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Unit 1 – Chapters 1 and 4 Unit 1 • • • • • • • • Section 1.6-1.7 Section 4.1 Section 4.2 Section 4.3 Section 4.4 Section 4.5 Section 4.6-4.7 Review Ch. 4 Warm-Up – 1.6-1.7 Lesson 1.6, For use with pages 35-41 3. Gabe spent $4 more than twice as much as Casey at a store. If Casey spent $6, how much did Gabe spend? ANSWER $16 Lesson 1.7, For use with pages 42-48 1. Make a table for y = 2x + 3 with x-values of 0, 3, 6, and 9. ANSWER x y 0 3 3 6 9 9 15 21 2. Write a rule for the function. Input, x 0 Output, y 1 ANSWER y = 3x + 1 2 4 9 7 13 28 Vocabulary – 1.6-1.7 • Domain • Quadrants • Set of INPUTS to a function • 4 regions of a coordinate plane • Sometimes these are considered the X variables • Function • AKA the “independent” variable • Range • Set of OUTPUTS of a function • Sometimes these are considered the Y variables • AKA the “dependent” variable • A numerical relationship where ONE input has EXACTLY ONE output • Relation • A set of inputs and corresponding outputs Notes – 1.6-1.7 – Functions – Intro. •To be a Function 1. Each INPUT must go to EXACTLY ONE output! 2. Graph must pass the Vertical Line Test •Is a function a relation? • Yes •Is a relation a function? • Sometimes!! • Only if it passes the two tests above! •There are several ways to sketch graphs of equations, but the most common is THIS 1. GET Y BY ITSELF!!! 2. Build a table with at least 3 values 3. Sketch the graph Notes – 1.6-1.7 – Functions – Intro. •4 Different Ways to view Functions 1. Verbal Rule – In English – Rarely used. 2. Graphs 3. Equations or Rules 4. Tables Examples 1.6-1.7 EXAMPLE 1 Identify the domain and range of a function The input-output table shows the cost of various amounts of regular unleaded gas from the same pump. Identify the domain and range of the function. Input gallons 10 Output dollars 19.99 23.99 12 13 17 25.99 33.98 ANSWER The domain is the set of inputs: 10, 12, 13, and 17. The range is the set of outputs: 19.99, 23.99, 25.99, and 33.98. GUIDED PRACTICE 1. for Example 1 Identify the domain and range of the function. Input Output 0 5 1 2 2 2 4 1 SOLUTION The domain is the set of inputs: 0, 1, 2, and 4 The range is the set of outputs: 1, 2, and 5 EXAMPLE 2 Identify a function Tell whether the pairing is a function. a. The pairing is not a function because the input 0 is paired with both 2 and 3. EXAMPLE 2 Identify a function b. Input Output 0 0 1 2 4 8 6 12 The pairing is a function because each input is paired with exactly one output. for Example 2 GUIDED PRACTICE Tell whether the pairing is a function. 2. Input Output SOLUTION 3 1 3 6 9 12 6 2 9 2 12 1 1 2 2 1 The pairing is a function because each input is paired with exactly one output. for Example 2 GUIDED PRACTICE Tell whether the pairing is a function. 3. Input Output SOLUTION 2 0 2 4 7 2 1 4 2 7 3 0 1 2 3 The pairing is not a function because each input is not paired with exactly one output. IT IS A RELATION, THOUGH!! EXAMPLE 3 Make a table for a function The domain of the function y = 2x is 0, 2, 5, 7, and 8. Make a table for the function, then identify the range of the function. SOLUTION x y = 2x 0 2 5 7 2 0 = 0 2 2 = 4 2 5 = 10 2 7 = 14 The range of the function is 0, 4, 10, 14, and 16. 8 2 8 = 16 EXAMPLE 4 Write a function rule Write a rule for the function. Input 0 1 4 6 10 Output 2 3 6 8 12 SOLUTION Let x be the input, or independent variable, and let y be the output, or dependent variable. Notice that each output is 2 more than the corresponding input. So, a rule for the function is y = x + 2. GUIDED PRACTICE 4. for Examples 3,4 and 5 Make a table for the function y – x = 5 with domain 10, 12, 15, 18, and 29. Then identify the range of the function. HINT: GET Y BY ITSELF FIRST!! SOLUTION x 10 12 15 18 29 y- x = 5 10 – 5 =5 12 – 5 =7 15 – 5 =10 18 – 5 =13 18 – 29 =24 The range of the function is 5,7,10,13 and 24. GUIDED PRACTICE 5. for Examples 3,4 and 5 Write a rule for the function. Identify the domain and the range. Time (hours) 1 2 3 4 Pay (dollars) 8 16 24 32 SOLUTION Let x be the input ,or independent variable and let y be the output, or dependent variable. Notice that each output is 8 times more than corresponding input . So as a rule of function y = 8x; domain 1, 2, 3 and 4; range 8, 16, 24 and 32. EXAMPLE 5 Write a function rule for a real-world situation Concert Tickets You are buying concert tickets that cost $15 each. You can buy up to 6 tickets. Write the amount (in dollars) you spend as a function of the number of tickets you buy. Identify the independent and dependent variables. Then identify the domain and the range of the function. EXAMPLE 5 Write a function rule for a real-world situation SOLUTION Write a verbal model. Then write a function rule. Let n represent the number of tickets purchased and A represent the amount spent (in dollars). Amount spent (dollars) A = Cost per ticket (dollars/ticket) = 15 • Tickets purchased (tickets) n So, the function rule is A = 15n. The amount spent depends on the number of tickets bought, so n is the independent variable and A is the dependent variable. EXAMPLE 5 Write a function rule for a real-world situation Because you can buy up to 6 tickets, the domain of the function is 0, 1, 2, 3, 4, 5, and 6. Make a table to identify the range. Number of tickets, n 0 1 2 3 4 5 6 Amount (dollars), A 0 15 30 45 60 75 90 The range of the function is 0, 15, 30, 45, 60, 75, and 90. EXAMPLE 1 Graph a function 1 Graph the function y = 2 x with domain 0, 2, 4, 6, and 8. SOLUTION STEP 1 Make an input-output table. x 0 2 4 6 8 y 0 1 2 3 4 EXAMPLE 1 Graph a function STEP 2 Plot a point for each ordered pair (x, y). WHY IS THE GRAPH A SCATTER PLOT AND NOT A LINE?? for Example 1 GUIDED PRACTICE 1. Graph the function y = 2x - 1 with domain 1, 2, 3, 4, and 5. SOLUTION STEP 1 Make an input-output table. x 1 2 3 4 5 y 1 3 5 7 9 STEP 2 Plot a point for each ordered pair (x, y). EXAMPLE 2 Graph a function Sat Scores The table shows the average scores on the mathematics section of the Scholastic Aptitude Test (SAT) in the United States from 1997 to 2003 as a function of the time t in years since 1997. In the table, 0 corresponds to the year 1997, 1 corresponds to 1998, and so on. Graph the function. Years since 1997, t 0 1 2 3 4 5 6 Average score, s 511 512 511 514 514 516 519 EXAMPLE 2 Graph a function STEP 2 Plot the points Does this graph pass the straight line test? Is it a function?? EXAMPLE 2 Graph a function Use the Vertical Line Test to determine if the graphs are Functions. Function Function NOT a Function Warm-Up – 4.1 Prerequisite Skills SKILLS CHECK Write the equation so that y is a function of x. 3 9. 6x + 4y = 16 y=– x+4 ANSWER 2 10. x + 2y = 5 11. –12x - y = –12 ANSWER ANSWER y=– 1 5 x+ 2 2 y = -12x + 12 Prerequisite Skills SKILLS CHECK Graph the function on a coordinate plane and give the Input/ Output table. 3. y = x + 6; when x = 0, 2, 4, 6, and 8 ANSWER Input Output 0 6 2 8 4 10 6 12 8 14 EXAMPLE 2 Graph a function Use the Vertical Line Test to determine if the graphs are Functions. Function Function NOT a Function Vocabulary – 4.1 • Coordinate Plane • Y-coordinate • Two dimensional plane • The VERTICAL used to graph ordered pairs component of an of numbers (x,y) ordered pair • X-coordinate • The HORIZONTAL component of an ordered pair • Sometimes called the “abscissa” • Sometimes called the “ordinate” Notes – 4.1 – Plot Pts. and Graphs • To plot points, move along the X axis first, and then the Y axis • You have to run before you jump (or drop!). • Domain = Inputs • Range = Outputs •Usually in Table format •Quadrants labeled With Roman Numerals Examples 4.1 GUIDED PRACTICE 1. for Example 1 Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E. SOLUTION C. C = (0,2) D = (3,1) E = (-2,-3) GUIDED PRACTICE 2. for Example 1 What is the y-coordinate of any point on the x-axis? SOLUTION y-coordinate of any point on the x-axis is 0 EXAMPLE 3 Graph a function Graph the function y = 2x – 1 with domain – 2, – 1, 0, 1, and 2. Then identify the range of the function. SOLUTION STEP 1 Make a table by substituting the domain values into the function. EXAMPLE 3 Graph a function STEP 2 List the ordered pairs: (– 2, – 5),(– 1, – 3), (0, – 1), (1, 1), (2, 3).Then graph the function. Identify the range. The range consists of the y-values from the table: – 5, – 3, – 1, 1, and 3. GUIDED PRACTICE for Examples 2 and 3 1 7. Graph the function y = – 3 x + 2 with domain – 6, –3, 0, 3, and 6. Then identify the range of the function. STEP 1 Make a table by substituting the domain values into the function. GUIDED PRACTICE x –6 –3 0 3 6 for Examples 2 and 3 y=– 1 x+2 3 y = – 1 (– 6) + 2 = 4 3 y = – 1 (– 3) + 2 = 3 3 y = – 1 (0) + 2 = 2 3 y = – 1 (3) + 2 = 1 3 y = – 1 (6) + 2 = 0 3 GUIDED PRACTICE for Examples 2 and 3 STEP 2 List the ordered pairs: (– 6, 4),(– 3, 3), (0, 2), (3, 1), (6,0). Then graph the function. STEP 3 Identify the range. The range consists of the y-values from the table: 0, 1, 2, 3 and 4. EXAMPLE 4 Graph a function represented by a table VOTING In 1920 the ratification of the 19th amendment to the United States Constitution gave women the right to vote. The table shows the number (to the nearest million) of votes cast in presidential elections both before and since women were able to vote. EXAMPLE 4 Graph a function represented by a table Years before or since 1920 – 12 –8 –4 0 4 8 12 Votes (millions) 15 15 19 27 29 37 40 a. Explain how you know that the table represents a function. b. Graph the function represented by the table. c. Describe any trend in the number of votes cast. EXAMPLE 4 Graph a function represented by a table SOLUTION a. The table represents a function because each input has exactly one output. b. To graph the function, let x be the number of years before or since 1920. Let y be the number of votes cast (in millions). The graph of the function is shown. EXAMPLE 4 Graph a function represented by a table SOLUTION c. In the three election years before 1920,the number of votes cast was less than 20 million. In 1920, the number of votes cast was greater than 20 million. The number of votes cast continued to increase in the three election years since 1920. Warm-Up – 4.2 You may use a calculator on every assignment from this point forward unless otherwise told not to! Lesson 4.2, For use with pages 215-222 1. Graph y = –x – 2 with domain –2, –1, 0, 1, and 2. ANSWER Lesson 4.2, For use with pages 215-222 Rewrite the equation so y is a function of x. 2. 3x + 4y = 16 ANSWER y=– 3 x+4 4 3. –6x – 2y = –12 ANSWER y = –3x + 6 Vocabulary – 4.2 • Linear Equation – The graph of the solutions to the function form a STRAIGHT LINE! Notes – 4.2 – Graph Linear Equations • Standard Form of a Linear Equation looks like this: •Ax + By = C, where A, B, and C are real numbers and A and B are not both = 0 • There are several ways to sketch graphs of linear equations, but the most common is THIS 1. GET Y BY ITSELF!!! 2. Build a table with at least 3 values (negative #, positive #, and zero) 3. Sketch the graph •The graphs of y = constant and x = constant are special cases of linear equations. •We’ll check those out in a minute! Examples 4.2 EXAMPLE 1 Standardized Test Practice Which ordered pair is a solution of 3x – y = 7? A (3, 4) B (1, –4) C (5, –3) D (–1, –2) SOLUTION Check whether each ordered pair is a solution of the equation. Test (3, 4): 3x – y = 7 ? 3(3) – 4 = 7 5=7 Write original equation. Substitute 3 for x and 4 for y. Simplify. EXAMPLE 1 Test (1, – 4): Standardized Test Practice 3x – y = 7 ? 3(1) – (– 4) = 7 7 =7 Write original equation. Substitute 1 for x and – 4 for y. Simplify. So, (3, 4) is not a solution, but (1, – 4) is a solution of 3x – y = 7. ANSWER The correct answer is B. A B C D EXAMPLE 2 Graph an equation Graph the equation – 2x + y = – 3. SOLUTION STEP 1 Solve the equation for y. – 2x + y = – 3 y = 2x – 3 STEP 2 Make a table by choosing a few values for x and finding the values of y. x –2 –1 0 1 2 y –7 –5 –3 –1 1 EXAMPLE 2 Graph an equation STEP 3 Plot the points. Notice that the points appear to lie on a line. EXAMPLE 3 Graph y = b and x = a Graph (a) y = 2 and (b) x = – 1. y=2 x = -1 GUIDED PRACTICE for Examples 2 and 3 Graph the equation 2. y + 3x = – 2 SOLUTION STEP 1 Solve the equation for y. y + 3x = – 2 y = – 3x – 2 GUIDED PRACTICE for Examples 2 and 3 STEP 2 Make a table by choosing a few values for x and finding the values of y. x y –2 4 –1 1 0 –2 1 –5 2 –8 STEP 3 Plot the points. Notice that the points appear to lie on a line. STEP 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end. GUIDED PRACTICE 3. for Examples 2 and 3 y = 2.5 SOLUTION For every value of x, the value of y is 2.5. The graph of the equation y = 2.5 is a horizontal line 2.5 units above the x-axis. 4. x=–4 SOLUTION For every value of y, the value of x is – 4. The graph of the equation x = – 4 is a vertical line 4 units to the left of the y-axis. EXAMPLE 4 Graph a linear function 1 – Graph the function y = 2 x + 4 with domain x > 0. – Then identify the range of the function. SOLUTION STEP 1 Make a table. x 0 2 4 6 8 y 4 3 2 1 0 EXAMPLE 4 Graph a linear function STEP 2 Plot the points. STEP 3 Connect the points with a ray because the domain is restricted. STEP 4 Identify the range. From the graph, you can see that all points have a y-coordinate of 4 or less, so the range of the function is y ≤ 4. GUIDED PRACTICE for Example 4 5. Graph the function y = – 3x + 1 with domain x <– 0. Then identify the range of the function. SOLUTION STEP 1 Make a table. x 0 y 1 –1 –2 –3 –4 4 7 10 13 GUIDED PRACTICE for Example 4 STEP 2 Plot the points. STEP 3 Connect the points with a ray because the domain is restricted. STEP 4 Identify the range. From the graph, you can see that all points have a y-coordinate of 1 or more, so the range of the function is y > – 1. Warm-Up – 4.3 Daily Homework Quiz 1. Graph y + 2x = 4 ANSWER For use after Lesson 4.2 Daily Homework Quiz For use after Lesson 4.2 2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range. ANSWER domain: 0 < – 2.5 –t< range: 0 < – d –< 12.5 Warm-up 1) Graph 5 points on the X-axis and label them. What conclusion can you make about every point on the X-axis? 2) Graph 5 points on the Y-axis and label them. What conclusion can you make about every point on the Y-axis? Vocabulary – 4.3 • X-intercept • Where a graph crosses the X axis • Y-intercept • Where a graph crosses the Y axis Notes – 4.3 – Graph using Intercepts. •The primary reason to use the Standard Form of a linear equation is b/c it does make finding the x and y intercepts VERY easy! •To find the X-intercept of a function • Set Y=0 and solve for X • To find the Y-intercept of a function • Set X=0 and solve for Y •Since you only need two points to make a line •Graph the X and Y intercepts and connect them! Examples 4.3 EXAMPLE 1 Find the intercepts of the graph of an equation Find the x-intercept and the y-intercept of the graph of 2x + 7y = 28. SOLUTION To find the x-intercept, substitute 0 for y and solve for x. 2x + 7y = 28 2x + 7(0) = 28 28 x = 2 = 14 Write original equation. Substitute 0 for y. Solve for x. EXAMPLE 1 Find the intercepts of the graph of an equation To find the y-intercept, substitute 0 for x and solve for y. 2x +7y = 28 2(0) + 7y = 28 28 y= 7 =4 Write original equation. Substitute 0 for x. Solve for y. ANSWER The x-intercept is (14,0). The y-intercept is (0,4). GUIDED PRACTICE for Example 1 Find the x-intercept and the y-intercept of the graph of the equation. 1. 3x + 2y = 6 SOLUTION To find the x-intercept, substitute 0 for y and solve for x. 3x + 2y = 6 3x + 2(0) = 6 x=2 Write original equation. Substitute 0 for y. Solve for x. EXAMPLE 1 for Example 1 graph of an equation Find the intercepts of the GUIDED PRACTICE To find the y-intercept, substitute 0 for x and solve for y. 3x +2y = 6 3(0) + 2y = 6 y =3 Write original equation. Substitute 0 for x. Solve for y. ANSWER The x-intercept is (2,0). The y-intercept is (0,3). EXAMPLE 2 Use intercepts to graph an equation Graph the equation x + 2y = 4. SOLUTION STEP 1 Find the intercepts. x + 2y = 4 x + 2y = 4 x + 2(0) = 4 0 + 2y = 4 x = 4 x-intercept X intercept = (4,0) y = 2 y-intercept Y intercept = (0,2) EXAMPLE 2 Use intercepts to graph an equation STEP 2 Plot points. The x-intercept is 4, so plot the point (4, 0). The y- intercept is 2, so plot the point (0, 2). Draw a line through the points. EXAMPLE 4 Solve a multi-step problem EVENT PLANNING You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation. 4x + 6y = 180 where x is the number of small tables and y is the number of large tables. • Find the intercepts of the graph of the equation. EXAMPLE 4 Solve a multi-step problem • Graph the equation. • Give four possibilities for the number of each size table you could rent. SOLUTION STEP 1 Find the intercepts. 4x + 6y = 180 4x + 6(0) = 180 x = 45 x-intercept 4x + 6y = 180 4(0) + 6y = 180 y = 30 y-intercept EXAMPLE 4 Solve a multi-step problem STEP 2 Graph the equation. The x-intercept is 45, so plot the point (45, 0).The y-intercept is 30, so plot the point (0, 30). Since x and y both represent numbers of tables, neither x nor y can be negative. So, instead of drawing a line, draw the part of the line that is in Quadrant I. EXAMPLE 4 Solve a multi-step problem STEP 3 Find the number of tables. For this problem, only whole-number values of x and y make sense. You can see that the line passes through the points (0, 30),(15,20),(30, 10), and (45, 0). EXAMPLE 4 Solve a multi-step problem So, four possible combinations of tables that will seat 180 people are: 0 small and 30 large, 15 small and 20 large, 30 small and 10 large,and 45 small and 0 large. Warm-Up – 4.4 1) Do pages 10-13 from the “Classified” ads packet as a group. 2) You have ~15 minutes to work on this. Vocabulary – 4.4 • Rate of Change • Ratio of How much something changed over how long did it take to change. • Slope – The STEEPNESS of a line – Same thing as UNIT RATE!!!! – Same thing as Rate of Change!!!! – HOW FAST SOMETHING IS CHANGING!!! • Rise – Vertical or UP/DOWN change – Change in Y’s • Run – Horizontal or LEFT/RIGHT change – Change in X’s Notes – 4.4–Slope and Rate of Change NOTES Slope = RISE RUN 4 Kinds of Slope 1. Positive = Slants UP 2. Negative = Slants DOWN 3. Zero = Horizontal line 4. No Slope = Vertical line Finding slope with a Graph 1. Draw a right triangle connecting the points 2. Calculate RISE and RUN 3. Use Slope = RISE/RUN Notes – Continued NOTES - CONTINUED Finding slope with a Table Slope = Change in Y = How much change = 2ND – 1ST Change in X = How long did it take = 2ND – 1ST SAME as UNIT RATE and RATE OF CHANGE!!! Pay attention to positives/negatives!! Finding slope Using Coordinates Variable for slope is usually m. 1. Slope = m = Y2 – Y1 = RISE X2 – X1 = RUN 2. Plug in what you know and solve for what you don’t! Examples 4.4 EXAMPLE 2 Find a negative slope Find the slope of the line shown. Let (x1, y1) = (3, 5) and (x2, y2) = (6, –1). y 2 – y1 Write formula for slope. m= x 2 – x1 –1 – 5 Substitute. = 6–3 –6 –2 Simplify. = 3 = EXAMPLE 3 Find the slope of a horizontal line Find the slope of the line shown. Let (x1, y1) = (– 2, 4) and (x2, y2) = (4, 4). y 2 – y1 m= x 2 – x1 4– 4 = 4 – (– 2) 0 = = 0 6 Write formula for slope. Substitute. Simplify. EXAMPLE 4 Find the slope of a vertical line Find the slope of the line shown. Let (x1, y1) = (3, 5) and (x2, y2) = (3, 1). y 2 – y1 m= x 2 – x1 1– 5 = 3–3 –4 = 0 Write formula for slope. Substitute. Division by zero is undefined. ANSWER Because division by zero is undefined, the slope of a vertical line is undefined. from 2, a graph EXAMPLE 2 Write an equation for Examples 3 and 4 GUIDED PRACTICE Find the slope of the line that passes through the points. 4. (5, 2) and (5, – 2) SOLUTION Let (x1, y1) = (5, 2) and (x2, y2) = (5, –2). y2 – y1 m= x2 – x1 –2 – 2 = 5–5 –4 = 0 ANSWER Write formula for slope. Substitute. Division by zero is undefined. The slope is undefined. EXAMPLE 5 Find a rate of change INTERNET CAFE The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time. Time(hours) 2 4 6 Cost (dollars) 7 14 21 EXAMPLE 5 Find a rate of change SOLUTION change in cost Rate of change = change in time 14 –7 7 = = = 3.5 2 4–2 ANSWER The rate of change in cost is $3.50 per hour. GUIDED PRACTICE 7. for Example 5 EXERCISE The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. SOLUTION Time(minute) 30 60 90 Distance (miles) 1.5 3 4.5 EXAMPLE 5 for of Example 5 Find a rate change GUIDED PRACTICE change in distance Rate of change = change in time 3 – 1.5 = 0.05 = 60 – 30 ANSWER The rate of change in cost is $0.05 mile/minute. EXAMPLE 6 Use a graph to find and compare rates of change COMMUNITY THEATER A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time. EXAMPLE 6 Use a graph to find and compare rates of change SOLUTION Find the rates of change using the slope formula. Weeks 1–4: Weeks 4–6: Weeks 6–10: 108 232 – 124 = 36 people per week = 4–1 3 204 – 232 = – 28 = – 14 people per week 6–4 2 – 132 72 – 204 = = – 33 people per week 10 – 6 4 ANSWER Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly. EXAMPLE 7 Interpret a graph COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words. EXAMPLE 7 Interpret a graph SOLUTION The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus. Warm-Up – 4.5 Lesson 4.5, For use with pages 243-250 1. Rewrite 6x + 2y = 8 so y is a function of x. ANSWER y = –3x + 4 2. Find the slope of the line that passes through (–5, 6) and (0, 8). ANSWER 2 5 Lesson 4.5, For use with pages 243-250 3. Find the intercepts of the graph of the function. (USE THE COORDINATES OF THE POINTS!) 200x + 100y = – 600. ANSWER y-intercept: (0,– 6), x-intercept: (- 3,0) 4a Find the slope of -2x + y = 1. 4b Find the y-intercept of this equation as well. HINT 1: GET Y BY ITSELF. HINT 2: BUILD A TABLE! ANSWER Slope = 2 and y-intercept is 1. Do these numbers look familiar??????? Vocabulary – 4.5 • Parallel Lines – Lines that never intersect – Lines that have the SAME SLOPE! • slope-intercept form – Linear equation where y = mx + b – m = slope of the line – b = y-intercept Notes – 4.5 – Slope-Intercept Form Slope Intercept Form of an Equation – y = mx + b To use the slope intercept form 1. Solve the equation so that Y is by itself 2. The coefficient of X is the slope. 3. The constant number is the Y intercept. To graph a function using the slope intercept form 1. Graph the Y intercept 2. Use the slope = rise/run to find the next point 3. Graph the second point and connect the two points If two lines are parallel, their slopes are ????? Identical! BrainPop: Slope and Intercept Examples 4.5 Examples 4.5 – Using the Graphing Calculator 1. Open up the graphing application. 2. Graph the equation f1(x) = 2x+1 and press the Graph button. What happens? 3. Graph f2(x) = 2x + 3 What happens? 4. Press the Table button (you have to find it first!). What happens? 5. On the menu, press the Window/Zoom option button and choose 5- Zoom standard. What happens? 6. Graph the function f3(x)= x^2. Look familiar? 7. Clear all the functions by pressing Actions Delete All EXAMPLE 1 Identify slope and y-intercept Identify the slope and y-intercept of the line with the given equation. 1. y = 3x + 4 2. 3x + y = 2 SOLUTION a. The equation is in the form y = m x + b. So, the slope of the line is 3, and the y-intercept is 4. b. Rewrite the equation in slope-intercept form by solving for y. EXAMPLE 1 Identify slope and y-intercept 3x + y = 2 y = – 3x + 2 Write original equation. Subtract 3x from each side. ANSWER The line has a slope of – 3 and a y-intercept of 2. EXAMPLE 1 for Example 1 Identify slope and y-intercept GUIDED PRACTICE Identify the slope and y-intercept of the line with the given equation. 1. y = 5x – 3 SOLUTION The equation is in the form y = mx + b. So, the slope of the line is 5, and the y-intercept is –3. EXAMPLE 1 for Example 1 Identify slope and y-intercept GUIDED PRACTICE Identify the slope and y-intercept of the line with the given equation. 2. 3x – 3y = 12 EXAMPLE 1 for Example 1 Identify slope and y-intercept GUIDED PRACTICE SOLUTION Rewrite the equation in slope-intercept form by solving for y. Write original equation. 3x – 3y = 12 Rewrite original equation. 3x – 12 = 3y Divide 3 by equation. y = 3x + 12 3 Simplify. y =x – 4 ANSWER The line has a slope of –1 and a y-intercept of –4. EXAMPLE 1 for Example 1 Identify slope and y-intercept GUIDED PRACTICE Identify the slope and y-intercept of the line with the given equation. 3. x + 4y = 6 EXAMPLE 1 for Example 1 Identify slope and y-intercept GUIDED PRACTICE SOLUTION Rewrite the equation in slope-intercept form by solving for y. Write original equation. x + 4y = 6 4y = – x + 6 y= – x + 6 4 –x 6 = 4 + 4 Rewrite original equation. Divide 3 by equation. Simplify. ANSWER – The line has a slope of 1 and a y-intercept of 1 1 . 4 2 EXAMPLE 2 Graph an equation using slope-intercept form Graph the equation 2x + y = 3. SOLUTION STEP 1 Rewrite the equation in slope-intercept form. y = – 2x + 3 EXAMPLE 2 Graph an equation using slope-intercept form STEP 2 Identify the slope and the y-intercept. m= – 2 and b =3 STEP 3 Plot the point that corresponds to the y-intercept,(0, 3). STEP 4 Use the slope to locate a second point on the line. Draw a line through the two points. EXAMPLE 3 Change slopes of lines ESCALATORS To get from one floor to another at a library, you can take either the stairs or the escalator. You can climb stairs at a rate of 1.75 feet per second, and the escalator rises at a rate of 2 feet per second. You have to travel a vertical distance of 28 feet. The equations model the vertical distance d (in feet) you have left to travel after t seconds. Stairs: d = – 1.75t + 28 Escalator: d = – 2t + 28 EXAMPLE 3 Change slopes of lines a. Graph the equations in the same coordinate plane. b. How much time do you save by taking the escalator? SOLUTION a. Draw the graph of d = – 1.75t + 28 using the fact that the d-intercept is 28 and the slope is – 1.75. Similarly, draw the graph of d = – 2t + 28. The graphs make sense only in the first quadrant. EXAMPLE 3 b. Change slopes of lines The equation d = – 1.75t + 28 has a t-intercept of 16. The equation d = – 2t + 28 has a t-intercept of 14. So, you save 16 – 14 = 2 seconds by taking the escalator. EXAMPLE 2 Graph an equation for Examples 2 and 3 using slope-intercept form GUIDED PRACTICE 4. Graph the equation y = – 2x + 5. SOLUTION STEP 1 Identify the slope and the y- intercept. m= – 2 and b =5 STEP 2 Plot the point that corresponds to the y-intercept, (0, 5). STEP 3 Use the slope to locate a second point on the line. Draw a line through the two points. EXAMPLE 5 Identify parallel lines Determine which of the lines are parallel. Find the slope of each line. –1– 0 –1 1 Line a: m = = –3 = –1– 2 3 Line b: m = – 3 – (–1 ) 0 – 5 – 5 – (–3) Line c: m = – 2 – 4 –2 2 = –5 = 5 –2 1 = –6 = 3 ANSWER Line a and line c have the same slope, so they are parallel. EXAMPLE 5 for Examples Identify parallel lines 4 and 5 GUIDED PRACTICE 7. Determine which lines are parallel: line a through (-1, 2) and (3, 4); line b through (3, 4) and (5, 8); line c through (-9, -2) and (-1, 2). SOLUTION Find the slope of each line. 4– 2 2 1 Line a: m = 3 –(1) = 3+1 = 2 8– 4 –4 Line b: m = 5 – 3 = –2 = 2 2 – (–2) 2+2 Line c: m = – 1 – (– 9) =–1+ 9 4 = 8 1 = 2 Warm-Up – 4.6 and 4.7 Lesson 4.6, For use with pages 253-259 Write the equation in slope intercept form and sketch the graph 1. 4x – y = –8 ANSWER y=4x+8 2. -9x - 3y = 21 ANSWER y = -3x - 7 Lesson 4.6, For use with pages 253-259 Write the equation and sketch the graph 1. Slope = 1 and y-int = 2 ANSWER Y=x+2 Lesson 4.6, For use with pages 253-259 Find the unit rate. 3. You are traveling by bus. After 4.5 hours, the bus has traveled 234 miles. Use the formula d = rt where d is distance, r is a rate, and t is time to find the average rate of speed of the bus. ANSWER 52 mi/h Lesson 4.6, For use with pages 253-259 Sketch the graph 1. y = 4x ANSWER Vocabulary – 4.6-4.7 • Direct Variation – Linear equation where y = kx • Constant of Variation – In a Direct Variation, the letter k is the constant of variation – It’s the unit rate, the rate of change and … – SAME AS SLOPE!! • Function Notation – Different way of writing functions – F(x) means the “the function of x” • Family of Functions – A group of functions with similar characteristics (e.g. their graphs are all linear) • Parent Linear Function – Simplest form of a family of functions – F(x) = x is the parent linear function Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions A Direct variation has the form y = kx k = the constant of variation and is AKA THE SLOPE! A direct variation graph ALWAYS goes through the origin. A direct variation is ALWAYS PROPORTIONAL! There are two ways to find the constant, k 1.Find the UNIT RATE 2.If they give you a coordinate (x,y), 1. Plug in the numbers to y = kx 2. Solve for k. Notes – 4.6-4.7 – Direct Variations and Graphing Linear Functions – cont. A “function” is usually written as F(x) and we read it as “F of x” OR y = F(x) To evaluate functions Plug in what you know and …..??? Examples 4.6-4.7 EXAMPLE 1 Identify direct variation equations Tell whether the equation represents direct variation. If so, identify the constant of variation. a. 2x – 3y = 0 b. – x + y = 4 EXAMPLE 1 Identify direct variation equations SOLUTION To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax. 2x – 3y = 0 – 3y = – 2x 2x y= 3 Write original equation. Subtract 2x from each side. Simplify. ANSWER Because the equation 2x – 3y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 2 3 EXAMPLE 1 b. Identify direct variation equations –x+y=4 y = x+4 Write original equation. Add x to each side. ANSWER Because the equation – x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation. GUIDED PRACTICE for Example 1 Tell whether the equation represents direct variation. If so, identify the constant of variation. 1. –x+y=1 GUIDED PRACTICE for Example 1 SOLUTION To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax. –x+y=1 y = x+1 Write original equation. Add x each side. ANSWER Because the equation – x + y = 1 cannot be rewritten in the form y = ax, it does not represent direct variation. GUIDED PRACTICE 2. for Example 1 2x + y = 0 SOLUTION 2x + y = 0 y = – 2x Write original equation. Subtract 2x from each side. ANSWER Because the equation 2x + y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. – 2 GUIDED PRACTICE 3. for Example 1 4x – 5y = 0 SOLUTION To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax. 4x – 5y = 0 Write original equation. 4x = 5y Subtract Add 5y each side. 4x Simplify. y= 5 ANSWER Because the equation 4x – 5y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 4 5 EXAMPLE 2 Graph direct variation equations Graph the direct variation equation. a. y = 2 x 3 b. y = – 3x SOLUTION a. Plot a point at the origin. The slope is equal to the constant of variation, or 2 Find and plot a second 3 point, then draw a line through the points. EXAMPLE 2 b. Graph direct variation equations Plot a point at the origin. The slope is equal to the constant of variation, or – 3. Find and plot a second point, then draw a line through the points. EXAMPLE 3 Write and use a direct variation equation The graph of a direct variation equation is shown. a. Write the direct variation equation. b. Find the value of y when x = 30. SOLUTION a. Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 2 when x = – 1 to find a. y = ax 2 = a (– 1) –2=a Write direct variation equation. Substitute. Solve for a. EXAMPLE 3 Graph direct variation equations ANSWER A direct variation equation that relates x and y is y = – 2x. b. When x = 30, y = – 2(30) = – 60. GUIDED PRACTICE 4. for Examples 2 and 3 Graph the direct variation equation. y = 2x SOLUTION Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points. GUIDED PRACTICE for Examples 2 and 3 5. The graph of a direct variation on equation passes through the point (4,6). Write the direct variation equation and find the value of y when x =24. SOLUTION Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 6 when x = 4 to find a. y = ax Write direct variation equation. 6 = a (4) Substitute. 3 6 = 2 4 Solve for a. a= GUIDED PRACTICE for Examples 2 and 3 ANSWER A direct variation equation that relates x and y is y = 3 x when x = 24. y = 3 (24) = 36 2 2 EXAMPLE 1 Standardized Test Practice SOLUTION f (x)= 3x – 15 f (– 3) = 3(– 3) – 15 = 24 Write original function. Substitute -3 for x. Simplify. ANSWER The correct answer is A. A B C D GUIDED PRACTICE 1. for Example 1 Evaluate the function h(x) = – 7x when x = 7. SOLUTION h(x) = – 7x Write original function. h(7) = – 7(7) Substitute 7 for x. = – 49 Simplify. EXAMPLE 2 Find an x-value For the function f(x) = 2x – 10, find the value of x so that f(x) = 6. f(x) = 2x – 10 Write original function. 6 = 2x – 10 Substitute 6 for f(x). 8=x Solve for x. ANSWER When x = 8, f(x) = 6. EXAMPLE 3 Graph a function GRAY WOLF The gray wolf population in central Idaho was monitored over several years for a project aimed at boosting the number of wolves. The number of wolves can be modeled by the function f(x) = 37x + 7 where x is the number of years since 1995. Graph the function and identify its domain and range. Review – Chap. 4 Daily Homework Quiz 1. Write the equation 15 = 5y – 4x so that y is a function of x ANSWER 2. y= 4 x+3 5 Solve C = 2pr for r ANSWER 3. For use after Lesson 3.8 r= C p 2 Solve V = 1 Bh for B. 3 ANSWER B = 3V h Daily Homework Quiz For use after Lesson 3.8 4. On a round-trip bicycle trip from Santa Barbara to Canada, Phil rode 2850 miles in 63 days. Find his average miles per day for the trip. Use the formula d = rt where d is distance, r is rate, and t is time.Solve for r to find the rate in miles per day to the nearest mile. ANSWER about 45 miles per day Daily Homework Quiz For use after Lesson 4.1 Give the coordinates of the points. Plot the points in a coordinate plane 1. and 2. ANSWER A(-3,2) ANSWER B(0,-1) Daily Homework Quiz 3. A(-2,-4) ANSWER For use after Lesson 4.1 Daily Homework Quiz For use after Lesson 4.1 4. B(3,0) ANSWER 5. 1 Graph y = 2 x – 1 with domain – 4, – 2, 0, 2, 4. Then identify the range ANSWER – 3, – 2, – 1, 0, 1 Daily Homework Quiz 1. Graph y + 2x = 4 ANSWER For use after Lesson 4.2 Daily Homework Quiz For use after Lesson 4.2 2. The distance in miles an elephant walks in t hours is given by d = 5t. The elephant walks for 2.5 hours. Graph the function and identify its domain and range. ANSWER domain: 0 < – 2.5 –t< range: 0 < – d –< 12.5 Daily Homework Quiz 1. For use after Lesson 4.3 Find the x-intercept and the y-intercept of the graph of 3x – y = 3 ANSWER x-int: 1, y-int: – 3 Daily Homework Quiz 2. For use after Lesson 4.3 A recycling company pays $1 per used ink jet cartridge and $2 per used cartridge. The company paid a customer $14.This situation is given by x + 2y = 14 where x is the number of inkjet cartridges and y the number of laser cartridges. Use intercepts to graph the equation. Give four possibilities for the number of each type of cartridge that could have been recycled. ANSWER (0,7),(6,4),(10,2),(14,0) Daily Homework Quiz For use after Lesson 4.4 Find the slope of the line that passes through the points 1. (12, – 1) and (– 3, – 1) ANSWER 2. 0 (–2, 6) and (4, –3) ANSWER – 3 2 Daily Homework Quiz 3. For use after Lesson 7.2 The graph shows the ticket sales for a school dance on day 1, day 3, day 6, and day 9,of ticket sales. Describe the rates of change in ticket sales with respect to time. ANSWER Ticket sales grew moderately, declined slightly, and then had another moderate rate of increase. Daily Homework Quiz 1. Identify the slope and y-intercept of the line 2x + 4y = –16. ANSWER 2. For use after Lesson 4.5 y = 2 x +1 3 ANSWER 1 Slope: – ,y-intercept: –4 2 Daily Homework Quiz 3. For use after Lesson 7.2 Determine which of the lines are parallel. ANSWER Lines a and c Daily Homework Quiz 1. Identify the slope and y-intercept of the line 2x + 4y = –16. ANSWER 2. For use after Lesson 4.5 y = 2 x +1 3 ANSWER 1 Slope: – ,y-intercept: –4 2 Daily Homework Quiz 3. For use after Lesson 7.2 Determine which of the lines are parallel. ANSWER Lines a and c Daily Homework Quiz For use after Lesson 4.6 Tell whether the equation represents direct variation. If so, identify the constant of variation. 1. 5x – 6y = 2 ANSWER 2. no x+y = 0 ANSWER yes, – 1 Daily Homework Quiz 3. For use after Lesson 4.6 The number p of parts a machine produces varies directly with the time t (in minutes) the machine is in operation. The machine produces 84 parts in 14 minutes. Write a direct variation equation that relates t and p. how many parts does the machine produce in 25 minutes? ANSWER p = 6t; 150 parts Daily Homework Quiz For use after Lesson 4.7 1. Evaluate f (x) = 8x – 4 when x = –3, 0, and 2. ANSWER –28, –4, 12 2. Find the value of x so g (x) = –2x+ 1 has the value –3. ANSWER 2 Daily Homework Quiz For use after Lesson 4.7 3. A stable charges $25 for feed and $50 per day to stable horses. The cost is given by f(x) = 50x + 25. Recently, the stable raised its fee for food to $50. The new fee is given by g(x) = 50x + 50.Graph the functions and then compare the two graphs. ANSWER The graphs have the same slope.The y-intercept of g is 25 units greater than that of f, so g is a vertical translation. Warm-Up – X.X Vocabulary – X.X • Holder • Holder 2 • Holder 3 • Holder 4 Notes – X.X – LESSON TITLE. • Holder •Holder •Holder •Holder •Holder Examples X.X