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Applications Equations Graphs McDougal Littell Larson Boswell Kanold Stiff Copyright 2004 Chapter 1 Chapter 3 Chapter 5 Chapter 7 Chapter 9 Chapter 11 Chapter 2 Chapter 4 Chapter 6 Chapter 8 Chapter 10 Chapter 12 Chapter 1 Connections to Algebra 1.1 Variables in Algebra 1.2 Exponents and Powers 1.3 Order of Operations 1.4 Equations and Inequalities 1.5 Problem Solving plane using Models 1.6 Tables and Graphs 1.7 And Introduction to Functions Chapter 1.1 Chapter 1.2 Expressions Containing Exponents An expression like 4 6 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor. base exponent 46 = 4 • 4 • 4 • 4 • 4 • 4 power 6 factors of 4 Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM 10 1 WORDS MEANING ten to the first power 10 Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM 10 1 42 WORDS MEANING ten to the first power 10 four to the second power or four squared 4•4 Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM WORDS MEANING ten to the first power 10 42 four to the second power or four squared 4•4 53 five to the third power, or five cubed 5•5•5 10 1 Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM WORDS MEANING ten to the first power 10 42 four to the second power or four squared 4•4 53 five to the third power, or five cubed 5•5•5 76 seven to the sixth power 7•7•7•7•7•7 10 1 Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM WORDS MEANING ten to the first power 10 42 four to the second power or four squared 4•4 53 five to the third power, or five cubed 5•5•5 76 seven to the sixth power 7•7•7•7•7•7 xn x to the nth power x•x•x•x…x 10 1 and so on Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. SOLUTION EXPONENTIAL FORM WORDS MEANING ten to the first power 10 42 four to the second power or four squared 4•4 53 five to the third power, or five cubed 5•5•5 76 seven to the sixth power 7•7•7•7•7•7 xn x to the nth power x•x•x•x…x 10 1 and so on For a number raised to the first power, you usually do not write the exponent 1. For instance, you write 5 1 simply as 5. Evaluating Powers Evaluate the expression x 3 when x = 5. SOLUTION x3 = 53 Substitute 5 for x. =5•5•5 Write factors. = 125 Multiply. The value of the expression is 125. Evaluating an Exponential Expression GROUPING SYMBOLS For problems that have more than one operation, it is important to know which operations to do first. Grouping symbols, such as parentheses ( ) or brackets [ ], indicate the order in which the operations should be performed. Operations within the innermost set of grouping symbols are done first. For instance, the value of the expression (3 • 4) + 7 is not the same as the value of the expression 3 • (4 + 7). Evaluating an Exponential Expression GROUPING SYMBOLS For problems that have more than one operation, it is important to know which operations to do first. Grouping symbols, such as parentheses ( ) or brackets [ ], indicate the order in which the operations should be performed. Operations within the innermost set of grouping symbols are done first. For instance, the value of the expression (3 • 4) + 7 is not the same as the value of the expression 3 • (4 + 7). Multiply. Then add. Add. Then multiply. (3 • 4) + 7 = 12 + 7 = 19 3 • (4 + 7) = 3 • 11 = 33 Evaluating an Exponential Expression Evaluate the expression when a = 1 and b = 2. (a + b) 2 SOLUTION (a + b) 2 = (1 + 2) 2 Substitute 1 for a and 2 for b. = 32 Add within parentheses. =3•3 Write factors. =9 Multiply. Evaluating an Exponential Expression Evaluate the expression when a = 1 and b = 2. (a 2 ) + (b 2 ) SOLUTION (a 2 ) + (b 2 ) = (1 2 ) + (2 2 ) Substitute 1 for a and 2 for b. =1+4 Evaluate power. =5 Add. Evaluating an Exponential Expression Evaluate the expression when a = 1 and b = 2. (a 2 ) + (b 2 ) SOLUTION (a 2 ) + (b 2 ) = (1 2 ) + (2 2 ) Substitute 1 for a and 2 for b. =1+4 Evaluate power. =5 Add. An exponent applies only to the number, variable, or expression immediately to its left. In the expression 2x 3, the base is x, not 2x. In the expression (2x) 3, the base is 2x, as indicated by the parentheses. Exponents and Grouping Symbols Evaluate the expression when x = 4. 2x 3 SOLUTION 2x 3 = 2(4 3) Substitute 4 for x. = 2(64) Evaluate power. = 128 Multiply. Exponents and Grouping Symbols Evaluate the expression when x = 4. 2x 3 (2x) 3 SOLUTION 2x 3 = 2(4 3) SOLUTION Substitute 4 for x. (2x) 3 = (2 • 4) 3 Substitute 4 for x. = 2(64) Evaluate power. = (8) 3 Multiply within parentheses. = 128 Multiply. = 512 Evaluate power. Real-Life Applications of Exponents Exponents often are used in the formulas for area and volume. In fact, the words squared and cubed come from the formula for the area of a square, A = s 2, and the formula for the volume of a cube, V = s 3. Area of Square: A = s 2 Units of area, such as square feet, ft 2, can be written using a second power. Real-Life Applications of Exponents Exponents often are used in the formulas for area and volume. In fact, the words squared and cubed come from the formula for the area of a square, A = s 2, and the formula for the volume of a cube, V = s 3. Area of Square: A = s 2 Volume of Cube: V = s 3 Units of area, such as square feet, ft 2, can be written using a second power. Units of volume, such as cubic centimeters, cm 3, can be written using a third power. Making a Table You can find the volume of cubes that have edge lengths of 1 inch, 2 inches, 3 inches, 4 inches, and 5 inches by using the formula V = s 3. Edge, s 1 2 3 4 5 s3 13 23 33 43 53 Volume, V 1 in. 3 8 in. 3 27 in. 3 64 in. 3 125 in. 3 Chapter 1.3 Using the Order of Operations Mathematicians have established an order of operations to evaluate an expression involving more than one operation. Start with operations within grouping symbols. Then evaluate powers. Then do multiplications and divisions from left to right. Finally, do additions and subtractions from left to right. Evaluating Without Grouping Symbols Evaluate the expression when x = 4. 3x 2 + 1 SOLUTION 3x 2 + 1 = 3 • 4 2 + 1 Substitute 4 for x. = 3 • 16 + 1 Evaluate power. = 48 + 1 Evaluate product. = 49 Evaluate sum. Evaluating Without Grouping Symbols Evaluate the expression when x = 4. 32 x 2 – 1 SOLUTION 32 x 2 – 1 = 32 4 2 – 1 Substitute 4 for x. = 32 16 – 1 Evaluate power. =2–1 Evaluate quotient. =1 Evaluate difference. Using the Order of Operations When you want to change the established order of operations for an expression, you must use parentheses or other grouping symbols. ACTIVITY Developing Concepts INVESTIGATING GROUPING SYMBOLS Without grouping symbols, the expression 3 • 4 2 + 8 4 has a value of 50. You can insert grouping symbols to produce a different value. For example: 3 • (4 2 + 8) 4 = 3 • (16 + 8) 4 = 3 • 24 4 = 72 4 = 18 Insert grouping symbols in the expression 3 • 4 2 + 8 4 to produce the indicated values. 146 54 38 Using the Left-to-Right Rule THE LEFT-TO-RIGHT RULE Operations that have the same priority, such as multiplication and division or addition and subtraction, are performed using the left-to-right rule, as shown below. 24 – 8 – 6 = (24 – 8) – 6 = 16 – 6 = 10 Work from left to right. Using the Left-to-Right Rule THE LEFT-TO-RIGHT RULE Operations that have the same priority, such as multiplication and division or addition and subtraction, are performed using the left-to-right rule, as shown below. 24 – 8 – 6 = (24 – 8) – 6 Work from left to right. = 16 – 6 = 10 15 • 2 6 = (15 • 2) 6 = 30 6 =5 Work from left to right. Using the Order of Operations ORDER OF OPERATIONS First do operations that occur within grouping symbols. Using the Order of Operations ORDER OF OPERATIONS First do operations that occur within grouping symbols. Then evaluate powers. Using the Order of Operations ORDER OF OPERATIONS First do operations that occur within grouping symbols. Then evaluate powers. Then do multiplications and divisions from left to right. Using the Order of Operations ORDER OF OPERATIONS First do operations that occur within grouping symbols. Then evaluate powers. Then do multiplications and divisions from left to right. Finally, do additions and subtractions from left to right. Using a Fraction Bar 1+2 A fraction bar can act as a grouping symbol: (1 + 2) (4 – 1) = 4–1 7•4 7•4 = 2 8+7 –1 8 + 49 – 1 Evaluate power. = 28 8 + 49 – 1 Simplify the numerator. = 28 57 – 1 Work from left to right. = 28 56 Subtract. = 1 2 Simplify. Evaluating Expressions with a Calculator Many calculators use the established order of operations, but some do not. When you enter the following in your calculator, does the calculator display 6.1 or 0.6? 10.5 – 6.3 2.1 – 1.4 Enter SOLUTION If your calculator uses order of operations, it will display 10.5 – 6.3 ÷ 2.1 – 1.4 = 10.5 – (6.3 ÷ 2.1) – 1.4 = 10.5 – (3) – 1.4 = 6.1 6.1 . Evaluating Expressions with a Calculator Many calculators use the established order of operations, but some do not. When you enter the following in your calculator, does the calculator display 6.1 or 0.6? 10.5 – 6.3 2.1 – 1.4 Enter SOLUTION If your calculator uses order of operations, it will display 6.1 . 10.5 – 6.3 ÷ 2.1 – 1.4 = 10.5 – (6.3 ÷ 2.1) – 1.4 = 10.5 – (3) – 1.4 = 6.1 SOLUTION If your calculator displays as they are entered. 0.6 , it performs the operations [(10.5 – 6.3) ÷ 2.1] – 1.4 = ( 4.2 ÷ 2.1) – 1.4 = 2 – 1.4 = 0.6 Chapter 1.4 Chapter 1.5 TRANSLATING VERBAL PHRASES Verbal Phrase Expression The sum of six and a number 6+x Eight more than a number y+8 A number plus five n+5 A number increased increased by seven x+7 A number decreased by nine n–9 Ten times a number Seven divided by a number 10 • n or (10n) 10n) (10 (10n 7 x USING A VERBAL MODEL In Mathematics there is a difference between a phrase and a sentence. Phrases translate into expressions; sentences translate into equations or inequalities. Phrases Expressions Sentences Equations or Inequalities USING A VERBAL MODEL Phrase Expression The sum of six and a number The sum of six and a number is Sentence 6+x 6+x= In this sentence, “is” says that one quantity is equal to one another. Equation The sum of six and a number number is is twelve. twelve. Sentence In this sentence, the words “is less than” indicate an inequality. The sum of six and a number is less than thantwelve. twelve. 6 + x = 12 Inequality 6 + x < 12 USING A VERBAL MODEL Writing algebraic expressions, equations, or inequalities that represent real-life situations is called modeling. The expression, equation, or inequality is a mathematical model. Use three steps to write a mathematical model. WRITE A ASSIGN WRITE AN VERBAL MODEL. LABELS. ALGEBRAIC MODEL. Writing an Algebraic Model You and three friends are having a dim sum lunch at a Chinese restaurant that charges $2 per plate. You order lots of plates. The waiter gives you a bill for $25.20, which includes tax of $1.20. Use mental math to solve the equation for how many plates your group ordered. SOLUTION Understand the problem situation before you begin. For example, notice that tax is added after the total cost of the dim sum plates is figured. Writing an Algebraic Model VERBAL MODEL Cost per plate LABELS Cost per plate = 2 (dollars) Number of plates = p (plates) • Number of plates = Bill – Tax Amount of bill = 25.20 (dollars) Tax = 1.20 (dollars) ALGEBRAIC MODEL 2 p = 25.20 – 1.20 2p = 24.00 p = 12 Your group ordered 12 plates of food costing $24.00. Writing an Algebraic Model A PROBLEM SOLVING PLAN USING MODELS VERBAL MODEL Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know. LABELS Assign labels to each part of your verbal problem. ALGEBRAIC Use the labels to write an algebraic model based on MODEL your verbal model. SOLVE Solve the algebraic model and answer the original question. CHECK Check that your answer is reasonable. Using a Verbal Model JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land. The pilot knows the speed of the jet must be greater then 322 miles per hour or the jet could stall. a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? b. Is it reasonable for the pilot to fly directly to Chicago at the reduced speed from part (a) or must the pilot take some other action? Using a Verbal Model a. At what speed would the jet have to fly to arrive in Chicago in 2 hours? SOLUTION You can use the formula (rate)(time) = (distance) to write a verbal model. VERBAL MODEL Speed of jet LABELS Speed of jet = x (miles per hour) Time = 2 (hours) Distance to travel = 600 (miles) ALGEBRAIC MODEL • Time = Distance to travel 2 x = 600 x = 300 To arrive in 2 hours, the pilot would have to slow the jet down to 300 miles per hour. Using a Verbal Model b. Is it reasonable for the pilot to fly directly to Chicago at 300 miles per hour or must the pilot take some other action? It is not reasonable for the pilot to fly at 300 miles per hour, because the jet could stall. The pilot should take some other action, such as circling in a holding pattern, to use some of the time. Chapter 1.6 Chapter 1.7 Chapter 2 Properties of Real Numbers 2.1 Real Number Line 2.2 Addition of Real Numbers 2.3 Subtraction of Real Numbers 2.4 Adding and Subtracting Matrices 2.5 Multiplication of Real Numbers 2.6 Distributive Property 2.7 Division of Real Numbers 2.8 Probability and Odds Chapter 2.1 Chapter 2.2 Modeling Addition of Integers To model addition problems involving positive and negative integers, you can use tiles labeled + and – . Each + represents positive 1, and each – represents negative 1. Combining a + with a – gives 0. You can use algebra tiles to find the sum of –8 and 3. 1 2 3 Model negative 8 and positive 3 using algebra tiles. – Group pairs of positive and negative tiles. Count the remaining tiles. – – – + + + The remaining tiles show the sum of –8 and 3. – – – – – – – 0–8 – + 3 – – – – Each pair has a sum of 0. – + – – – –8 + 3 = –5 – + Adding Real Numbers Addition can be modeled with movements on a number line. You add a positive number by moving to the right. Model –2 + 5. Start at –2. –4 –3 Move 5 units to the right. • –2 –1 0 1 The sum can be written as –2 + 5 = 3. 2 End at 3. • 3 4 Adding Real Numbers Addition can be modeled with movements on a number line. You add a positive number by moving to the right. Model –2 + 5. Start at –2. –4 –3 Move 5 units to the right. • –2 –1 0 1 2 End at 3. • 3 4 The sum can be written as –2 + 5 = 3. You add a negative number by moving to the left. Model 2 + (–6). Move 6 units to the left. End at – 4. –5 • –4 –3 –2 –1 0 Start at 2. 1 The sum can be written as 2 + (–6) = – 4. • 2 3 Adding Real Numbers The rules of addition show how to add two real numbers without a number line. RULES OF ADDITION TO ADD TWO NUMBERS WITH THE SAME SIGN: 1 Add their absolute values. 2 Attach the common sign. Example: – 4 + (–5) – 4 + –5 = 9 –9 Adding Real Numbers The rules of addition show how to add two real numbers without a number line. RULES OF ADDITION TO ADD TWO NUMBERS WITH THE SAME SIGN: 1 Add their absolute values. 2 Attach the common sign. Example: – 4 + (–5) – 4 + –5 = 9 –9 TO ADD TWO NUMBERS WITH OPPOSITE SIGNS: 1 Subtract the smaller absolute value from the larger absolute value. 2 Attach the sign of the number with the larger absolute value. Example: 3 + (–9) –9 – 3 = 6 –6 Adding Real Numbers The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition. PROPERTIES OF ADDITION COMMUTATIVE PROPERTY The order in which two numbers are added does not change the sum. a+b=b+a Example: 3 + (–2) = –2 + 3 ASSOCIATIVE PROPERTY The way you group three numbers when adding does not change the sum. (a + b) + c = a + (b + c) Example: (–5 + 6) + 2 = –5 + (6 + 2) Adding Real Numbers The rules of addition on the previous slide will help you find sums of positive and negative numbers. It can be shown that these rules are a consequence of the following Properties of Addition. PROPERTIES OF ADDITION IDENTITY PROPERTY The sum of a number and 0 is the number. a+0=a Example: –4 + 0 = –4 PROPERTY OF ZERO (INVERSE PROPERTY) The sum of a number and its opposite is 0. a + (–a) = 0 Example: 5 + (–5) = 0 Adding Three Real Numbers Use a number line to find the following sum. –3 + 5 + (– 6) SOLUTION Start at –3. Move 5 units to the right. –5 • • –4 End at – 4. –3 –2 –1 0 1 Move 6 units to the left. The sum can be written as –3 + 5 + (–6) = – 4. –3 + 5 = 2 • 2 3 Finding a Sum Find the following sums. 1.4 + (–2.6) + 3.1 = 1.4 + (–2.6 + 3.1) = 1.4 + 0.5 = 1.9 Use associative property. Simplify. Finding a Sum Find the following sums. 1.4 + (–2.6) + 3.1 = 1.4 + (–2.6 + 3.1) = 1.4 + 0.5 Use associative property. Simplify. = 1.9 – 1 1 1 1 +3+ = – + +3 2 2 2 2 = (– 12 + 12 ) + 3 = 0+3=3 Use commutative property. Use associative property. Use identity property and property of zero. Using Addition in Real Life SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions? Aluminum: 13 electrons, 13 neutrons, 13 protons SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al. Using Addition in Real Life SCIENCE CONNECTION Atoms are composed of electrons, neutrons, and protons. Each electron has a charge of –1, each neutron has a charge of 0, and each proton has a charge of +1. The total charge of an atom is the sum of all the charges of its electrons, neutrons, and protons. An atom is an ion if it has a positive or a negative charge. If an atom has a charge of zero, it is not an ion. Are the following atoms ions? Aluminum: 13 electrons, 13 neutrons, 13 protons SOLUTION The total charge is –13 + 0 + 13 = 0, so the atom is not an ion. In chemistry, this aluminum atom is written as Al. Aluminum: 10 electrons, 13 neutrons, 13 protons SOLUTION The total charge is –10 + 0 + 13 = 3, so the atom is an ion. In chemistry, this aluminum atom is written as Al 3 +. Finding the Total Profit A consulting company had the following monthly results after comparing income and expenses. Add the monthly profits and losses to find the overall profit or loss during the six-month period. SOLUTION January February March –$13,142.50 –$6783.16 –$4734.86 April May June $3825.01 $7613.17 $12,932.54 With this many large numbers, you may want to use a calculator. 13142.50 +/– + 3825.01 + + 6783.16 7613.17 +/– + + 4734.86 12932.54 = +/– –289.8 The display is –298.8. This means the company had a loss of $298.80. Chapter 2.3 Chapter 2.4 Chapter 2.5 Chapter 2.6 MODELING THE DISTRIBUTIVE PROPERTY You can use algebra tiles to model algebraic expressions. 1 1 x-tile 1-tile 1 x This 1-by-1 square tile has an area of 1 square unit. This 1-by-x square tile has an area of x square units. Model the Distributive Property using Algebra Tiles x+2 x+2 Area = 3(x + 2) 3 3+ x 2 Area = 3(x ) + 3(2) 3 USING THE DISTRIBUTIVE PROPERTY THE DISTRIBUTIVE PROPERTY The product of a and (b + c): a(b + c) = ab + ac 2(x + 5) = 2(x) + 2(5) = 2x + 10 (b + c)a = ba + ca (x + 5)2 = (x)2 + (5)2 = 2x + 10 y(1 – y) = y(1) – y(y) = y – y 2 (1 + 5x)2 = (1)2 + (5x)2 = 2 + 10x USING THE DISTRIBUTIVE PROPERTY Remember that a factor must multiply each term of an expression. Distribute the –3. (–3)(1 + x) = (–3)(1) + (–3)(x) = –3 – 3x Simplify. (y – 5)(–2) = (y)(–2) + (–5)(–2) Distribute the –2. = –2y + 10 –(7 – 3x) = (–1)(7) + (–1)(–3x) = –7 + 3x Simplify. –a = –1 • a Simplify. Forgetting to distribute the negative sign when multiplying by a negative factor is a common error. MENTAL MATH CALCULATIONS SOLUTION You are shopping for CDs. You want to buy six CDs for $11.95 each. 6(11.95) = 6(12 – 0.05) The mental math is easier if you think of $11.95 as $12.00 – $.05. Write 11.95 as a difference. = 6(12) – property 6(0.05) Use the distributive to calculate the total cost = 72 – 0.30 mentally. Use the distributive property. = 71.70 Find the difference mentally. Find the products mentally. The total cost of 6 CDs at $11.95 each is $71.70. SIMPLIFYING BY COMBINING LIKE TERMS Each of these terms terms is the product of a number number and a variable. variable. – x + 3y2 x isis the –1 the y3 is the 2. coefficient variable. of x. coefficient variable. of y Like Liketerms terms have the samevariable variable raised to the same power. power. y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y The constant2 terms 3 –5 and 3 2are also 3 like terms. x y x y SIMPLIFYING BY COMBINING LIKE TERMS 8x + 3x = (8 + 3)x = 11x 4x2 + 2 – x2 = 4x2 – x2 + 2 = 3x2 + 2 3 – 2(4 + x) = 3 + (–2)(4 + x) Use the distributive property. Add coefficients. Group like terms. Combine like terms. Rewrite as addition expression. = 3 + [(–2)(4) + (–2)(x)] Distribute the –2. = 3 + (–8) + (–2x) Multiply. = –5 + (–2x) = –5 – 2x Combine like terms and simplify. Chapter 2.7 Chapter 2.8 Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Chapter 3.1 Modeling One-Step Equations Algebra tiles can be used to solve one-step equations. 1 x 2 x Model the equation x + 5 = –2. A + + + + + = represents the unknown value x. – – To find the value of x, get the x-tile by itself on the left side of the equation. You can undo the addition by subtracting 5. Be sure to subtract the same amount on each side of the equation, so that the two sides stay equal. x + + + + + = – – – – – – – + + + + + 3 The resulting equation shows the value of x. So, x = –7. First add 5 zero pairs so you can subtract 5 Addition and Subtraction Equations You can solve an equation by using the transformations on the next slide to isolate the variable on one side of the equation. When you rewrite an equation using these transformations, you produce an equation with the same solutions as the original equation. These equations are called equivalent equations. For example, x + 3 = 5 and x = 2 are equivalent because x + 3 = 5 can be transformed into x = 2 and each equation has 2 as its only solution. To change, or transform, an equation into an equivalent equation, think of an equation as having two sides that are “in balance.” Original Equation: x + 3 = 5 Subtract 3 from both sides to isolate x on the left. Scale stays in balance Simplify both sides. Equivalent equation: x = 2 Addition and Subtraction Equations You can use transformations to produce equivalent equations. Any transformation you apply to an equation must keep the equation in balance. For instance, if you subtract 3 from one side of the equation, you must also subtract 3 from the other side of the equation. Simplifying one side of the equation does not affect the balance, so you don’t have to change the other side. Addition and Subtraction Equations You can use transformations to produce equivalent equations. TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS ORIGINAL EQUATION EQUIVALENT EQUATION Add the same number to each side. x–3=5 Add 3. x=8 Subtract the same number from each side. x + 6 = 10 Subtract 6. x=4 Simplify one or both sides of the equation. x=8–3 Simplify. x=5 Interchange the sides. 7=x Interchange. x=7 Addition and Subtraction Equations You can use transformations to produce equivalent equations. TRANSFORMATIONS THAT PRODUCE EQUIVALENT EQUATIONS ORIGINAL EQUATION EQUIVALENT EQUATION Add the same number to each side. x–3=5 Add 3. x=8 Subtract the same number from each side. x + 6 = 10 Subtract 6. x=4 Simplify one or both sides of the equation. x=8–3 Simplify. x=5 Interchange the sides. 7=x Interchange. x=7 Inverse operations are operations that undo each other, such as addition and subtraction. Inverse operations help you to isolate the variable in an equation. Adding to Each Side Solve the following equation by adding to each side of the equation. x – 5 = –13 SOLUTION x – 5 = –13 x – 5 + 5 = –13 + 5 x = –8 Write original equation. Add 5 to each side. Simplify. The solution is –8. Check by substituting –8 for x in the original equation. Adding to Each Side Solve the following equation by adding to each side of the equation. x – 5 = –13 SOLUTION x – 5 = –13 x – 5 + 5 = –13 + 5 x = –8 Write original equation. Add 5 to each side. Simplify. The solution is –8. Check by substituting –8 for x in the original equation. CHECK x – 5 = –13 Write original equation. x – 5 = –13 –8 ? Substitute –8 for x. –13 = –13 Solution is correct. Simplifying First Solve the following equation. –8 = n – (– 4) SOLUTION –8 = n – (– 4) Write original equation. –8 = n + 4 Simplify. –8 – 4 = n + 4 – 4 –12 = n Subtract 4 from each side. Simplify. Using Linear Equations in Real-Life Problems The equations used in this chapter are called linear equations. In a linear equation the variable is raised to the first power and does not occur in a denominator, inside a square root symbol, or inside absolute value symbols. LINEAR EQUATION x+5=9 – 4 + n = 2n – 6 NOT A LINEAR EQUATION x2 + 5 = 9 x+3 =7 Later on you will see that linear equations get their name from the fact that their graphs are straight lines. Modeling a Real-Life Problem Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54˚F at 9:00 A.M. to –4˚F at 9:27 A.M. By how many degrees did the temperature fall? SOLUTION Verbal Model Labels Temperature at 9:27 A.M. = Temperature at 9:00 A.M. Temperature at 9:27 A.M. = – 4 Temperature at 9:00 A.M. = 54 Degrees fallen … =T – Degrees fallen Modeling a Real-Life Problem Several record temperature changes have taken place in Spearfish, South Dakota. On January 22, 1943, the temperature in Spearfish fell from 54˚F at 9:00 A.M. to –4˚F at 9:27 A.M. By how many degrees did the temperature fall? SOLUTION … Algebraic Model – 4 = 54 – T – 4 – 54 = 54 – T – 54 –58 = –T 58 = T Write algebraic model. Subtract 54 from each side. Simplify. T is the opposite of –58. The temperature fell by 58˚F. Check this in the original statement of the problem. Translating Verbal Statements Match the real-life problem with an equation. x – 4 = 16 x + 16 = 4 16 – x = 4 You owe $16 to your cousin. You paid x dollars back and you now owe $4. How much did you pay back? SOLUTION Original amount owed (16) – Amount paid back (x) = Amount now owed (4) Translating Verbal Statements Match the real-life problem with an equation. x – 4 = 16 x + 16 = 4 16 – x = 4 You owe $16 to your cousin. You paid x dollars back and you now owe $4. How much did you pay back? The temperature was x˚F. It rose 16˚F and is now 4˚F. What was the original temperature? SOLUTION Original amount owed (16) – Amount paid back (x) = Amount now owed (4) Original temperature (x) + Degrees risen (16) = Temperature now (4) Translating Verbal Statements Match the real-life problem with an equation. x – 4 = 16 x + 16 = 4 16 – x = 4 You owe $16 to your cousin. You paid x dollars back and you now owe $4. How much did you pay back? The temperature was x˚F. It rose 16˚F and is now 4˚F. What was the original temperature? A telephone pole extends 4 feet below ground and 16 feet above ground. What is the total length x of the pole? SOLUTION Original amount owed (16) – Amount paid back (x) = Amount now owed (4) Original temperature (x) + Degrees risen (16) = Temperature now (4) Total length (x) – Length below ground (4) = Length above ground (16) Chapter 3.2 Solving Equations Using Multiplication and Division Goals Solve linear equations using multiplication and division. Use multiplication and division equations to solve problems. VOCABULARY Properties of equality The rules of algebra used to transform equations into equivalent equations. Ratio of a to b If a and b are two quantities measured in the same units, then the ratio of a to b is a/b. Similar triangles Two triangles are similar triangles if they have equal corresponding angles. Chapter 3.3 Solving Multi-Step Equations Goals Use two or more transformations to solve an equation. Use multi-step equations to solve real-life problems. USING TWO OR MORE TRANSFORMATIONS Solving a Linear Equation SOLUTION 1 Solve x + 6 = –8 To isolate the 3 variable, undo the addition and then the multiplication. 1 3 x + 6 = –8 1 3 x + 6 – 6 = –8 – 6 1 x = –14 3 1 3( 3 x) = 3(–14) x = – 42 Write original equation. Subtract 6 from each side. Simplify. Multiply each side by 3. Simplify. Combining Like Terms First Solve 7x – 3x – 8 = 24. 7x – 3x – 8 = 24 Write original equation. 4x – 8 = 24 Combine like terms. 4x – 8 + 8 = 24 + 8 Add 8 to each side. 4x = 32 Simplify. 4x = 32 4 4 Divide each side by 4. x=8 Simplify. The solution is 8. When you check the solution, substitute 8 for each x in the original equation. Using the Distributive Property Solve 5x + 3(x + 4) = 28. METHOD 1 Show All Steps METHOD 2 Do Some Steps Mentally 5x + 3(x + 4) = 28 5x + 3x + 12 = 28 5x + 3(x + 4) = 28 5x + 3x + 12 = 28 8x + 12 = 28 8x + 12 = 28 8x + 12 – 12 = 28 – 12 8x = 16 8x 16 = 8 8 x=2 8x = 16 x=2 Multiplying by a Reciprocal First 6 (x + 3) Solve 66 = – 5 6 = – systematically (x + 3) Solving 66 equations 5 is an example of deductive 6 (x + 3) reasoning. – 5 66 =Notice – 5how – each 6 6 5 solution step is based on number properties or properties of –55 = x + 3 equality. –58 = x It is easier to solve this equation if you don’t distribute – 6 first. 5 Write original equation. 6 Multiply by reciprocal of – 5 . Simplify. Subtract 3 from each side. SOLVING REAL-LIFE PROBLEMS Using a Known Formula SOLUTION Use the formula to convert 95 °F to Celsius. 9 C + 32 F = The Fahrenheit 5 and Celsius scales are related by the equation 95 = 9 C + 32 5 9 F =9 5CC + 32 63 = 5 35 = C A body temperature of 95 °F or lower may indicate the medical condition called hypothermia. What temperature on the Celsius scale may indicate hypothermia? Write known formula. Substitute 95° for F. Subtract 32 from each side. 5 9 Multiply by 9 , the reciprocal of . 5 A temperature of 35°C or lower may indicate hypothermia. Chapter 3.4 Chapter 3.5 Linear Equations and Problem Solving Drawing a Diagram You have learned to use a verbal model to solve real-life problems. Verbal models can be used with other problem solving strategies, such as drawing a diagram. Visualizing a Problem 1 YEARBOOK DESIGN A page of your school yearbook is 8 inches by 11 2 3 inches. The left margin is inch and the space to the right of the pictures is 4 2 7 inches. The space between pictures is 3 inch. How wide can each 8 16 picture be to fit three across the width of the page? SOLUTION DRAW A DIAGRAM The diagram shows that the page width is made up of the width of the left margin, the space to the right of the pictures, two spaces between pictures, and three picture widths Visualizing a Problem 1 YEARBOOK DESIGN A page of your school yearbook is 8 inches by 11 2 3 inches. The left margin is inch and the space to the right of the pictures is 4 2 7 inches. The space between pictures is 3 inch. How wide can each 8 16 picture be to fit three across the width of the page? Verbal Model Labels Left margin + Left margin = Space to right of pictures + 2 • 3 4 Space between pictures = + 3 • (inches) Space to right of pictures = 2 … Space between pictures 7 8 3 16 (inches) (inches) Picture width = w (inches) 1 2 (inches) Page width = 8 Picture width = Page width Visualizing a Problem 1 YEARBOOK DESIGN A page of your school yearbook is 8 inches by 11 2 3 inches. The left margin is inch and the space to the right of the pictures is 4 2 7 inches. The space between pictures is 3 inch. How wide can each 8 16 picture be to fit three across the width of the page? … Algebraic Model 3 7 3 1 +2 +2• +3•w=8 4 8 16 2 1 When you solve for w, you will find that each picture can be 1 inches 2 wide. Using Tables and Graphs as Checks At East High School, 579 students take Spanish. This number has been increasing at a rate of about 30 students per year. The number of students taking French is 217 and has been decreasing at a rate of about 2 students per year. At these rates, when will there be three times as many student taking Spanish as taking French? SOLUTION First write an expression for the number of students taking each language after some number of years. Then set the expression for students taking Spanish equal to three times the expression for students taking French. Subtraction is used in the expression for students taking French, because there is a decrease. Verbal Model Number taking Spanish now 3• … + ( Rate of increase for Spanish Number taking French now – • Number of years = Rate of decrease for French • Number of years ) Using Tables and Graphs as Checks At East High School, 579 students take Spanish. This number has been increasing at a rate of about 30 students per year. The number of students taking French is 217 and has been decreasing at a rate of about 2 students per year. At these rates, when will there be three times as many student taking Spanish as taking French? … Labels … Number of students taking Spanish now = 579 (students) Rate of increase for Spanish = 30 (students per year) Number of years = x (years) Number of students taking French now = 217 (students) Rate of decrease for French = 2 (students per year) Using Tables and Graphs as Checks At East High School, 579 students take Spanish. This number has been increasing at a rate of about 30 students per year. The number of students taking French is 217 and has been decreasing at a rate of about 2 students per year. At these rates, when will there be three times as many student taking Spanish as taking French? … Algebraic Model 579 + 30 • x = 3(217 – 2 • x) 579 + 30 x = 651 – 6 x Use distributive property. 579 + 36 x = 651 Add 6 x to each side. 36 x = 72 x=2 Subtract 579 from each side. Divide each side by 36. At these rates, the number of students taking Spanish will be three times the number of students taking French in two years. Using Tables and Graphs as Checks At East High School, 579 students take Spanish. This number has been increasing at a rate of about 30 students per year. The number of students taking French is 217 and has been decreasing at a rate of about 2 students per year. At these rates, when will there be three times as many student taking Spanish as taking French? CHECK Use the information above to make a table. For example, the number of students taking Spanish in Year 1 is 579 + 30 = 609 Year 0 (now) 1 2 Number taking Spanish 579 609 639 Number taking French 217 215 213 3 • Number taking French 651 645 639 equal Using a Graph as a Check GAZELLE AND CHEETAH A gazelle can run 73 feet per second for several minutes. A cheetah can run faster (88 feet per second) but can only sustain its top speed for about 20 seconds before it is worn out. How for away from the cheetah does the gazelle need to stay for it be be safe? SOLUTION You can use a diagram to picture the situation. If the gazelle is too far away for the cheetah to catch it within 20 seconds, the gazelle is probably safe. If the gazelle is closer to the cheetah than x feet, the cheetah can catch the gazelle. If the gazelle is farther away than x feet, the gazelle is in the safety zone. “Safety zone” 0 x gazelle caught in 20 seconds Using a Graph as a Check GAZELLE AND CHEETAH A gazelle can run 73 feet per second for several minutes. A cheetah can run faster (88 feet per second) but can only sustain its top speed for about 20 seconds before it is worn out. How for away from the cheetah does the gazelle need to stay for it be be safe? To find the “safety zone,” you can first find the starting distance for which the cheetah reaches the gazelle in 20 seconds. To find the distance each animal runs in 20 seconds, use the formula d = r t. Verbal Model … Distance gazelle runs in 20 sec Gazelle’s + starting distance from cheetah = Distance cheetah runs in 20 sec Using a Graph as a Check GAZELLE AND CHEETAH A gazelle can run 73 feet per second for several minutes. A cheetah can run faster (88 feet per second) but can only sustain its top speed for about 20 seconds before it is worn out. How for away from the cheetah does the gazelle need to stay for it be be safe? … Labels Algebraic Model Gazelle’s distance = 73 • 20 (feet) Gazelle’s starting distance from cheetah = x (feet) Cheetah’s distance = 88 • 20 (feet) 73 • 20 + x = 88 • 20 You find that x = 300, so the gazelle needs to stay more than 300 feet away from the cheetah to be safe. Using a Graph as a Check GAZELLE AND CHEETAH A gazelle can run 73 feet per second for several minutes. A cheetah can run faster (88 feet per second) but can only sustain its top speed for about 20 seconds before it is worn out. How for away from the cheetah does the gazelle need to stay for it be be safe? CHECK You can use a table or a graph to check your answer. A graph helps you see the relationships. To make a graph, first make a table and then plot the points. You can just find and mark the points for every 5 seconds. Distance from cheetah's starting point (feet) Gazelle and Cheetah 2000 1500 1000 Gazelle 500 300 Cheetah 0 0 5 10 Gazelle starting at 300 feet from cheetah is caught in 20 seconds. 15 Time (seconds) 20 Chapter 3.6 Solving Decimal Equations Goals Find exact and approximate solutions of equations that contain decimals. Solve real-life problems that use decimals. Round-off error Round-off error is the error produced when a decimal result is rounded in order to provide a meaningful answer. Chapter 3.7 Formulas and Functions Goals Solve a formula for one of its variables. Rewrite an equation in function form. Formula A formula is an algebraic equation that relates two or more real-life quantities. Chapter 3.8 Rates, Ratios, and Percents Goals Use rates and ratios to model and solve real-life problems. Use percents to model and solve real-life problems. Rate of a per b If a and b are two quantities measured in different units, then the rate of a per b is a/b Unit rate A unit rate is a rate per one given unit. Chapter 4 Chapter 4.1 Chapter 4.2 Graphing a Linear Equation A solution of an equation in two variables x and y is an ordered pair (x, y) that makes the equation true. The graph of an equation in x and y is the set of all points (x, y) that are solutions of the equation. In this lesson you will see that the graph of a linear equation is a line. Verifying Solutions of an Equation Use the graph to decide whether the point lies on the graph of x + 3y = 6. Justify your answer algebraically. y (1, 2) 3 1 SOLUTION –1 O –1 x + 3y = 6 ? 1 + 3(2) = 6 76 1 3 Write original equation. Substitute 1 for x and 2 for y. Simplify. Not a true statement. (1, 2) is not a solution of the equation x + 3y = 6, so it is not on the graph. 5 x Verifying Solutions of an Equation Use the graph to decide whether the point lies on the graph of x + 3y = 6. Justify your answer algebraically. y (–3, 3) 3 1 SOLUTION –1 O –1 x + 3y = 6 ? –3 + 3(3) = 6 6=6 1 Write original equation. Substitute –3 for x and 3 for y. Simplify. True statement. (–3, 3) is a solution of the equation x + 3y = 6, so it is on the graph. 3 5 x Graphing a Linear Equation In the previous example you learned that the point (–3, 3) is on the graph of x + 3y = 6, but how many points does a graph have in all? The answer is that most graphs have too many points to list. y 3 Then how can you ever graph an equation? 1 One way is to make a table or a list of a few values, plot enough solutions to recognize a pattern, and then connect the points. –1 O –1 1 3 5 Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input. x Graphing a Linear Equation In the previous example you learned that the point (–3, 3) is on the graph of x + 3y = 6, but how many points does a graph have in all? The answer is that most graphs have too many points to list. y 3 Then how can you ever graph an equation? 1 One way is to make a table or a list of a few values, plot enough solutions to recognize a pattern, and then connect the points. –1 O –1 1 3 5 Even then, the graph extends without limit to the left of the smallest input and to the right of the largest input. When you make a table of values to graph an equation, you may want to choose values for x that include negative values, zero, and positive values. This way you will see how the graph behaves to the left and right of the y -axis. x Graphing an Equation Use a table of values to graph the equation y + 2 = 3x. SOLUTION Rewrite the equation in function form by solving for y. y + 2 = 3x y = 3x – 2 Write original equation. Subtract 2 from each side. Choose a few values for x and make a table of values. Choose x. Substitute to find the corresponding y-value. –2 y = 3(–2) – 2 = – 8 –1 y = 3(–1) – 2 = –5 0 y = 3(0) – 2 = –2 1 y = 3(1) – 2 = 1 2 y = 3(2) – 2 = 4 Graphing an Equation Use a table of values to graph the equation y + 2 = 3x. SOLUTION x y –2 –8 –1 –5 0 1 –2 2 4 1 With the table of values you have found the five solutions (–2, –8), (–1, –5), (0, –2), (1, 1), and (2, 4). y 3 1 –3 –1 –1 Plot the points. Note that they appear to lie on a straight line. The line through the points is the graph of the equation. –6 –8 1 3x Graphing a Linear Equation You may remember examples of linear equations in one variable. The solution of an equation such as 2 x – 1 = 3 is a real number. Its graph is a point on the real number line. The equation y + 2 = 3x in the previous example is a linear equation in two variables. Its graph is a straight line. GRAPHING A LINEAR EQUATION STEP 1 Rewrite the equation in function form, if necessary. STEP 2 Choose a few values of x and make a table of values. STEP 3 Plot the points from the table of values. A line through these points is the graph of the equation. Graphing a Linear Equation Use a table of values to graph the equation 3x + 2y = 1. SOLUTION 1 Rewrite the equation in function form by solving for y. This will make it easier to make a table of values. 3x + 2y = 1 2y = –3x + 1 y=– 3x+ 1 2 2 2 Choose a few values for x and make a table of values. Write original equation. Subtract 3x from each side. Divide each side by 2. Choose x. –2 0 2 Evaluate y. 7 2 1 2 – 5 2 Graphing a Linear Equation Use a table of values to graph the equation 3x + 2y = 1. SOLUTION With the table of values you have found three solutions. ( –2, 3 7 2 ) ( )( , 0, 1 2 5 , 2, – 2 ) Plot the points and draw a line through them. Choose x. –2 0 2 Evaluate y. 7 2 1 2 – 5 2 y 3 2 1 O The graph of 3x + 2y = 1 is shown at the right. –2 –3 –4 1 2 3 x Using the Graph of a Linear Model An Internet Service Provider estimates that the number of households h (in millions) with Internet access can be modeled by h = 6.76 t + 14.9, where t represents the number of years since 1996. Graph this model. Describe the graph in the context of the real-life situation. SOLUTION Make a table of values. Use 0 t 6, for 1996 – 2002. From the table and the graph, you can see that the number of households with Internet access is projected to increase by about 7 million households per year. t 0 1 2 3 4 5 6 h 14.90 21.66 28.42 35.18 41.94 48.70 55.46 Horizontal and Vertical Lines All linear equations in x and y can be written in the form A x + B y = C. When A = 0 the equation reduces to B y = C and the graph is a horizontal line. When B = 0 the equation reduces to A x = C and the graph is a vertical line. EQUATIONS OF HORIZONTAL AND VERTICAL LINES y=b In the coordinate plane, the graph y = b is a horizontal line. x=a In the coordinate plane, the graph x = a is a horizontal line. Graphing y = b Graph the following equation. y=2 SOLUTION y The equation does not have x as a variable. (–3, 2) The y -value is always 2, regardless of the value of x. For instance, here are some points that are solutions of the equation: (–3, 2), (0, 2), (3, 2) 3 (0, 2) 2 (3, 2) y=2 1 O 1 –2 –3 –4 The graph of the equation is a horizontal line 2 units above the x-axis. 2 3 x Graphing x = a Graph the following equation. x = –3 SOLUTION y The x-value is always –3, regardless of the value of y. (–3, 3) 2 x = –3 For instance, here are some points that are solutions of the equation: (–3, –2), (–3, 0), (–3, 3) 3 (–3, 0) 1 O 1 2 (–3, –2) –2 –3 –4 The graph of the equation is a vertical line 3 units to the left of the y-axis. 3 x Chapter 4.3 FINDING THE SLOPE OF A L INE The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. FINDING THE SLOPE OF A L INE y nonvertical line The slope m of a nonvertical line is the number of units the line rises or falls for each unit of horizontal change from left to right. 9 7 The slanted line at the right rises 3 units for each 2 units of horizontal change from left to right. So, the slope m of the line is 3 . 2 Two points on a line are all that is needed to find its slope. rise = 5 - 2 = 3 units 3 (0, 2) run = 2 - 0 = 2 units 1 -1 rise slope = = run (2, 5) 5 5-2 2-0 = 1 -1 3 2 3 5 7 9 x FINDING THE SLOPE OF A L INE y The slope m of the nonvertical line passing through the points (x1, y1) and (x2, y2) is (x2, y2) (x1, y1) Read y1 as “y sub one” Read x1 as “x sub one” (y2 - y1 ) (x2 - x1 ) x m = rise run = change in y change in x = xy22 -- yx11 FINDING THE SLOPE OF A L INE When you use the formula for the slope, m y2 - y1 m = x1 y -y rise changexin 2 -y = run = change in x = x2 - x1 2 1 The order of subtraction important. You label either point as (x1, y1) theis numerator andcan denominator and the other point as (x2,use y2).the However, both the numerator numerator denominator must same subtraction order. and denominator must use the same order. CORRECT y2 - y 1 x2 - x1 Subtraction order is the same INCORRECT y2 - y1 x1 - x2 Subtraction order is different A Line with a Zero Slope is Horizontal (-1, 2) 2) and (3, Find the slope of a line passing through (-1, (3, 2). 2). y SOLUTION 9 (x11,, yy11)) = (-1, 2) and Let (x (x22, y2) = (3, (3, 2) 2) 7 m= y2 - y1 x2 - x1 2 = 32- -(-1) = 0 4 = 0 Rise: difference of y-values 5 rise = 2 - 2 = 0 units Run: difference of x-values (-1, 2) 3 (3, 2) Substitute values. run = 3 - ( -1) = 4 units 1 Simplify. -1 -1 1 3 Slope is zero. Line is horizontal. 5 7 9 x INTERPRETING SLOPE AS A RATE OF CHANGE Slope as a Rate of Change You are parachuting. At time tt == 00 seconds, 2500 feet feet seconds,you open your parachute at hh == 2500 above the ground. At t = 35 seconds, seconds, you are at h = 2115 feet. y 2700 a. What is your rate of change in height? (0, 2500) 2500 b. About when will you reach the ground? Height (feet) 2300 2100 (35, 2115) 1900 5 15 25 Time (seconds) 35 45 x Slope as a Rate of Change SOLUTION a. Use the formula for slope to find the rate of change. change. The change change in time is 35 - 0 = 35 seconds. Subtract in the same order. The change change in height height is 2115 - 2500 = -385 feet. VERBAL MODEL LABELS Rate of Change = Change in Height Change in Time Rate of Change = m (ft/sec) Change in Height = -- 385 (ft) Change in Time = ALGEBRAIC MODEL m = 35 (sec) - 385 35 = -11 Your rate of change is -11 ft/sec. The negative value indicates you are falling. Slope as a Rate of Change b. Falling at a rate of -ll ft/sec, find the time it will take you to fall 2500 ft. Time Time Time = = Distance Distance Rate Rate -2500 ft -11 ft/sec 227 sec You will reach the ground about 227 seconds after opening your parachute. Chapter 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Chapter 5.1 Chapter 5.2 Chapter 5.3 Chapter 5.4 FITTING A LINE TO DATA 8 There Usually, arethere several is no ways single to find line the thatbest-fitting passes through lineall forthe a given data set points, of data so you points. try to In find this lesson, the line you that will best use fits athe graphical data. This is best-fitting line. line. approach. called the best-fitting 6 4 2 –8 –6 –4 –2 0 –2 –4 –6 –8 2 4 6 Approximating a Best-Fitting Line 250 DISCUS THROWS Write an equation of your line. 230 220 210 200 Distance (ft) The winning Olympic discus throws from 1908 to 1996 are plotted in the graph. Approximate the best-fitting line for these throws. 240 190 180 170 160 150 140 130 120 110 100 0 8 16 24 32 40 48 56 64 72 Years since 1900 80 88 96 104 Approximating a Best-Fitting Line 250 SOLUTION Find the slope of the line through these points. (96, 230) 230 220 210 200 Distance (ft) Find two points that lie on the best-fitting line, such as (8, 138) and (96, 230)).. 240 190 180 170 160 150 140 (8, 138) 130 120 110 100 0 8 16 24 32 40 48 56 64 72 Years since 1900 80 88 96 104 Approximating a Best-Fitting Line 250 y2 – y1 230 – 138 m = x – x = 96 – 8 230= 92 88 2 1 220 240 y138 = mx + b+ b = 8.4 Distance (ft) – 138 = y230 = mx 96 – 8 + b 138 = (1.05) = 92 1.05(8) + b 88 1.05 (96, 230) 210 200 190 180 Write slope intercept form. Substitute 1.05 for m, 8 for x, 138 for y. 170 160 Simplify. 150 129.6 = b 140 130 Solve for b. (8, 138) An equation of the best-fitting line is y = 1.05 x + 129.6. 120 110 In most years, the winner of 100 the discus throw was able to 16 24 32 40 48 56 64 72 80 throw the discus farther than 0the8 previous winner. Years since 1900 88 96 104 DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have a positive correlation, which means that the points can be approximated by a line with a positive slope. DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have a negative correlation, which means that the points can be approximated by a line with a negative slope. DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have relatively no correlation, which means that the points cannot be approximated by a line. DETERMINING THE CORRELATION OF X AND Y TYPES OF CORRELATION Positive Correlation Negative Correlation No Correlation Chapter 5.5 Using the Point-Slope Form Previously you learned one strategy for writing a linear equation when given the slope and a point on the line. In this lesson you will learn a different strategy – one that uses the point-slope form of an equation of a line. POINT-SLOPE FORM OF THE EQUATION OF A LINE The point-slope form of the equation of the nonvertical line that passes through a given point (x 1, y 1) with a slope of m is y – y 1 = m (x – x 1). Developing the Point-Slope Form Write an equation of the line. Use the points (2, 5) and (x, y). SOLUTION You are given one point on the line. Let (x, y) be any point on the line. Because (2, 5) and (x, y) are two points on the line, you can write the following expression for the slope of the line. m= y–5 x–2 y–5 2 = x–2 3 y–5= 2 (x – 2) 3 The equation y – 5 = Use formula for slope. 2 Substitute for m. 3 Multiply each side by (x – 2). The graph shows that the slope 2 2 is . Substitute for m in the 3 3 formula for slope. 2 (x – 2) is written in point-slope form. 3 Using the Point-Slope Form You can use the point-slope form when you are given the slope and a point on the line. In the point-slope form, (x 1, y 1) is the given point and (x, y) is any other point on the line. You can also use the point-slope form when you are given two points on the line. First find the slope. Then use either given point as (x 1, y 1). Using the Point-Slope Form Write an equation of the line shown below. SOLUTION First find the slope. Use the points (x 1, y 1) = (–3, 6) and (x 2, y 2) = (1, –2). y2 – y1 m= x –x 2 1 = –2 – 6 1 – (–3) = –8 4 = –2 Using the Point-Slope Form Write an equation of the line shown below. SOLUTION Then use the slope to write the point-slope form. Choose either point as (x 1, y 1). y – y 1 = m (x – x 1) Write point-slope form. y – 6 = –2[x – (–3)] Substitute for m, x 1 and y 1. y – 6 = –2(x + 3) Simplify. y – 6 = –2 x – 6 Use distributive property. y = –2 x Add 6 to each side. Modeling a Real-Life Situation MARATHON The information below was taken from an article that appeared in a newspaper. Use the information to write a linear model for optimal running pace, then use the model to find the optimal running pace for a temperature of 80°F. Writing and Using a Linear Model SOLUTION Let T represent the temperature in degrees Fahrenheit. Let P represent the optimal pace in feet per second. From the article, you know that the optimal running pace at 60°F is 17.6 feet per second so one point on the line is (T 1, P 1) = (60, 17.6). Find the slope of the line. m= change in P – 0.3 = = – 0.06 change in T 5 Use the point-slope form to write the model. P – P 1 = m (T – T 1) Write the point-slope form. P – 17.6 = (– 0.06)(T – 60) Substitute for m, T1, and P1. P – 17.6 = – 0.06T + 3.6 Use distributive property. P = –0.06T + 21.2 Add 17.6 to each side. Writing and Using a Linear Model SOLUTION Use the model P = –0.06T + 21.2 to find the optimal pace at 80°F. P = – 0.06(80) + 21.2 = 16.4 At 80°F the optimal running pace is 16.4 feet per second. CHECK A graph can help you check this result. You can see that as the temperature increases the optimal running pace decreases. Chapter 5.6 Chapter 5.7 Chapter 6 Chapter 6.1 Chapter 6.2 Chapter 6.3 Solving Compound Inequalities Previously you studied four types of simple inequalities. In this lesson you will study compound inequalities. A compound inequality consists of two inequalities connected by and or or. Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. SOLUTION 0<x4 –1 0 1 2 3 4 5 This inequality is also written as 0 < x and x 4. Solving Compound Inequalities Previously you studied four types of simple inequalities. In this lesson you will study compound inequalities. A compound inequality consists of two inequalities connected by and or or. Write an inequality that represents the set of numbers and graph the inequality. All real numbers that are greater than zero and less than or equal to 4. SOLUTION 0<x4 –1 0 1 2 3 4 5 This inequality is also written as 0 < x and x 4. All real numbers that are less than –1 or greater than 2. SOLUTION x < –1 or x > 2 –2 –1 0 1 2 3 4 Compound Inequalities in Real Life Write an inequality that describes the elevations of the regions of Mount Rainier. Timber region below 6000 ft 14,140 ft Glacier and permanent snow field region SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000 y < 6000 7500 ft 6000 ft Alpine meadow region Timber region 2000 ft 0 ft Compound Inequalities in Real Life Write an inequality that describes the elevations of the regions of Mount Rainier. Timber region below 6000 ft Alpine meadow region below 7500 ft 14,140 ft Glacier and permanent snow field region SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000 y < 6000 Alpine meadow region: 6000 y < 7500 7500 ft 6000 ft Alpine meadow region Timber region 2000 ft 0 ft Compound Inequalities in Real Life Write an inequality that describes the elevations of the regions of Mount Rainier. Timber region below 6000 ft Alpine meadow region below 7500 ft 14,140 ft Glacier and permanent snow field region SOLUTION Let y represent the approximate elevation (in feet). Timber region: 2000 y < 6000 Alpine meadow region: 6000 y < 7500 Glacier and permanent snow field region: 7500 y 14,410 Glacier and permanent snow field region 7500 ft 6000 ft Alpine meadow region Timber region 2000 ft 0 ft Solving a Compound Inequality with And Solve –2 3x – 8 10. Graph the solution. SOLUTION Isolate the variable x between the two inequality symbols. –2 3 x – 8 10 Write original inequality. 6 3 x 18 Add 8 to each expression. 2x6 Divide each expression by 3. The solution is all real numbers that are greater than or equal to 2 and less than or equal to 6. 0 1 2 3 4 5 6 7 Solving a Compound Inequality with Or Solve 3x + 1 < 4 or 2x – 5 > 7. Graph the solution. SOLUTION A solution of this inequality is a solution of either of its simple parts. You can solve each part separately. 3x + 1 < 4 or 2x – 5 > 7 3x < 3 or 2x > 12 x<1 or x>6 The solution is all real numbers that are less than 1 or greater than 6. –1 0 1 2 3 4 5 6 7 Reversing Both Inequality Symbols Solve –2 < –2 – x < 1. Graph the solution. SOLUTION Isolate the variable x between the two inequality signs. –2 < –2 – x < 1 Write original inequality. 0 < –x < 3 Add 2 to each expression. 0 > x > –3 Multiply each expression by –1 and reverse both inequality symbols. To match the order of numbers on a number line, this compound is usually written as –3 < x < 0. The solution is all real numbers that are greater than –3 and less than 0. –5 – 4 –3 –2 –1 0 1 2 3 Modeling Real-Life Problems You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the smallest value d might have? SOLUTION Problem Solving Strategy DRAW A DIAGRAM A good way to begin this problem is to draw a diagram with the school at the center of a circle. Bill’s home is somewhere on the circle with radius 3 miles and center at the school. Mary’s home is somewhere on the circle with radius 2 miles and center at the school. Modeling Real-Life Problems You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the smallest value d might have? SOLUTION If both homes are on the same line going toward school, the distance is 1 mile. Modeling Real-Life Problems You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the largest value d might have? SOLUTION If both homes are on the same line but in opposite directions from school, the distance is 5 miles. Modeling Real-Life Problems You have a friend Bill who lives three miles from school and another friend Mary who lives two miles from the same school. You wish to estimate the distance d that separates their homes. What is the largest value d might have? SOLUTION If both homes are on the same line but in opposite directions from school, the distance is 5 miles. Write an inequality that describes all the possible values that d might have. SOLUTION The values of d can be described by the inequality 1 d 5. Chapter 6.4 SOLVING ABSOLUTE-VALUE EQUATIONS You can solve some absolute-value equations using mental math. For instance, you learned that the equation | x | 8 has two solutions: 8 and 8. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative. Solving an Absolute-Value Equation Solve | x 2 | 5 SOLUTION The expression x 2 can be equal to 5 or 5. x 2 IS IS POSITIVE POSITIVE x 2 IS NEGATIVE || xx 22 || 55 xx 22 5 5 |x2|5 xx 77 x 2 5 x 3 The equation has two solutions: 7 and –3. CHECK |72||5|5 | 3 2 | | 5 | 5 Solving an Absolute-Value Equation Solve | 2x 7 | 5 4 SOLUTION Isolate the absolute value expression on one side of the equation. 2x 7 IS IS POSITIVE POSITIVE 2x 7 IS IS NEGATIVE NEGATIVE | 2x 7 | 5 4 | 2x 7 | 5 4 | 2x 7 | 9 | 2x 7 | 9 2x 7 +9 2x 16 x8 2x 7 9 9 2x 2 TWO SOLUTIONS x 1 Solving an Absolute-Value Equation Recall that |xx is | isthe thedistance distancebetween betweenxxand and0.0.IfIf x | x | 8,8,then then any number between 8 and 8 is a solution of the inequality. 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 You can use the following properties to solve absolute-value inequalities and equations. 7 8 SOLVING ABSOLUTE-VALUE INEQUALITIES SOLVING ABSOLUTE-VALUE EQUATIONS AND INEQUALITIES | ax b | c means ax b c and a x b c. means valueais | a xWhen b | an c absolute x less b than c and a x the b c. a number, inequalities are connected by and. When an absolute means | a xvalue b | iscgreater a x b cthe inequalities or a x bare c. than a number, connected by or. | ax b | c means ax b c or a x b c. | ax b | c means ax b c or a x b c. Solving an Absolute-Value Inequality Solve | x 4 | < 3 x 4 IS POSITIVE |x4|3 x 4 3 x7 x 4 IS NEGATIVE |x4|3 x 4 3 x1 Reverse inequality symbol. The solution is all real numbers greater than 1 and less than 7. This can be written as 1 x 7. Solving an Absolute-Value Inequality Solve 1 | 3 6 and graph 2x + 1| 2x IS POSITIVE 2x +the 1 ISsolution. NEGATIVE | 2x 1 | 3 6 | 2x 1 | 3 6 2x + 1 IS POSITIVE | 2x|2x1 | 13| 69 2x + 1 IS NEGATIVE 1 | 69 | 2x|2x1 | 3 2x 1 9 | 2x 1 | 9 2x 10 2x 8 2x 1 9 2x 1 +9 x4 x 5 2x 10 2x 8 The solution is all real numbers greater than or equal x4 x 5 to 4 or less than or equal to 5. This can be written as the compound inequality x 5 or x 4. Reverse 2x11| +9 | 2x 9 inequality symbol. 6 5 4 3 2 1 0 1 2 3 4 5 6 Writing an Absolute-Value Inequality You work in the quality control department of a manufacturing company. The diameter of a drill bit must be between 0.62 and 0.63 inch. a. Write an absolute-value inequality to represent this requirement. b. Does a bit with a diameter of 0.623 inch meet the requirement? Writing an Absolute-Value Inequality The diameter of a drill bit must be between 0.62 and 0.63 inch. a. Write an absolute-value inequality to represent this requirement. Let d represent the diameter (in inches) of the drill bit. Write a compound inequality. 0.62 d 0.63 Find the halfway point. 0.625 Subtract 0.625 from each part of the compound inequality. 0.62 0.625 d 0.625 0.63 0.625 0.005 d 0.625 0.005 Rewrite as an absolute-value inequality. | d 0.625 | 0.005 This inequality can be read as “the actual diameter must differ from 0.625 inch by no more than 0.005 inch.” Writing an Absolute-Value Inequality The diameter of a drill bit must be between 0.62 and 0.63 inch. b. Does a bit with a diameter of 0.623 meet the requirement? | d 0.625 | 0.005 | 0.623 0.625 | 0.005 | 0.002 | 0.005 0.002 0.005 Because 0.002 0.005, the bit does meet the requirement. Chapter 6.5 Chapter 6.6 Chapter 6.7 Chapter 7 Systems of Linear Equations 7.1 Solving Linear Systems by Graphing 7.2 Solving Linear Systems by Substitution 7.3 Solving Linear Systems by Linear Combinations 7.4 Applications of Linear Systems 7.5 Special Types of Linear Systems 7.6 Solving Systems of Linear Inequalities 7.1 Solving Linear Systems by Graphing Goals Solve a system of linear equations by graphing. Model a real-life problem using a linear system. System of linear equations - Two or more linear equations in the same variables form a system of linear equations, or simply a linear system. Solution of a system of linear equations A solution of a system of linear equations in two variables x and y is an ordered pair (x, y) that satisfies each equation in the system. SOLVING A LINEAR SYSTEM USING GRAPHAND-CHECK To use the graph-and-check method to solve a system of linear equations in two variables, use the following steps. Step 1 Write each equation in a form that is easy to graph Step 2 Graph both equations in the same coordinate plane Step 3 Estimate the coordinates of the point of intersection Step 4 Check the coordinates algebraically by substituting into each equation of the original linear system. 7.2 Solving Linear Systems by Substitution Goals Use substitution to solve a linear system. Model a real-life situation using a linear system. SOLVING A LINEAR SYSTEM BY SUBSTITUTION Step 1 Solve one of the equations for one of its variables . Step 2 Substitute the expression from Step 1 into the other equation and solve for the other variable . Step 3 Substitute the value from Step 2 into the revised equation from Step 1 and solve. Step 4 Check the solution in each of the original equations. IDENTIFYING THE NUMBER OF SOLUTIONS NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y Lines intersect one solution x IDENTIFYING THE NUMBER OF SOLUTIONS NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y Lines are parallel no solution x IDENTIFYING THE NUMBER OF SOLUTIONS NUMBER OF SOLUTIONS OF A LINEAR SYSTEM y Lines coincide infinitely many solutions (the coordinates of every point on the line) x IDENTIFYING THE NUMBER OF SOLUTIONS CONCEPT NUMBER OF SOLUTIONS OF A LINEAR SYSTEM SUMMARY y y y x x x Lines intersect Lines are parallel Lines coincide one solution no solution infinitely many solutions A Linear System with No Solution Show that this linear system METHOD 1: GRAPHING has no solution. 2x y 5 2x y 1 Equation 1 Equation 2 Rewrite each equation in slope-intercept form. y –2 x 5 y –2 x 1 Revised Equation 1 Revised Equation 2 6 Graph the linear system. 5 y 2x 5 4 The lines are parallel; they have the same slope but different y-intercepts. Parallel lines never intersect, so the system has no solution. 3 y 2x 1 5 4 3 2 2 1 1 1 0 1 2 3 4 5 A Linear System with No Solution Show that this linear METHOD 2: SUBSTITUTION system has no solution. 2x y 5 2x y 1 Equation 1 Equation 2 Because Equation 2 can be revised to y –2 x 1, you can substitute –2 x 1 for y in Equation 1. 2x y 5 2 x (–2 x 1) 5 15 Write Equation 1. Substitute –2 x 1 for y. Simplify. False statement! Once the variables are eliminated, the statement is not true regardless of the values of x and y. This tells you the system has no solution. A Linear System with Many Solutions Show that this linear system has infinitely many solutions. METHOD 1: GRAPHING Rewrite each equation in slope-intercept form. –2x y 3 – 4 x 2y 6 y 2x 3 y 2x 3 Equation 1 Equation 2 Revised Equation 1 Revised Equation 2 6 Graph the linear system. –4x 2y 6 From these graphs you can see that the equations represent the same line. Any point on the line is a solution. –2x y 3 5 4 3 2 1 5 4 3 2 1 1 0 1 2 3 4 5 A Linear System with Many Solutions Show that this linear system M ETHOD 2: LINEAR COMBINATIONS has infinitely many solutions. –2x y 3 – 4 x 2y 6 Equation 1 Equation 2 You can multiply Equation 1 by –2. 4x – 2y – 6 – 4 x 2y 6 0 0 Multiply Equation 1 by –2. Write Equation 2. Add Equations. True statement! The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This result tells you that the linear system has infinitely many solutions. MODELING A REAL-LIFE PROBLEM Error Analysis GEOMETRY CONNECTION Two students are visiting the mysterious statues on Easter Island in the South Pacific. To find the heights of two statues that are too tall to measure, they tried a technique involving proportions. They measured the shadow lengths of the statues at 2:00 P.M. and again at 3:00 P.M. 3:00 2:00 Error Analysis SOLUTION They let a and b represent the heights of the two statues. Because the ratios of corresponding sides of similar triangles are equal, the students wrote the following two equations. a bb 27 30 aa = bb = 2:00 3:00 27 30 27 ft 30 18 ft 20 20 18 18 20 30 aa = 27 b 18 20 a = 3 b 2 Error Analysis They got stuck because the equations that they wrote are equivalent. All that the students found from these measurements was that a = 3 b. In other words, the height of the first statue is one and 2 one-half times the height of the second. a To findb the heights of the two statues, the a b = = = students need to measure the shadow length of something else whose height 27 30 18 20 they know or can measure. Then they 3 a = 3 b a = b can compare those measurements 2with 2 the measurements of the statue. Solving Systems of Linear Inequalities The graph of a linear inequality in two variables is a half-plane. The boundary line of the half-plane is dashed if the inequality is < or > and solid if the inequality is or . Two or more linear inequalities form a system of linear inequalities or simply a system of inequalities. A solution of a system of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system. The graph of a system of linear inequalities is the graph of all solutions of the system. A Triangular Solution Region Graph the system of linear inequalities. y<2 x –1 y>x–2 Inequality 1 Inequality 2 Inequality 3 SOLUTION Graph all three inequalities in the same coordinate plane. The graph of the system is the overlap, or the intersection, of the three half-planes shown. CHECK You can see from the graph that the point (2, 1) is a solution of the system. To check this, substitute the point into each inequality. 1<2 2 –1 1>2–2 True. True. True. A Triangular Solution Region Graph the system of linear inequalities. The point (0, 3) is not in the graph of the system. Notice (0, 3) is not a solution of inequality 1. This point is not a solution of the system. When graphing a system of inequalities, it is helpful to find each corner point (or vertex). For instance, this graph has three corner points: (–1, 2), (–1, –3), and (4, 2). y<2 x –1 y>x–2 Inequality 1 Inequality 2 Inequality 3 Solution Region Between Parallel Lines Write a system of inequalities that defines the shaded region shown. SOLUTION The graph of one inequality is the half-plane below the line y = 3. The graph of the other inequality is the half-plane above the line y = 1. The shaded region of the graph is the horizontal band that lies between the two horizontal lines, y = 3 and y = 1, but not on the lines. The system of linear inequalities at the right defines the shaded region. y<3 y>1 Inequality 1 Inequality 2 A Quadrilateral Solution Region Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. x0 The graph of the first inequality is the half-plane on and to the right of the y-axis. A Quadrilateral Solution Region Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. x0 y0 The graph of the first inequality is the half-plane on and to the right of the y-axis. The graph of the second inequality is the half-plane on and above of the x-axis. A Quadrilateral Solution Region Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. y2 The graph of the third inequality is the half-plane on and below the horizontal line y = 2. A Quadrilateral Solution Region Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. y2 The graph of the third inequality is the half-plane on and below the horizontal line y = 2. 1 y – 2 x + 3. The graph of the fourth inequality is the half-plane on and below the line 1 x + 3. y= – 2 A Quadrilateral Solution Region Graph the system of linear inequalities. Label each vertex of the solution region. Describe the shape of the region. The region that lies in all four half-planes is a quadrilateral with vertices at (0, 2), (0, 0), (6, 0), and (2, 2). Note that (0, 3) is not a vertex of the solution region even though two boundary lines meet at that point. Modeling A Real-Life Problem You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1000 and a longthrow spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many mediumthrow spotlights and long-throw spotlights should be ordered to get the free delivery. Verbal Model Number of medium-throws Number of long-throws Number of mediumthrows … • Price of a mediumthrow 3 1 + Number of longthrows • Price of a longthrow 10,000 Modeling A Real-Life Problem You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1000 and a longthrow spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many mediumthrow spotlights and long-throw spotlights should be ordered to get the free delivery. … Labels … Number of medium-throws = x (no units) Number of long-throws = y (no units) Price of a medium-throw = 1000 (dollars) Price of a long-throw = 3500 (dollars) Modeling A Real-Life Problem You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1000 and a longthrow spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many mediumthrow spotlights and long-throw spotlights should be ordered to get the free delivery. … Algebraic Model x3 Inequality 1 y1 Inequality 2 1000x + 3500y 10,000 Inequality 3 Modeling A Real-Life Problem You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1000 and a longthrow spotlight costs $3500. The minimum order for free delivery is $10,000. Write and graph a system of linear inequalities that shows how many mediumthrow spotlights and long-throw spotlights should be ordered to get the free delivery. The graph of the system of inequalities is shown. Any point in the shaded region of the graph is a solution to the system. A fraction of a spotlight cannot be ordered, so only ordered pairs of integers in the shaded region will correctly answer the problem. Modeling A Real-Life Problem You are ordering lighting for a theater so the spotlights can follow the performers. The lighting technician needs at least 3 medium-throw spotlights and at least 1 long-throw spotlight. A medium-throw spotlight costs $1000 and a longthrow spotlight costs $3500. The minimum order for free delivery is $10,000. Will an order of 4 medium-throw spotlights and 1 long-throw spotlight be delivered free? The point (4, 1) is outside the solution region, so an order of 4 medium-throw spotlights and 1 long-throw spotlight would not be delivered free. Chapter 8 Exponents and Exponential Functions 8.1 Multiplication Properties of Exponents 8.2 Zero and Negative Exponents 8.3 Division properties of Exponents 8.4 Scientific Notation 8.5 Exponential Growth Functions 8.6 Exponential Decay Functions Chapter 8.1 Multiplication Properties of Exponents INVESTIGATING POWERS How can you use addition to multiply exponential expressions? 1 Complete the table. To simplify an expression, expand the product. Then count the factors. Product of powers 2 Expanded product Number Product as of factors a power 73 · 72 (7 · 7 · 7) · (7 · 7) 5 75 24 · 24 (2 · 2 · 2 · 2) · (2 · 2 · 2 · 2) 8 28 x4 · x5 (x · x · x · x) · (x · x · x · x · x) 9 x9 Add a column to your table that shows the sum of the exponents that are in the first column. What pattern do you notice? INVESTIGATING POWERS How can you use multiplication to raise an exponential expression to a power? 3 Complete the table. To simplify an expression, expand the product. Then count the factors. Power of Expanded product a power (5 2) 3 (5 2) · (5 2) · (5 2) [(–3) 2] 2 [(–3) 2] · [(–3) 2] (b 2) 4 4 Expanded product Number Product as of factors a power (5 · 5) · (5 · 5) · (5 · 5) 6 56 [(–3) · (–3)] · [(–3) · (–3)] 4 (–3) 4 8 b8 (b 2) · (b 2) · (b 2) · (b 2) (b · b) · (b · b) · (b · b) · (b · b) Add a column to your table that shows the product of the exponents that are in the first column. What pattern do you notice? Multiplying Exponential Expressions To multiply two powers that have the same base, you add the exponents. Here is an example. 5 factors a2 · a3 = a · a · a · a · a = a5 = a2 + 3 2 factors 3 factors Multiplying Exponential Expressions To multiply two powers that have the same base, you add the exponents. Here is an example. 5 factors a2 · a3 = a · a · a · a · a = a5 = a2 + 3 2 factors 3 factors To find a power of a power, you multiply the exponents. Here is an example. 3 factors 6 factors (a 2) 3 = a 2 · a 2 · a 2 = a · a · a · a · a · a = a 6 = a 2 · 3 Multiplying Exponential Expressions CONCEPT SUMMARY MULTIPLICATION PROPERTIES OF EXPONENTS Let a and b be numbers and let m and n be positive integers. PRODUCT OF POWERS PROPERTY To multiply powers having the same base, add the exponents. a m · a n = a m+n Example: 3 2 · 3 7 = 3 2+7 = 3 9 POWER OF A POWER PROPERTY To find the power of a power, multiply the exponents. (a m) n = a m · n Example: (5 2) 4 = 5 2 · 4 = 5 8 POWER OF A PRODUCT PROPERTY To find the power of a product, find the power of each factor and multiply (a · b) m = a m · b m Example: (2 · 3) 6 = 2 6 · 3 6 Using the Product of Powers Property Use the product of powers property to simplify the expression. 53 · 56= 53 + 6 = 59 Using the Product of Powers Property Use the product of powers property to simplify the expression. 53 · 56= 53 + 6 = 59 x2 · x3 · x4 = x2 + 3 + 4 = x9 Using the Power of a Power Property When you use the power of a power property, it is the quantity within the parentheses that is raised to the power, not the individual terms. Correct (a + 1) 3 = (a + 1)(a + 1)(a + 1) Incorrect (a + 1) 3 = a 3 + 1 3 Using the Power of a Power Property When you use the power of a power property, it is the quantity within the parentheses that is raised to the power, not the individual terms. Correct (a + 1) 3 = (a + 1)(a + 1)(a + 1) Incorrect (a + 1) 3 = a 3 + 1 3 Use the power of a power property to simplify the expression. ( y 2) 4 = y 2 · 4 = y8 Using the Power of a Power Property When you use the power of a power property, it is the quantity within the parentheses that is raised to the power, not the individual terms. Correct (a + 1) 3 = (a + 1)(a + 1)(a + 1) Incorrect (a + 1) 3 = a 3 + 1 3 Use the power of a power property to simplify the expression. ( y 2) 4 = y 2 · 4 = y8 [(a + 1) 2 ] 5 = (a+1) 2 · 5 = (a+1) 10 Using the Power of a Product Property Use the power of a product property to simplify the expression. (4yz) 3 = (4 · y · z)3 Identify factors. = 43 · y3 · z3 Raise each factor to a power. = 64y 3z 3 Simplify. Using the Power of a Product Property Use the power of a product property to simplify the expression. (4yz) 3 = (4 · y · z)3 Identify factors. = 43 · y3 · z3 Raise each factor to a power. = 64y 3z 3 Simplify. – (2w) 2= – (2 · w) 2 Identify factors. = – (2 2 · w 2) Raise each factor to a power. = – 4w 2 Simplify. Using All Three Properties Simplify (4x 2y) 3 · x 5. SOLUTION (4x 2y) 3 · x 5 = 4 3 · (x 2) 3 · y 3 · x 5 Power of a product = 64 · x 6 · y 3 · x 5 Power of a power = 64x 11y 3 Product of powers Using Powers in Real Life Some farmers use center-pivot irrigation, which is a sprinkler system that revolves around a center pivot. The large irrigation circles help farmers to conserve water, maximize crop yield, and reduce cost of pesticides. Find the ratio of the area of the larger irrigation circle to the area of the smaller irrigation circle. SOLUTION (2r) 2 Ratio = r 2 · 22 · r2 = · r2 · 4 · r2 = · r2 4 = 1 Using Powers in Real Life Some farmers use center-pivot irrigation, which is a sprinkler system that revolves around a center pivot. The large irrigation circles help farmers to conserve water, maximize crop yield, and reduce cost of pesticides. Write a general statement about doubling the radius of an irrigation circle. SOLUTION Doubling the radius of an irrigation circle makes the area four times as large. Chapter 8.2 Zero and Negative Exponents Chapter 8.3 Division Properties of Exponents Chapter 8.4 Scientific Notation Chapter 8.5 Exponential Growth Functions WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. y = C (1 + r)t (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 1 SOLVE A SIMPLER PROBLEM Find the account balance A1 after 1 year and multiply by the growth factor to find the balance for each of the following years. The growth rate is 0.08, so the growth factor is 1 + 0.08 = 1.08. A1 = 500(1.08) = 540 Balance after one year A2 = 500(1.08)(1.08) = 583.20 Balance after two years A3 = 500(1.08)(1.08)(1.08) = 629.856 Balance after three years A6 = Balance after six years • • • 500(1.08) 6 793.437 • • • Finding the Balance in an Account COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 2 USE A FORMULA Use the exponential growth model to find the account balance A. The growth rate is 0.08. The initial value is 500. EXPONENTIAL GROWTH MODEL is the initial amount. C 500 is the initial amount. 6tisisthe period. thetime time period. A6y==500 C (1 (1++0.08) r)t 6 (1(1++0.08) thegrowth growth factor, factor, 0.08 is the growthrate. rate. r) isisthe r is the growth 6 793.437 The)percent of increase is 100r. A6 = 500(1.08 Balance after 6 years Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. a. What is the percent of increase each year? SOLUTION The population triples each year, so the growth factor is 3. 1+r = 3 So, the growth rate r is 2 and the percent of increase each year is 200%. Reminder: percent increase is 100r. Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. b. What is the population after 5 years? SOLUTION After 5 years, the population is P = C(1 + r) t Exponential growth model = 20(1 + 2) 5 Substitute C, r, and t. = 20 • 3 5 Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years. Help GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor Graph the growth of the rabbit population. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. SOLUTION t 0 P 20 1 2 3 60 180 540 4 5 1620 4860 6000 Population 5000 4000 P = 20 ( 3 ) t 3000 Here, the large growth factor of 3 corresponds to a rapid increase 2000 1000 0 1 2 3 4 Time (years) 5 6 7 Chapter 8.6 Exponential Decay Functions WRITING EXPONENTIAL DECAY MODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. EXPONENTIAL DECAY MODEL C is the initial amount. t is the time period. y = C (1 – r)t (1 – r ) is the decay factor, r is the decay rate. The percent of decrease is 100r. Writing an Exponential Decay Model From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? COMPOUND INTEREST SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. y = C(1 – r) t Exponential decay model = (1)(1 – 0.035) t Substitute 1 for C, 0.035 for r. = 0.965 t Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.96515 Substitute 15 for t. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59. GRAPHING EXPONENTIAL DECAY MODELS Graphing the Decay of Purchasing Power Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. SOLUTION Help Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t 0 1 2 3 4 5 6 7 8 9 10 y 1.00 0.965 0.931 0.899 0.867 0.837 0.808 0.779 0.752 0.726 0.7 y = 0.965t 0.8 (dollars) Purchasing Power 1.0 0.6 0.4 0.2 0 1 2 3 Your dollar of today will be worth about 70 cents in ten years. 4 5 6 7 8 Years From Now 9 10 11 12 GRAPHING EXPONENTIAL DECAY MODELS CONCEPT EXPONENTIAL GROWTH AND DECAY MODELS SUMMARY EXPONENTIAL GROWTH MODEL y = C (1 + r)t EXPONENTIAL DECAY MODEL y = C (1 – r)t An exponential model y = a • b t represents exponential b > 1 and exponential (1 decay 0 <decay b < 1.factor, (1 growth + r) is theifgrowth factor, – r) isifthe Ct is is the the initial time period. amount. C) (0, C) rate. r is the(0,growth rate. r is the decay 1+r>1 0<1–r<1 Chapter 9 Quadratic Equations and Functions 9.1 Solving Quad. Equations by Finding Sq. Roots 9.2 Simplifying Radicals 9.3 Graphing Quad. Functions 9.4 Solving Quad. Equations by Graphing 9.5 Solving Quad. Equations by the Quadratic Formula 9.6 Applications of the Discriminant 9.7 Graphing Quad. Inequalities 9.8 Comparing Linear, Exponential, & Quad. Models Chapter 9.1 Chapter 9.2 Chapter 9.3 Chapter 9.4 SOLVING QUADRATIC EQUATIONS GRAPH ICALLY SOLVING QUADRATIC EQUATIONS USING GRAPHS The solution of a quadratic equation in one variable x can be solved or checked graphically with the following steps: STEP 1 Write the equation in the form ax 2 + bx + c = 0. STEP 2 Write the related function y = ax 2 + bx + c. STEP 3 Sketch the graph of the function y = ax 2 + bx + c. The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts. Checking a Solution Using a Graph Solve 1 2 x = 8 algebraically. 2 Check your solution graphically. SOLUTION 1 2 x = 8 2 CHECK Write original equation. x 2 = 16 Multiply each side by 2. x= 4 Find the square root of each side. Check these solutions using a graph. Checking a Solution Using a Graph CHECK Check these solutions using a graph. 1 Write the equation in the form ax 2 + bx + c = 0 1 2 x =8 2 1 2 x –8=0 2 2 Rewrite original equation. Subtract 8 from both sides. Write the related function y = ax2 + bx + c. y = 1 x2 – 8 2 Checking a Solution Using a Graph CHECK Check these solutions using a graph. 2 Write the related function y = ax2 + bx + c. –4, 0 y = 1 x2 – 8 2 3 1 Sketch graph of y = x2 – 8. 2 The x-intercepts are 4, which agrees with the algebraic solution. 4, 0 Solving an Equation Graphically Solve x 2 – x = 2 graphically. Check your solution algebraically. SOLUTION 1 Write the equation in the form ax 2 + bx + c = 0 x2 – x = 2 x2 – x – 2 = 0 2 Write original equation. Subtract 2 from each side. Write the related function y = ax2 + bx + c. y = x2 – x – 2 Solving an Equation Graphically 2 Write the related function y = ax2 + bx + c. y = x2 – x – 2 3 Sketch the graph of the function y = x2 – x – 2 From the graph, the x-intercepts appear to be x = –1 and x = 2. – 1, 0 2, 0 Solving an Equation Graphically From the graph, the x-intercepts appear to be x = –1 and x = 2. – 1, 0 CHECK You can check this by substitution. Check x = –1: Check x = 2: x2 – x = 2 x2 – x = 2 ? ? (–1) = 2 22 – 2 = 2 1+1=2 4–2=2 (–1) 2 – 2, 0 USING QUADRATIC MODELS IN REAL LIFE Comparing Two Quadratic Models A shot put champion performs an experiment for your math class. Assume that both times he releases the shot with the same initial speed but at different angles. The path of each put can be modeled by one of the equations below, where x represents the horizontal distance (in feet) and y represents the height (in feet). Initial angle of 35º: y = – 0.010735x 2 + 0.700208x + 6 Initial angle of 65º: y = – 0.040330x 2 + 2.144507x + 6 Which angle results in a farther throw? Comparing Two Quadratic Models SOLUTION Begin by graphing both models in the same coordinate plane. Use only positive x-values because x represents the distance of the throw. Which angle results in a farther throw? Comparing Two Quadratic Models PARABOLIC MOTION An angle of 45 º produces the maximum distance when an object is propelled from ground level. An angle smaller than 45 º is better when the shot is released above the ground. If the shot is released from 6 feet above the ground, a 43 º angle produces a maximum distance of 74.25 feet. 45º 65º 43º 35º Chapter 9.5 Using the Quadratic Formula In this lesson you will learn how to use the quadratic formula to solve any quadratic equation. THE QUADRATIC FORMULA The solutions of the quadratic equation a x 2 + b x + c = 0 are x= –b b2 – 4 a c 2a when a 0 and b 2 – 4 a c ≥ 0. You can read this formula as: x equals the opposite of b, plus or minus the square root of b squared minus 4 a c, all divided by 2 a. Using the Quadratic Formula Solve x 2 + 9x + 14 = 0. SOLUTION Use the quadratic formula. 1x 2 + 9x + 14 = 0 x= x= x= – 9b 9b 2 – 4( 1a )(14 c) 2( a1 ) Identify a = 1, b = 9, and c = 14. Substitute values in the quadratic formula. –9 81 – 56 2 Simplify. –9 25 Simplify. 2 The equation has two solutions: –9 + 5 = –2 x= 2 –9 5 x= 2 –9 – 5 = –7 x= 2 Using the Quadratic Formula Check the solutions to x 2 + 9x + 14 = 0 in the original equation. Check x = –2 : Check x = –7 : x 2 + 9x + 14 = 0 ? (–2) 2 + 9(–2) + 14 = 0 ? x 2 + 9x + 14 = 0 ? (–7) 2 + 9(–7) + 14 = 0 ? 4 + –18 + 14 = 0 49 + –63 + 14 = 0 0=0 0=0 Finding the x-Intercepts of a Graph Find the x-intercepts of the graph of y = –x 2 – 2 x + 5. SOLUTION The x-intercepts occur when y = 0. y = –x 2 – 2 x + 5 Write original equation. 0 = –1x 2 – 2 x + 5 Substitute 0 for y, and identify a = 1, b = –2, and c = 5. –(–2 b) x= x= x= 2 2 (–2 b ) 2 – 4(–1 a )( 5c ) 2(–1 a) 4 + 20 –2 24 –2 Substitute values in the quadratic formula. The equation has two solutions: x= Simplify. x= Solutions x= 2+ 24 –3.45 –2 2 24 –2 2– 24 1.45 –2 Finding the x-Intercepts of a Graph Check your solutions to the equation y = –x 2 – 2 x + 5 graphically. Check y –3.45 and y 1.45. You can see that the graph shows the x-intercepts between –3 and –4 and between 1 and 2. Using Quadratic Models in Real Life Problems involving models for the dropping or throwing of an object are called vertical motion problems. VERTICAL MOTION MODELS OBJECT IS DROPPED: h = –16 t 2 + s OBJECT IS THROWN: h = –16 t 2 + v t + s h = height (feet) t = time in motion (seconds) s = initial height (feet) v = initial velocity (feet per second) In these models the coefficient of t 2 is one half the acceleration due to gravity. On the surface of Earth, this acceleration is approximately 32 feet per second per second. Remember that velocity v can be positive (object moving up), negative (object moving down), or zero (object not moving). Speed is the absolute value of velocity. Modeling Vertical Motion BALLOON COMPETITION You are competing in the Field Target Event at a hot-air balloon festival. You throw a marker down from an altitude of 200 feet toward a target. When the marker leaves your hand, its speed is 30 feet per second. How long will it take the marker to hit the target? SOLUTION Because the marker is thrown down, the initial velocity is v = –30 feet per second. The initial height is s = 200 feet. The marker will hit the target when the height is 0. v = –30, s = 200, h = 0. Modeling Vertical Motion BALLOON COMPETITION SOLUTION v = –30, s = 200, h = 0. h = –16 t 2 + v t + s Choose the vertical motion model for a thrown object. h = –16 t 2 + (–30) v t +200 s Substitute values for v and s into the vertical motion model. 0h = –16 t 2 – 30t + 200 Substitute 0 for h. Write in standard form. t= t= –(–30 b ) 30 (–30 b ) 2 – 4(–16 a )(200 c ) 2(–16 a ) 13,700 –32 t 2.72 or –4.60 Substitute values in the quadratic formula. Simplify. Solutions Modeling Vertical Motion BALLOON COMPETITION SOLUTION t 2.72 or –4.60 As a solution, –4.60 does not make sense in the context of the problem. Therefore, the weighted marker will hit the target about 2.72 seconds after it was thrown. Chapter 9.6 Chapter 9.7 Chapter 9.8 Chapter 10 Chapter 10.1 MODELING ADDITION OF POLYNOMIALS Algebra tiles can be used to model polynomials. + – 1 –1 These 1-by-1 square tiles have an area of 1 square unit. + – + – x –x x2 –x 2 These 1-by-x rectangular tiles have an area of x square units. These x-by-x rectangular tiles have an area of x 2 square units. MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 1 Form the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1 with algebra tiles. x2 + 4x + 2 + + + + + + + – 2 x2 + + – 1 3x – – – – MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 2 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles. x 2 + 4x + 2 + + + + + + + + 2x 2 – 3x – 1 + + – + + + + + + + + + = + – – – – – – – MODELING ADDITION OF POLYNOMIALS You can use algebra tiles to add the polynomials x 2 + 4x + 2 and 2 x 2 – 3x – 1. 2 3 To add the polynomials, combine like terms. Group the x 2-tiles, the x-tiles, and the 1-tiles. 2 + 4x + 2 xFind and remove the zero pairs. + + + sum is + 3x+2 + x+ + 1. + The + 2x 2 – 3x – 1 + + – + + + + + + + + + = + – – – – – – – Adding and Subtracting Polynomials An expression which is the sum of terms of the form a x k where k is a nonnegative integer is a polynomial. Polynomials are usually written in standard form. Standard form means that the terms of the polynomial are placed in descending order, from largest degree to smallest degree. Polynomial in standard form: 2 x 3 + 5x 2 – 4 x + 7 Leading coefficient Degree Constant term The degree of each term of a polynomial is the exponent of the variable. The degree of a polynomial is the largest degree of its terms. When a polynomial is written in standard form, the coefficient of the first term is the leading coefficient. Classifying Polynomials A polynomial with only one term is called a monomial. A polynomial with two terms is called a binomial. A polynomial with three terms is called a trinomial. Identify the following polynomials: Degree Classified by degree Classified by number of terms 6 0 constant monomial –2 x 1 linear monomial 3x + 1 1 linear binomial –x 2 + 2 x – 5 2 quadratic trinomial 4x 3 – 8x 3 cubic binomial 2 x 4 – 7x 3 – 5x + 1 4 quartic polynomial Polynomial Adding Polynomials Find the sum. Write the answer in standard format. (5x 3 – x + 2 x 2 + 7) + (3x 2 + 7 – 4 x) + (4x 2 – 8 – x 3) SOLUTION Vertical format: Write each expression in standard form. Align like terms. 5x 3 + 2 x 2 – x + 7 3x 2 – 4 x + 7 3 2 + – x + 4x –8 4x 3 + 9x 2 – 5x + 6 Adding Polynomials Find the sum. Write the answer in standard format. (2 x 2 + x – 5) + (x + x 2 + 6) SOLUTION Horizontal format: Add like terms. (2 x 2 + x – 5) + (x + x 2 + 6) = (2 x 2 + x 2) + (x + x) + (–5 + 6) = 3x 2 + 2 x + 1 Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 No change Add the opposite –2 x 3 + 5x 2 – x + 8 + 2 x3 – 3x + 4 Subtracting Polynomials Find the difference. (–2 x 3 + 5x 2 – x + 8) – (–2 x 2 + 3x – 4) SOLUTION Use a vertical format. To subtract, you add the opposite. This means you multiply each term in the subtracted polynomial by –1 and add. –2 x 3 + 5x 2 – x + 8 – –2 x 3 + 3x – 4 –2 x 3 + 5x 2 – x + 8 + 2 x3 – 3x + 4 5x 2 – 4x + 12 Subtracting Polynomials Find the difference. (3x 2 – 5x + 3) – (2 x 2 – x – 4) SOLUTION Use a horizontal format. (3x 2 – 5x + 3) – (2 x 2 – x – 4) = (3x 2 – 5x + 3) + (–1)(2 x 2 – x – 4) = (3x 2 – 5x + 3) – 2 x 2 + x + 4 = (3x 2 – 2 x 2) + (– 5x + x) + (3 + 4) = x 2 – 4x + 7 Using Polynomials in Real Life You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Write a model for the area of the mat around the photograph as a function of the scale factor. Use a verbal model. Verbal Model Area of mat = Total Area – Labels … Area of photo 7x Area of mat = A (square inches) 5x Total Area = (10x)(14x – 2) (square inches) 10x Area of photo = (5x)(7x) (square inches) 14x – 2 SOLUTION Using Polynomials in Real Life You are enlarging a 5-inch by 7-inch photo by a scale factor of x and mounting it on a mat. You want the mat to be twice as wide as the enlarged photo and 2 inches less than twice as high as the enlarged photo. Write a model for the area of the mat around the photograph as a function of the scale factor. SOLUTION … = 140x 2 – 20x – 35x 2 5x = 105x 2 – 20x 10x 14x – 2 Algebraic Model 7x A = (10x)(14x – 2) – (5x)(7x) A model for the area of the mat around the photograph as a function of the scale factor x is A = 105x 2 – 20x. Chapter 10.2 Multiplying Polynomials You have already learned how to multiply a polynomial by a monomial by using the distributive property. (3x)(2x 2 – 5x + 3) = (3x)(2x 2) + (3x)(–5x) + (3x)(3) = 6x 3 – 15x 2 + 9x In the following slide, you will see how an area model illustrates the multiplication of two binomials Multiplying Polynomials ACTIVITY Developing Concepts INVESTIGATING BINOMIAL MULTIPLICATION The rectangle shown at the right has a width of (x + 2) and a height of (2x + 1) 1 What is the area of each part of the rectangle? x 2 x 1 1 x2 x x Find the product of (x + 2) and (2 x + 1) by adding the areas of the parts to get an expression for total area. 2x + 1 x x2 x x 1 x 1 1 (x + 2)(2x + 1) = x 2 + x 2 + x + x + x + x + x + 1 + 1 = 2x 2 + 5x + 2 x+2 Multiplying Polynomials Another way to multiply two binomials is to use the distributive property twice. First use (a + b)(c + d) = a(c + d) + b(c + d). Then use a(c + d) = ac + ad and b(c + d) = bc + bd. This shows that (a + b)(c + d) = ac + ad + bc + bd. This property can also be applied to the binomials of the form a – b or c – d. Using the Distributive Property Find the product (x + 2)(x – 3). SOLUTION (x + 2)(x – 3) = x(x – 3) + 2(x – 3) (b + c) a = ba + ca = x 2 – 3x + 2x – 6 a (b – c) = ab – ac = x2 – x – 6 Combine like terms. Multiplying Polynomials When you multiply two binomials, you can remember the results given by the distributive property by means of the FOIL pattern. Multiply the First, Outer, Inner, and Last terms. Product of First Terms (3x + 4)(x + 5) = 3x 2 Product of Outer Terms + 15x = 3x 2 + 19x + 20 Product of Inner Terms + 4x + 20 Product of Last Terms Multiplying Binomials Using the FOIL Pattern Find the product (3x – 4)(2x + 1). SOLUTION First, Outer, Inner, and Last. F (3x – 4)(2x + 1) O I L = 6x 2 + 3x – 8x – 4 Mental math. = 6x 2 – 5x – 4 Simplify. Multiplying Polynomials Vertically To multiply two polynomials that have three or more terms, remember that each term of one polynomial must be multiplied by each term of the other polynomial. Use a vertical or a horizontal format. Write each polynomial in standard form. Find the product (x – 2)(5 + 3x – x 2). SOLUTION Align like terms in columns. – x 2 + 3x + 5 Standard form x–2 Standard form 2x 2 – 6x – 10 –2(– x2 + 3x + 5) –x 3 + 3x 2 + 5x –x 3 + 5x 2 – x – 10 x(– x2 + 3x + 5) Combine like terms. Multiplying Polynomials Horizontally Find the product (4x 2 – 3x – 1)(2x – 5). SOLUTION Multiply 2x – 5 by each term of 4x 2 – 3x – 1. (4x 2 – 3x – 1)(2x – 5) = 4x 2(2x – 5) – 3x(2x – 5) – 1(2x – 5) = 8x 3 – 20x 2 – 6x 2 + 15x – 2x + 5 = 8x 3 – 26x 2 + 13x + 5 Using Polynomials in Real Life The diagram below shows the basic dimensions for a window. The glass portion of the window has a height-to-width ratio of 3 : 2. The framework adds 6 inches to the width and 10 inches to the height. Write a polynomial expression that represents the total area of the window, including the framework. SOLUTION 5 Use a verbal model. 3x Verbal Model Labels Total area = Height Width · of window of window 5 Total area = A (square inches) 3 … Height of window = 3x + 10 (inches) Width of window (inches) = 2x + 6 2x 3 Using Polynomials in Real Life The diagram below shows the basic dimensions for a window. The glass portion of the window has a height-to-width ratio of 3 : 2. The framework adds 6 inches to the width and 10 inches to the height. Write a polynomial expression that represents the total area of the window, including the framework. 5 SOLUTION 3x … Algebraic Model A = (3x + 10) · (2x + 6) = 6x 2 + 18x + 20x + 60 Area model 5 FOIL pattern 3 = 6x 2 + 38x + 60 Combine like terms. 2x 3 Using Polynomials in Real Life The diagram below shows the basic dimensions for a window. The glass portion of the window has a height-to-width ratio of 3 : 2. The framework adds 6 inches to the width and 10 inches to the height. Find the area when x = 10, 11, 12, 13, and 14. SOLUTION You can evaluate the polynomial expression 6x 2 + 38x + 60 by substituting x-values. For instance, to find the total area of the window when x = 10, substitute 10 for x. A = 6x 2 + 38x + 60 = 6(10) 2 + 38(10) + 60 = 600 + 380 + 60 = 1040 The areas for all five x-values are listed in the table. x (in.) 10 11 12 13 14 A (in. 2) 1040 1204 1380 1568 1768 Chapter 10.3 Chapter 10.4 Chapter 10.5 Chapter 10.6 FACTORING A QUADRATIC TRINOMIAL To factor quadratic polynomials whose leading factor is not 1: Find the factors of a (m and n). Find the factors of c (p and q). The sum of the outer and inner products (mq + pn) is b. a = mn b = mq + pn ax2 + bx + c = (mx + p)(nx + q) c = pq FACTORING A QUADRATIC TRINOMIAL a = mn c = pq b = mq + pn 22 = 3 • 4 + 5 • 2 6=3•2 6x2 + bx + 20 = (3x + 5)(2x + 4) c = pq One Pair of Factors for a and c Factor 2x2 + 11x + 5 SOLUTION Test combinations of factors for a c 1 and 2 1 and 5 Try a = 1 • 2 and c = 1 • 5 (1x + 1) (2x + 5) = 2x2 + 7x + 5 Not correct. Try a = 1 • 2 and c = 5 • 1 (1x + 5) (2x + 1) = 2x2 + 11x + 5 Correct. A Common Factor for a, b, and c Factor 6x 2 + 2x – 8 SOLUTION Begin by factoring out the common factor 2. 6x 2 + 2x – 8 The correct 2factorization is 2 (3x – x – 4) = 6x 2 – 2x – 8 = 2(x + 1)(3x – 4). Now factor 3x 2 – x – 4 by testing possible factors of a and c. FACTORS OF a AND c a = 1 • 3 and c = (2)(–2) PRODUCT When you factor, you canCORRECT? stop+testing (x – 2)(3x 2) = 3x2once – 4x –you 4 find No the correct factorization. 2 (x + 2)(3x – 2) = 3x + 4x – 4 No a = 1 • 3 and c = (–4)(1) (x – 4)(3x + 1) = 3x2 – 11x – 4 No a = 1 • 3 and c = (1)(–4) (x + 1)(3x – 4) = 3x2 – x – 4 Yes a = 1 • 3 and c = (–2)(2) SOLVING QUADRATIC EQUATIONS BY FACTORING Writing a Quadratic Model A cliff diver jumps from a ledge 48 feet above the ocean with an initial upward velocity of 8 feet per second. How long will it take until the diver enters the water? Use a vertical motion Let hv = 0. 8 and s = 48. When the diver entersmodel. the water, Solve the resulting equation for t. 2 h = – 16t + vt + s Vertical motion model 0 = – 16t 2 + 8t + 48 Write quadratic model. 2 = – 16t + 28t + 48 Substitute values. Factor out – 8. 0 = (– 8)(2t – t – 6) SOLUTION 0 = (– 8)(t – 2)(2t + 3) The solutions are 2 and – Factor. 3 . 2 Negative values do not make sense, so the only reasonable solution is t = 2. It will take the diver 2 seconds. Chapter 11 Chapter 11.1 Chapter 11.2 Chapter 11.3 USING DIRECT AND INVERSE VARIATION DIRECT VARIATION The variables x and y vary directly if, for a constant k, y = k, x or y = kx, k 0. USING DIRECT AND INVERSE VARIATION INDIRECT VARIATION The variables x and y vary inversely, if for a constant k, k = y, x or xy = k, k 0. USING DIRECT AND INVERSE VARIATION MODELS FOR DIRECT AND INVERSE VARIATION DIRECT VARIATION y = kx k>0 INVERSE VARIATION y = k>0 k x Using Direct and Inverse Variation When x is 2, y is 4. Find an equation that relates x and y in each case. x and y vary directly SOLUTION y = k x 4 = k 2 2 = k Write direct variation model. Substitute 2 for x and 4 for y. Simplify. y An equation that relates x and y is = 2, or y = 2x. x Using Direct and Inverse Variation When x is 2, y is 4. Find an equation that relates x and y in each case. x and y vary inversely SOLUTION xy = k Write inverse variation model. (2)(4) = k Substitute 2 for x and 4 for y. 8=k Simplify. An equation that relates x and y is xy = 8, or y = 8 . x Comparing Direct and Inverse Variation Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4. SOLUTION Make a table using y = 2x and y = x8 . x 1 2 3 4 Direct Variation: k > 0. As x increases by 1, y increases by 2. y = 2x 2 4 6 8 Inverse Variation: k > 0. As x doubles (from 1 to 2), y is halved (from 8 to 4). y = x8 8 4 8 3 2 Comparing Direct and Inverse Variation Compare the direct variation model and the inverse variation model you just found using x = 1, 2, 3, and 4. SOLUTION Plot the points and then connect the points with a smooth curve. Direct Variation: the graph for this model is a line passing through the origin. Inverse Variation: The graph for this model is a curve that gets closer and closer to the x-axis as x increases and closer and closer to the y-axis as x gets close to 0. Direct y = 2x Inverse y= 8 x USING DIRECT AND INVERSE VARIATION IN REAL LIFE Writing and Using a Model BICYCLING A bicyclist tips the bicycle when making turn. The angle B of the bicycle from the vertical direction is called the banking angle. banking angle, B Writing and Using a Model BICYCLING The graph below shows a model for the relationship between the banking angle and the turning radius for a bicycle traveling at a particular speed. For the values shown, the banking angle B and the turning radius r vary inversely. Banking angle (degrees) r turning radius Turning Radius banking angle, B Writing and Using a Model Banking angle (degrees) r turning radius Turning Radius banking angle, B Find an inverse variation model that relates B and r. Use the model to find the banking angle for a turning radius of 5 feet. Use the graph to describe how the banking angle changes as the turning radius gets smaller. Find an inverse variation model that relates B and r. SOLUTION From the graph, you can see that B = 32° when r = 3.5 feet. k B= r k 32 = 3.5 112 = k The model is B = Banking angle (degrees) Writing and Using a Model Turning Radius Write direct variation model. Substitute 32 for B and 3.5 for r. Solve for k. 112 , where B is in degrees and r is in feet. r Use the model to find the banking angle for a turning radius of 5 feet. Banking angle (degrees) Writing and Using a Model Turning Radius SOLUTION Substitute 5 for r in the model you just found. 112 = 22.4 B= 5 When the turning radius is 5 feet, the banking angle is about 22°. Writing and Using a Model Use the graph to describe how the banking angle changes as the turning radius gets smaller. SOLUTION As the turning radius gets smaller, smaller, the thebanking bankingangle angle becomes greater. greater.The Thebicyclist bicyclistleans leansatatgreater greaterangles. angles. Notice that the increase in the banking angle becomes more rapid when the turning radius is small. Chapter 11.4 Chapter 11.5 Chapter 11.6 Chapter 11.7 Chapter 11.8 Solving Rational Equations A rational equation is an equation that contains rational expressions. The next two examples show the two basic strategies for solving a rational equation. Cross Multiplying Cross multiplying can be used to solve a rational equation. Solve y 5 = . 3 y+2 SOLUTION y 5 = y+2 3 5(3) = y ( y + 2) 15 = y 2 + 2y Write original equation. Cross multiply. Simplify. 0 = y 2 + 2y – 15 Write in standard form. 0 = ( y + 5)( y – 3) Factor right side. If you set each factor equal to 0, you see that the solutions are –5 and 3. Check both solutions in the original equation. Multiplying by the LCD Cross multiplying can be used only for equations in which each side is a single fraction. The second method, multiplying by the LCD, works for any rational equation. Multiplying by the LCD always leads to an equation with no fractions. 2 1 4 Solve x + 3 = x . SOLUTION 2 1 4 + = x 3 x 2 1 4 3x • x + 3x • 3 = 3x • x 6 + x = 12 x=6 CHECK 2 1 ? 4 6 + 3 = 6 4 4 = 6 6 The LCD is 3x. Multiply each side by 3x. Simplify. Subtract 6 from each side. Substitute 6 for x. Simplify. Factoring to Find the LCD Solve –4 –10 +1= 2 . y–3 y + y – 12 SOLUTION The denominator y 2 + y – 12 factors as ( y + 4)( y – 3), so the LCD is ( y + 4)( y – 3). Multiply each side of the equation by ( y + 4)( y – 3). –4 –10 • ( y + 4)( y – 3) + 1 • ( y + 4)( y – 3) = 2 • ( y + 4)( y – 3) y–3 y + y – 12 –10( y + 4)( y – 3) –4( y + 4)( y – 3) + ( y + 4)( y – 3) = ( y + 4)( y – 3) y–3 –4( y + 4) + ( y 2 + y – 12) = –10 Factoring to Find the LCD Solve –4 –10 +1= 2 . y–3 y + y – 12 SOLUTION The denominator y 2 + y – 12 factors as ( y + 4)( y – 3), so the LCD is ( y + 4)( y – 3). Multiply each side of the equation by ( y + 4)( y – 3). –4( y + 4) + ( y 2 + y – 12) = –10 –4 y – 16 + y 2 + y – 12 = –10 y 2 – 3y – 28 = –10 y 2 – 3y – 18 = 0 ( y – 6)( y + 3) = 0 The solutions are 6 and –3. Check both solutions in the original equation. Writing and Using a Rational Equation BATTING AVERAGE You have 35 hits in 140 times at bat. Your batting average is 35 = 0.250. How many consecutive hits must you get to increase 140 your batting average to 0.300? SOLUTION If your hits are consecutive, then you must get a hit each time you are at bat, so “future hits” is equal to “future times at bat.” Past hits + Future hits Verbal Model Batting average = Labels Batting average = 0.300 (no units) Past hits = 35 (no units) Future hits = x (no units) Past times at bat = 140 (no units) Future times at bat = x (no units) … Past times at bat + Future times at bat Writing and Using a Rational Equation BATTING AVERAGE You have 35 hits in 140 times at bat. Your batting average is 35 = 0.250. How many consecutive hits must you get to increase 140 your batting average to 0.300? SOLUTION … Algebraic Model 35 + x 0.300 = 140 + x 0.300(140 + x) = 35 + x 42 + 0.3x = 35 + x 7 = 0.7x 10 = x Write algebraic model. Multiply by LCD. Use distributive property. Subtract 0.3x and 35 from each side. Divide each side by 0.7. You need to get a hit in each of your next 10 times at bat. Graphing Rational Functions The inverse variation models you previously graphed are a type of rational function. A rational function is a function of the form f(x) = polynomial . polynomial Graphing Rational Functions The inverse variation models you previously graphed are a type of rational function. A rational function is a function of the form f(x) = polynomial . polynomial In this lesson you will learn to graph rational functions whose numerators and denominators are first-degree polynomials. Using long division, such a function can be written in the form y= a + k. x–h Graphing Rational Functions RATIONAL FUNCTIONS WHOSE GRAPHS ARE HYPERBOLAS a +k x–h is a hyperbola whose center is (h, k). The graph of the rational function y = The vertical and horizontal lines through the center are the asymptotes of the hyperbola. An asymptote is a line that the graph approaches. While the distance between the graph and the line approaches zero, the asymptote is not part of the graph. Graphing a Rational Function Sketch the graph of y = 1 + 2. x 1 + 2. You can see that the x–0 center is (0, 2). The asymptotes can be drawn as dashed lines through the SOLUTION Think of the function as y = center. Make a table of values. Then plot the points and connect them with two smooth branches. x y –4 1.75 –2 1.5 –1 1 – 0.5 0 0 undefined 0.5 4 1 3 2 2.5 4 2.25 If you have drawn your graph correctly, it should be symmetric about the center (h, k). For example, the points (0.5, 4) and (–0.5, 0) are the same distance from the center (0, 2) but in opposite directions. Graphing a Rational Function When h is Negative Sketch the graph of y = SOLUTION –1 + 1. Describe the domain. x+2 For this equation, x – h = x + 2, so h = –2. The graph is a hyperbola with center at (–2, 1). Plot this point and draw a horizontal and a vertical asymptote through it. Then make a table of values, plot the points, and connect them with two smooth branches. x y –6 1.25 –4 1.5 –3 2 –2.5 3 –2 undefined –1.5 –1 –1 0 0 0.5 2 0.75 The domain is all real numbers except x = –2. Rewriting before Graphing Sketch the graph of y = SOLUTION 2x + 1 . x+2 Begin by using long division to rewrite the rational function. 2 x + 2 2x + 1 2x + 4 –3 Quotient y=2+ Remainder –3 –3 , or y = +2 x+2 x+2 The graph is a hyperbola with center at (–2, 2). Plot this point and draw a horizontal and a vertical asymptote through it. Make a table of values, plot the points, and connect them with two smooth branches. When making a table of values, include values of x that are less than h and values of x that are greater than h. Chapter 12 Chapter 12.1 Chapter 12.2 Chapter 12.3 Chapter 12.4 MODELING COMPLETING THE SQUARE Use algebra tiles to complete a perfect square trinomial. Model the expression x 2 + 6x. Arrange the x-tiles to form part of a square. To complete the square, add nine 1-tiles. You have completed the square. x2 x x x x x x x x x 1 1 1 1 1 1 1 1 1 x2 + 6x + 9 = (x + 3)2 SOLVING BY COMPLETING THE SQUARE To complete the square of the expression x2 + bx, add the square of half the coefficient of x. x2 + bx + 2 ( ) ( b 2 = x+ b 2 2 ) Completing the Square What term should you add to x2 – 8x so that the result is a perfect square? SOLUTION The coefficient of x is –8, so you should add to the expression. –8 2 x – 8x + 2 2 –8 , or 16, ( 2) 2 ( )=x 2 – 8x + 16 = (x – 4)2 Completing the Square Factor 2x2 – x – 2 = 0 SOLUTION 2x2 – x – 2 = 0 Write original equation. 2x2 – x = 2 Add 2 to each side. x2 – 1 x = 1 Divide each side by 2. 2 x2 – 2 ( ) 1 1 x+ – 2 4 =1+ 1 16 ( 1 1 Add – 2 • 2 to each side. 2 1 2 1 = – , or ) ( 4) 16 Completing the Square x2 – 2 ( ) 1 1 x+ – 2 4 ( 1 x– 4 x– 2 ) =1+ ( 1 1 Add – 2 • 2 1 16 x= 16 Write left side as a fraction. 17 4 1 4 1 The solutions are + 4 ) ( 4) to each side. 17 = 16 1 = 4 2 1 2 1 = – , or Find the square root of each side. 17 4 Add 1 to each side. 4 1 17 – 1.28 and 4 4 17 – 0.78. 4 Completing the Square 1 The solutions are + 4 CHECK 1 17 – 1.28 and 4 4 17 – 0.78. 4 You can check the solutions on a graphing calculator. CHOOSING A SOLUTION METHOD Investigating the Quadratic Formula Perform the following steps on the general quadratic equation ax2 + bx + c = 0 where a 0. ax2 + bx = – c x2 Subtract c from each side. bx –c + a+ = a 2 Divide each side by a. 2 ( ) ( ) b –c + b x + = ( 2a) a 4a bx b x2 + + a 2a –c = a + 2 b 2a 2 2 ( b x+ 2a 2 ) Add the square of half the coefficient of x to each side. – 4ac + b 2 = 2 4a Write the left side as a perfect square. Use a common denominator to express the right side as a single fraction. CHOOSING A SOLUTION METHOD Investigating the Quadratic Formula ( b x+ 2a 2 ) 2 – 4ac + b = 2 4a b x+ = 2a 2 b 4ac 2a b2 4ac b x= – 2a 2a x= 2 –b b 4ac 2a Use a common denominator to express the right side as a single fraction. Find the square root of each side. Include ± on the right side. Solve for x by subtracting the same term from each side. Use a common denominator to express the right side as a single fraction. Chapter 12.5 Chapter 12.6 THE DISTANCE AND MIDPOINT FORMULAS Investigating Distance: 1 Plot A(2,1) and B(6,4) on a coordinate plane. Then draw a right triangle that has AB as its hypotenuse. 2 Find and label the coordinates of the vertex C. 3 Find the lengths of the legs of 4 Use the Pythagorean theorem to find AB. AB 5 ABC. Remember: 16 + 9 = C (6, 1) x 16 + 9 = c AB = 5 c2 y4B –3 y1A A (2, 1) x6B –4 x2A a2 + b2 = c2 42 + 32 = c2 B (6, 4) y 25 = c Finding the Distance Between Two Points The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2). Using the Pythagorean theorem B (x 2, y 2 ) y a2 + b2 = c2 You can write the equation (x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2 d A (x 1, y 1 ) Solving this for d produces the distance formula. THE DISTANCE FORMULA The distance d between the points (x 1, y 1) and (x 2, y 2) is d = (x 2 – x 1) 2 + ( y 2 – y 1) 2 y2 – y1 x2 – x1 C (x 2, y 1 ) x Finding the Distance Between Two Points Find the distance between (1, 4) and (–2, 3). SOLUTION To find the distance, use the distance formula. d = (x 2 – x 1) 2 + ( y 2 – y 1) 2 Write the distance formula. = 2 (x–2 y 1) 2 2 –– x11) + ( y32 – 4 Substitute. = 10 3.16 Simplify. Use a calculator. Applying the Distance Formula A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a goal line. The ball lands 45 yards from the same goal line and 40 yards from the same sideline. How far was the ball kicked? SOLUTION The ball is kicked from the point (10, 5), and lands at the point (40, 45). Use the distance formula. d = = (40 – 10) 2 + (45 – 5) 2 900 + 1600 = 2500 = 50 The ball was kicked 50 yards. Finding the Midpoint Between Two Points The midpoint of a line segment is the point on the segment that is equidistant from its end-points. The midpoint between two points is the midpoint of the line segment connecting them. THE MIDPOINT FORMULA The midpoint between the points (x 1, y 1) and (x 2, y 2) is ( x1 + x2 2 , y1 + y2 2 ) Finding the Midpoint Between Two Points Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result. SOLUTION ( –2 + 4 2 , Remember, the midpoint formula is 3+2 2 The midpoint is ) =( ( 1, 5 2 2 2 ). , 5 2 ) =( 1 , 5 2 ( ) x1 + x2 2 , y1 + y2 2 ). Finding the Midpoint Between Two Points Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check the result. CHECK (–2, 3) From the graph, you can see that the 5 point 1 , appears halfway 2 between (–2, 3) and (4, 2). You can also ( (1, 5 ) 2 (4, 2) ) use the distance formula to check that the distances from the midpoint to each given point are equal. Applying the Midpoint Formula You are using computer software to design a video game. You want to place a buried treasure chest halfway between the center of the base of a palm tree and the corner of a large boulder. Find where you should place the treasure chest. SOLUTION 1 Assign coordinates to the locations of the two landmarks. The center of the palm tree is at (200, 75). The corner of the boulder is at (25, 175). (25, 175) (112.5, 125) (200, 75) 2 Use the midpoint formula to find the point that is halfway between the two landmarks. ( 25 + 200 175 + 75 , 2 2 ) ( = 225 250 2 , 2 ) = (112.5, 125) Chapter 12.7