frequency, 710 identifying linear functions by, 301 of values in graphing quadratic functions, 611 Tangent, 928 Technology, 29, 142, 471, 534, 546, 766, 801 Technology Lab Connect Function Rules, Tables, and Graphs, 263 Create a Table to Evaluate Expressions, 12–13 Explore Roots, Zeros, and x-intercepts, 648–649 Families of Linear Functions, 368 Families of Quadratic Functions, 632 Graph Linear Functions, 359 Graph Radical Functions, 828 Graph Rational Functions, 893 Graphing to Solve Equations, AT9 Interpret Scatter Plots and Trend Lines, 274 Solve Equations by Graphing, 91 Solve Linear Equations by Using a Spreadsheet, 396 Solve Systems of Linear Inequalities, 441 Use a Graph to Factor Polynomials, 575 Use Random Numbers, 743 Use Technology to Make Graphs, 724–725 Index IN13 Technology Link, 142, 801 Telephone numbers, 765 Temperature, 116, 215, 217 Ten, powers of, 466–467 Terminating decimals, 33 Terms, 47, 276, 496, 841 Test Prep Test Prep questions are found in every exercise set. Some examples: 11, 19, 25, 31, 37; see also Assessment Tests, see Assessment Test Tackler, see also Assessment Any Question Type Read the Problem for Understanding, 450–451 Spatial Reasoning, 780–781 Translate Words to Math, 600–601 Use a Diagram, 536–537 Extended Response Explain Your Reasoning, 690–691 Understand the Scores, 290–291 Gridded Response Fill in Answer Grids Correctly, 68–69 Multiple Choice Choose Combinations of Answers, 936–937 Eliminate Answer Choices, 160–161 None of the Above or All of the Above, 862–863 Recognize Distracters, 388–389 Short Response Understand Short Response Scores, 226–227 Theoretical probability, 744–747 Third quartile, 718 Tip (amount of money), 140 Tolkowsky, Marcel, 881 Transcontinental railroad, 910 Transformations of absolute-value functions, 379–381 of linear functions, 369–372 of quadratic functions, 633–636 Transitive Property of Equality, 187 Translations, 369, 379 Transportation, 96, 124, 179, 209, 254, 261, 271, 304, 305, 341, 500, 851 Transportation Link, 254, 500 Trapezoid, area of, 335, 654 Travel, 18, 36, 90, 104, 185, 278, 279, 308, 319, 334, 630, 875, 909, 910, 924, AT8 Travel Link, 319, 630, 910 Tree diagram, 760 Trend lines finding equations for, 360 interpreting, 274, 360 scatter plots and, 266–269, 360 Trial, 737 Trial-and-error, AT10 IN14 Index Triangle(s) area of, 83 Pythagorean Theorem, 331, 661, 831 Triangle Inequality, 211 Triathlon, 46 Trigonometric ratios, 928–929 Trinomials, see also Factoring polynomials, Polynomial(s) definition of, 497 difference of two squares, 580–581, 585 perfect-square, 521, 578 Truth tables, 203 Tsunamis, 826 Turning points, 680 Two-step equations, solving, 92–95 Two-step inequalities, solving, 190–192 U Undefined slope, 316 Understanding the problem, 459 read a lesson for, 543 read the problem for, 450–451 Unfactorable polynomials, 587 Union, 206 Unit conversions, 121 Unit rate, 120 Unlike denominators, 907 Upper quartile, see Third quartile Use logical reasoning, PS9 Use a Venn diagram, PS10 V Van Dyk, Ernst, 107 Variable(s), 6 on both sides modeling equations with, 99 solving equations with, 100–103 solving inequalities with, 196–199 dependent, 250–254 expressions and, 6–8 independent, 250–254 solving for, 107–109 Variable relationships, model, 248 Variation constant of, 336, 871 direct, 336–339 inverse, 871–874, 896 Vertex of absolute-value functions, 378 of a parabola axis of symmetry through, 618 finding the, 621 Vertical line(s), 879 Vertical-line test, 247 Vertical method for multiplication of polynomials, 515 Vertical translations of linear functions, 369 of parabolas, 635 Vocabulary, 9, 17, 23, 29, 35, 43, 49, 57, 80, 103, 109, 123, 130, 136, 141, 147, 173, 208, 237, 243, 253, 270, 279, 304, 310, 318, 333, 339, 365, 373, 400, 423, 432, 438, 469, 491, 499, 525, 547, 614, 623, 645, 667, 675, 705, 712, 720, 728, 740, 747, 754, 764, 793, 800, 809, 825, 832, 837, 850, 875, 883, 890, 923 Vocabulary Connections, 4, 74, 168, 232, 298, 394, 458, 542, 608, 696, 788, 868 Volume of a cone, 520 of a cylinder, 520, 661 of a prism, 520 of a pyramid, 520 of a sphere, 493, AT4 Voluntary response sample, 733 Vomit Comet, 822 W Wadlow, Robert P., 88 Wages, 305, 339 War Admiral, 238 Waterfalls, 645 Weather, 23, 712, 713, 721 What if...?, 16, 18, 22, 28, 30, 55, 79, 86, 90, 98, 104, 137, 178, 272, 280, 303, 341, 354, 372, 374, 402, 414, 424, 430, 498, 552, 580, 644, 652, 673, 723, 901, 908 Whole numbers, 33 Width of a parabola, 633 Wildlife refuge, 271 Wind turbines, 844 Winter Sports, 876 Winter Sports Link, 876 Work backward, PS5 Write About It Write About It questions are found in every exercise set. Some examples: 9, 19, 24, 29, 31 Writing Math, 6, 32, 33, 55, 77, 102, 112, 170, 241, 258, 344, 407, 466, 482, 488, 505, 551, 709, 790, 879, AT19 Writing Strategies, Write a Convincing Argument/ Explanation, 395 X x-axis, 54 x-coordinate, 54 x-intercept, 307, 648–649 x-values, 240 Y y-axis, 54 y-coordinate, 54 y-intercept, 307, 344–347 Yosemite Falls, 645 y-values, 240 Z Zero(s) division by, 21–22 exploring, 648–649 finding the axis of symmetry of a parabola by using, 620 of a function, 619 multiplication by, 21–22 properties of, 21 of quadratic functions, 619 Zero exponent, 460 Zero Product Property, 650, 886 Zero slope, 316 Index IN15 Credits Abbreviations used: (t) top, (c) center, (b) bottom, (l) left, (r) right, (bkgd) background Photo All images HRW Photo unless otherwise noted. Master Icons—teens, authors (all), Sam Dudgeon/HRW. 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Bukaty; 856 (bl), PhotoDisc/Getty Images; 858 (tr), © Steve Hamblin/Alamy; 864 (bl), PhotoDisc/Getty Images; 866 (tr), Gerald C. Kelley/Photo Researchers, Inc.; 868 (bl), © Paul Barton/CORBIS; 869 (tl), © SuperStock; 871 (cl), AP Photo/Lou Krasky; 871 (bl), PhotoDisc/Getty Images; 876 (tl), PhotoDisc/Getty Images; 876 (br), AP Photo/Tyler Morning Telegraph, Tom Worner; 876 (sky), PhotoDisc/Getty Images; 878 (tr), © Stock Connection Distribution / Alamy; 885 (tr), Philippe Blondel/Photo Researchers, Inc.; 890 (tl), Photo Researchers, Inc.; 893 (tc), AP Photo/Dolores Ochoa; 893 (tr), © Nigel Francis/Robert Harding/Alamy Photos; 898 (cl), AP Photo /NASA, Nick Galante, PMRF; 900 (tr), © Tom Stewart/CORBIS; 904 (br), ON-PAGE CREDIT; 906 (b), © Dynamic Graphics Group/i2i/Alamy; 906 (movie screen), © Ralph Nelson/Imagine Entertainment/ZUMA/CORBIS; 920 (tr), Sam Dudgeon/HRW; 920 (bl), © David Bergman/Corbis; 920 (bc), AP Photo/Al Behrman; 921 (br), Bicycle Museum of America/New Bremen, Ohio/www.bicyclemuseum.com; 921 (tc), Courtesy of the Bicycle Museum of America. Student Handbook: S2 (tl), PhotoDisc/Getty Images; S3 (br), Sam Dudgeon/HRW. Credits CR3