Polynomials and Nonlinear Functions

Transcription

1 Polnomials and Nonlinear Functions Lesson 3- Identif and classif polnomials. Lessons 3-2 through 3-4 Add, subtract, and multipl polnomials. Lesson 3-5 Determine whether functions are linear or nonlinear. Lesson 3-6 Eplore different representations of quadratic and cubic functions. Ke Vocabular polnomial (p. 669) degree (p. 670) nonlinear function (p. 687) quadratic function (p. 688) cubic function (p. 688) You have studied situations that can be modeled b linear functions. Man real-life situations, however, are not linear. These can be modeled using nonlinear functions. You will use a nonlinear function in Lesson 3-6 to determine how far a skdiver falls in 4.5 seconds. 666 Chapter 3 Polnomials and Nonlinear Functions

2 Prerequisite Skills To be successful in this chapter, ou ll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter 3. For Lesson 3- Monomials Determine the number of monomials in each epression. (For review, see Lesson 4-.) a s 5t t For Lesson 3-4 Distributive Propert Use the Distributive Propert to write each epression as an equivalent algebraic epression. (For review, see Lesson 3-.) 7. 5(a 4) 8. 2(3 8) 9. 4( 8n) 0. 6( 2). (9b 9c)3 2. 5(q 2r 3s) For Lesson 3-5 Linear Functions Determine whether each equation is linear. (For review, see Lesson 8-2.) Polnomials Make this Foldable to help ou organize our notes. Begin with a sheet of " b 7" paper. Fold the short sides toward the middle. Fold Fold the top to the bottom. Fold Again Cut Label pen. Cut along the second fold to make four tabs. Label each of the tabs as shown. Functions Reading and Writing As ou read and stud the chapter, write eamples of each concept under each tab. Chapter 3 Polnomials and Nonlinear Functions 667

3 Prefies and Polnomials You can determine the meaning of man words used in mathematics if ou know what the prefies mean. In Lesson 4-, ou learned that the prefi mono means one and that a monomial is an algebraic epression with one term. Monomials Not Monomials 5 2 8n 2 n 3 a 3 4a 2 a 6 The words in the table below are used in mathematics and in everda life. The contain the prefies bi, tri, and pol. Prefi bi Words bisect to divide into two congruent parts biannual occurring twice a ear biccle a vehicle with two wheels X Y Z bisect tri triangle a figure with three sides triathlon an athletic contest with three phases trilog a series of three related literar works, such as films or books C A triangle B pol polhedron a solid with man flat surfaces polchrome having man colors polgon a figure with man sides polhedron 668 Reading to Learn. How are the words in each group of the table related? 2. What do the prefies bi, tri, and pol mean? 3. Write the definition of binomial, trinomial, and polnomial. 4. Give an eample of a binomial, a trinomial, and a polnomial. 5. RESEARCH Use the Internet or a dictionar to make a list of other words that have the prefies bi, tri, and pol. Give the definition of each word. 668 Chapter 3 Polnomials and Nonlinear Functions

4 Polnomials Vocabular polnomial binomial trinomial degree Identif and classif polnomials. Find the degree of a polnomial. Heat inde is a wa to describe how hot it feels outside with the temperature and humidit combined. Some eamples are shown below. Humidit (%) are polnomials used to approimate real-world data? Temperature ( F) Heat Inde To calculate heat inde, meteorologists use an epression similar to the one below. In this epression, is the percent humidit, and is the temperature a. How man terms are in the epression for the heat inde? b. What separates the terms of the epression? Stud Tip Classifing Polnomials Be sure epressions are written in simplest form. is the same as 2, so the epression is a monomial. 25 is the same as 5, so the epression is a monomial. CLASSIFY PLYNMIALS Recall that a monomial is a number, a variable, or a product of numbers and/or variables. An algebraic epression that contains one or more monomials is called a polnomial. In a polnomial, there are no terms with variables in the denominator and no terms with variables under a radical sign. A polnomial with two terms is called a binomial, and a polnomial with three terms is called a trinomial. Polnomial Number of Terms The terms in a binomial or a trinomial ma be added or subtracted. Eamples monomial 4,, 2 3 binomial 2, a 5b, c 2 d trinomial 3 a b c, 2 2 Eample Classif Polnomials Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial. a b. t t 2 This is a polnomial because The epression is not a it is the sum of three polnomial because t 2 has monomials. There are three a variable in the denominator. terms, so it is a trinomial. Concept Check Is a polnomial? Eplain. Lesson 3- Polnomials 669

5 DEGREES F PLYNMIALS The degree of a monomial is the sum of the eponents of its variables. The degree of a nonzero constant such as 6 or 0 is 0. The constant 0 has no degree. Stud Tip Degrees The degree of a is because a a. Eample 2 Degree of a Monomial Find the degree of each monomial. a. 5a b. 3 2 The variable a has degree, 2 has degree 2 and has degree. so the degree of 5a is. The degree of 3 2 is 2 or 3. A polnomial also has a degree. The degree of a polnomial is the same as that of the term with the greatest degree. Eample 3 Degree of a Polnomial Find the degree of each polnomial. a b. a 2 ab 2 b 4 term degree term degree a 2 2 ab 2 2 or 3 b 4 4 The greatest degree is 2. So The greatest degree is 4. So the the degree of is 2. degree of a 2 ab 2 b 4 is 4. Ecologist An ecologist studies the relationships between organisms and their environment. nline Research For more information about a career as an ecologist, visit: careers Eample 4 Degree of a Real-World Polnomial ECLGY In the earl 900s, the deer population of the Kaibab Plateau in Arizona was affected b hunters and b the food suppl. The population from 905 to 930 can be approimated b the polnomial , where is the number of ears since 900. Find the degree of the polnomial degree 5 degree 4 degree 0 So, has degree 5. Concept Check Find the degree of the polnomial at the beginning of the lesson. Concept Check. Eplain how to find the degree of a monomial and the degree of a polnomial. 2. PEN ENDED Write three binomial epressions. Eplain wh the are binomials. 670 Chapter 3 Polnomials and Nonlinear Functions

6 3. FIND THE ERRR Carlos and Tanisha are finding the degree of 5 2. Carlos 5 has degree. 2 has degree has degree + 2 or 3. Tanisha 5 has degree. 2 has degree has degree 2. Guided Practice Who is correct? Eplain our reasoning. Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial d a 5 a Application Find the degree of each polnomial. 0. 4b r 3 7r 5. d 2 c 4 GEMETRY For Eercises 6 and 7, refer to the square at the right with a side length of units. 6. Write a polnomial epression for the area of the small blue rectangle. 7. What is the degree of the polnomial ou wrote in Eercise 6? Practice and Appl For Eercises See Eamples , Etra Practice See page 755. Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial a w c k 24. r 4 r 2 s n n ab 2 3a b a cb c Find the degree of each polnomial ab 33. 2c z s 4 t n 37. g 5 5h d 2 c 4 d Tell whether each statement is alwas, sometimes, or never true. Eplain. 42. A trinomial has a degree of An integer is a monomial. 44. MEDICINE Doctors can stud a patient s heart b injecting de in a vein near the heart. In a normal heart, the amount of de in the bloodstream after t seconds is given b 0.006t t t 2.79t. Find the degree of the polnomial. Lesson 3- Polnomials 67

7 LANDSCAPING For Eercises 45 and 46, use the information below and the diagram at the right. Lee wants to plant flowers along the perimeter of his vegetable garden. 45. Write a polnomial that represents the perimeter of the garden in feet. 46. What is the degree of the polnomial? z 47. RESEARCH Suppose our grandparents deposited $00 in our savings account each ear on our birthda. n our fifth birthda, there would have been approimatel dollars, where is the annual interest rate plus. Research the current interest rate at our famil s bank. Using that interest rate, how much mone would ou have on our net birthda? 48. CRITICAL THINKING Find the degree of a 3 2 b 3 b 2. Standardized Test Practice 49. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are polnomials used to approimate real-world data? Include the following in our answer: a description of how the value of heat inde is found, and an eplanation of wh a linear equation cannot be used to approimate the heat inde data. 50. Choose the epression that is not a binomial. A 2 B a b C m 3 n 3 D State the degree of A B 2 C 3 D 4 Maintain Your Skills Mied Review A number cube is rolled. Determine whether each event is mutuall eclusive or inclusive. Then find the probabilit. (Lesson 2-9) 52. P(odd or greater than 3) 53. P(5 or even) 54. A number cube is rolled. Find the odds that the number is greater than 2. (Lesson 2-8) Find the volume of each solid. If necessar, round to the nearest tenth. (Lesson -3) m 7 in. 6 in. 5 in..3 m Getting Read for the Net Lesson PREREQUISITE SKILL Rewrite each epression using parentheses so that the terms having variables of the same power are grouped together. (To review properties of addition, see Lesson -4.) 57. ( 4) (6n 2) (3n 5) 60. (a 2b) (3a b) 6. (s t) (5s 3t) 62. ( 2 4) (7 2 3) 672 Chapter 3 Polnomials and Nonlinear Functions

8 A Follow-Up of Lesson 3- Modeling Polnomials with Algebra Tiles In a set of algebra tiles, represents the integer, represents the variable, and 2 represents 2. Red tiles are used to represent,, and 2. 2 You can use these tiles to model monomials You can also use algebra tiles to model polnomials. The polnomial is modeled below Model and Analze Use algebra tiles to model each polnomial Eplain how ou can tell whether an epression is a monomial, binomial, or trinomial b looking at the algebra tiles. 6. Name the polnomial modeled below Eplain how ou would find the degree of a polnomial using algebra tiles. Investigating Slope-Intercept Form 673 Algebra Activit Modeling Polnomials with Algebra Tiles 673

9 Adding Polnomials Add polnomials. can ou use algebra tiles to add polnomials? Consider the polnomials and 2 2 modeled below Follow these steps to add the polnomials. Step Combine the tiles that have the same shape. Step 2 When a positive tile is paired with a negative tile that is the same shape, the result is called a zero pair. Remove an zero pairs ( 2 ) 3 4 ( 2) a. Write the polnomial for the tiles that remain. b. Find the sum of and b using algebra tiles. c. Compare and contrast finding the sums of polnomials with finding the sum of integers. Stud Tip Algebra Tiles Tiles that are the same shape and size represent like terms. ADD PLYNMIALS Monomials that contain the same variables to the same power are like terms. Terms that differ onl b their coefficient are called like terms. Like Terms Unlike Terms 2 and 7 6a and 7b 2 and 5 2 4ab 2 and 4a 2 b 674 Chapter 3 Polnomials and Nonlinear Functions You can add polnomials b combining like terms. Eample Add Polnomials Find each sum. a. (3 5) (2 ) Method Add verticall. Method 2 Add horizontall. 3 5 (3 5) (2 ) ( ) 2 Align like terms. (3 2) (5 ) 5 6 Add. 5 6 The sum is 5 6. Associative and Commutative Properties

10 Stud Tip Negative Signs When a monomial has a negative sign, the coefficient is a negative number. 4 coefficient is 4. b coefficient is. When a term in a polnomial is subtracted, add its opposite b making the coefficient negative. 2 ( 2) b. (2 2 7) ( 2 3 5) Method ( ) Align like terms. Add. Method 2 (2 2 7) ( 2 3 5) Write the epression. (2 2 2 ) ( 3) ( 7 5) Group like terms Simplif. The sum is c. (9c 2 4c) ( 6c 8) (9c 2 4c) ( 6c 8) Write the epression. 9c 2 (4c 6c) 8 Group like terms. 9c 2 2c 8 Simplif. The sum is 9c 2 2c 8. d. ( ) (6 2 2 ) ( ) The sum is Leave a space because there is no other term like. Concept Check Name the like terms in b 2 5b ab 9b 2. Polnomials are often used to represent measures of geometric figures. Eample 2 Use Polnomials to Solve a Problem GEMETRY The lengths of the sides of golden rectangles are in the ratio :.62. So, the length of a golden rectangle is approimatel.62 times greater than the width..62 a. Find a formula for the perimeter of a golden rectangle. P 2 2w Formula for the perimeter of a rectangle P 2(.62) 2 Replace with.62 and w with. P or 5.24 Simplif. A formula for the perimeter of a golden rectangle is P 5.24, where is the measure of the width. Geometr The ancient Greeks often incorporated the golden ratio into their art and architecture. Source: b. Find the length and the perimeter of a golden rectangle if its width is 8.3 centimeters. length.62 Length of a golden rectangle.62(8.3) or Replace with 8.3 and simplif. perimeter 5.24 Perimeter of a golden rectangle 5.24(8.3) or Replace with 8.3 and simplif. The length of the golden rectangle is centimeters, and the perimeter is centimeters. Lesson 3-2 Adding Polnomials 675

11 Concept Check. Name the like terms in ( 2 5 2) ( ). 2. PEN ENDED Write two binomials that share onl one pair of like terms. 3. FIND THE ERRR Hai sas that 7z and 2z are like terms. Devin sas the are not. Who is correct? Eplain our reasoning. Guided Practice Find each sum a 2 9a 6 ( ) 3 ( )4a ( 3) (2 5) 7. (3 7) 3 8. (2 2 5) (9 7) 9. (3 2 2 ) ( 2 5 3) Application 0. GEMETRY Find the perimeter of the figure at the right Practice and Appl For Eercises See Eamples Etra Practice See page 756. Find each sum b 5 ( ) 8 ( ) 9b a 3 a 2 8a 8 ( ) ( ) 2a (3 9) ( 5) 6. (4 3) ( ) 7. (6 5r) (2 7r) 8. (8m 2n) (3m n) 9. ( 2 ) (4 2 ) 20. (3a 2 b 2 ) (3a b 2 ) 2. ( ) (2 2 3 ) 22. ( 2 2 5) ( 2 3 2) 676 Chapter 3 Polnomials and Nonlinear Functions Find each sum. Then evaluate if a 3, b 4, and c (3a 5b) (2a 9b) 24. (a 2 7b 2 ) (5 3b 2 ) (2a 2 7) 25. (3a 5b 4c) (2a 3b 7c) ( a 4b 2c) GEMETRY For Eercises 26 28, refer to the triangle. 26. Find the sum of the measures of the angles. 27. The sum of the measures of the angles in an triangle is 80. Find the value of. 28. Find the measure of each angle. (2 30) ( 4)

12 FINANCE For Eercises 29 3, refer to the information below. Jason and Will both work at the same supermarket and are paid the same hourl rate. At the end of the week, Jason s pacheck showed that he worked 23 hours and had $2 deducted for taes. Will worked 9 hours during the same week and had $0 deducted for taes. Let represent the hourl pa. 29. Write a polnomial epression to represent Jason s pa for the week. 30. Write a polnomial epression to represent Will s pa for the week. 3. Write a polnomial epression to represent the total weekl pa for Jason and Will. 32. CRITICAL THINKING In the figure at the right, 2 is the area of the larger square, and 2 is the area of each of the two smaller squares. What is the perimeter of the whole rectangle? Eplain WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can ou use algebra tiles to add polnomials? Include the following in our answer: a description of algebra tiles that represent like terms, and an eplanation of how zero pairs are used in adding polnomials. Standardized Test Practice 34. Choose the pair of terms that are not like terms. A 6cd, 2cd B 2, 5 C a2, b 2 D 2, What is the sum of 2 and 5? A 0 3 B 2 3 C 2 3 D 2 5 Maintain Your Skills Mied Review Find the degree of each polnomial. (Lesson 3-) 36. a 3 b z c 2 7c 3 4 A card is drawn from a standard deck of 52 plaing cards. Find each probabilit. (Lesson 2-9) 39. P(2 or jack) 40. P(0 or red) 4. P(ace or black 7) 42. Determine whether the prisms are similar. Eplain. (Lesson -6) 9 cm 3.5 cm 6 cm 4 cm 24 cm 6 cm Getting Read for the Net Lesson PREREQUISITE SKILL Rewrite each epression as an addition epression b using the additive inverse. (To review additive inverse, see Lesson 2-3.) 43. 5c b 3a (n rt) r (s t) 2s Lesson 3-2 Adding Polnomials 677

13 Subtracting Polnomials Subtract polnomials. is subtracting polnomials similar to subtracting measurements? At the North Pole, buo stations drift with the ice in the Arctic cean. The table shows the latitudes of two North Pole buos in April, Station 5 Latitude 'N = 89 degrees 35.4 minutes 'N = 85 degrees 27.3 minutes Reading Math Smbols The smbol " in 68 8' 2" is read as seconds. a. What is the difference in degrees and the difference in minutes between the two stations? b. Eplain how ou can find the difference in latitude between an two locations, given the degrees and minutes. c. The longitude of Station is and the longitude of Station 5 is Find the difference in longitude between the two stations. SUBTRACT PLYNMIALS When ou subtract measurements, ou subtract like units. Consider the subtraction of latitude measurements shown below. 89 degrees 35.4 minutes ( ) 85 degrees 27.3 minutes 4 degrees 8. minutes 89 degrees 85 degrees 35.4 minutes 27.3 minutes Similarl, when ou subtract polnomials, ou subtract like terms ( ) Eample Subtract Polnomials Find each difference. a. (5 9) (3 6) b. (4a 2 7a 4) (3a 2 2) 5 9 4a 2 7a 4 ( ) 3 6 Align like terms. ( ) 3a 2 2 Align like terms. 2 3 Subtract. a 2 7a 2 Subtract. The difference is 2 3. The difference is a 2 7a Chapter 3 Polnomials and Nonlinear Functions

14 Recall that ou can subtract a rational number b adding its additive inverse ( 8) The additive inverse of 8 is 8. You can also subtract a polnomial b adding its additive inverse. To find the additive inverse of a polnomial, multipl the entire polnomial b. Polnomial Multipl b Additive Inverse t (t) t 3 ( 3) 3 a 2 b 2 c ( a 2 b 2 c) a 2 b 2 c Stud Tip Zeros It can be helpful to add zeros as placeholders when a term in one polnomial does not have a corresponding like term in another polnomial ( ) Eample 2 Subtract Using the Additive Inverse Find each difference. a. (3 8) (5 ) The additive inverse of 5 is ( )(5 ) or 5. (3 8) (5 ) (3 8) ( 5 ) To subtract (5 ), add ( 5 ). (3 5) (8 ) Group the like terms. 2 7 Simplif. The difference is 2 7. b. (4 2 2 ) ( 3 2 ) The additive inverse of 3 2 is ( )( 3 2 ) or 3 2. Align the like terms and add the additive inverse ( ) 3 2 ( ) The difference is Concept Check What is the additive inverse of a 2 9a? Eample 3 Subtract Polnomials to Solve a Problem SHIPPING The cost for shipping a Shipping package that weighs pounds from Cost ($) Compan Dallas to Chicago is shown in the Atlas Service table at the right. How much more Bell Service 3 25 does the Atlas Service charge for shipping the package? difference in cost cost of Atlas Service cost of Bell Service (4 280) (3 25) Substitution (4 280) ( 3 25) Add additive inverse. (4 3) (280 25) Group like terms. 55 Simplif. The Atlas Service charges 55 dollars more for shipping a package that weighs pounds. Lesson 3-3 Subtracting Polnomials 679

15 Concept Check Guided Practice. Describe how subtraction and addition of polnomials are related. 2. PEN ENDED Write two polnomials whose difference is Find each difference. 3. r 2 5r ( ) r 2 r ( ) 2 5. (9 5) (4 3) 6. (2 4) ( 5) 7. (3 2 ) (8 2) 8. (6a 2 3a 9) (7a 2 5a ) Application 9. GEMETRY The perimeter of the isosceles trapezoid shown is 6 units. Find the length of the missing base of the trapezoid Practice and Appl For Eercises See Eamples , 27 3 Etra Practice See page 756. Find each difference. 0. 8k 9. n 2 n ( ) k 2 ( ) n 2 5n 2. 5a 2 9a ( ) 3a 2 5a 7 ( ) w 2 7 ( ) ( ) 6w 2 2w Chapter 3 Polnomials and Nonlinear Functions 6. (3 4) ( 2) 7. (7 5) (3 2) 8. (2 5) ( 8) 9. (3t 2) (5t 4) 20. (2 3) ( ) 2. (a 2 6b 2 ) ( 2a 2 4b 2 ) 22. ( 2 6) (3 2 7) 23. (9n 2 8) (n 4) 24. ( ) (2 2 8 ) 25. ( ) ( ) 26. GEMETRY Alssa plans to trim a picture to fit into a frame. The area of the picture is square units, but the area inside the frame is onl square units. How much of the picture will Alssa have to trim so that it will fit into the frame? 27. TEMPERATURE The highest recorded temperature in North Carolina occurred in 983. The lowest recorded temperature in North Carolina occurred two ears later. The difference between these two record temperatures is 68 F more than the sum of the temperatures. Write an equation to represent this situation. Then find the record low temperature in North Carolina. 28. CRITICAL THINKING Suppose A and B represent polnomials. If A B and A B 2 4 8, find A and B.

16 29. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is subtracting polnomials similar to subtracting measurements? Include the following in our answer: a comparison between subtracting measurements with two parts and subtracting polnomials with two terms, and an eample of a subtraction problem involving measurements that have two parts, and an eplanation of how to find the difference. Standardized Test Practice 30. What is (5 7) (3 4)? A 2 3 B 2 3 C 2 D 2 Maintain Your Skills Mied Review 3. Write the additive inverse of 4h 2 hk k 2. A 4h 2 hk k 2 B 4h 2 hk k 2 C 4h 2 hk k 2 D 4h 2 hk k 2 Find each sum. (Lesson 3-2) 32. (2 3) ( ) 33. ( 2) ( 5) 34. ( ) ( ) 35. (4t t 2 ) (8t 2) Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial. (Lesson 3-) a c 2 d 3 cd Getting Read for the Net Lesson 39. Make a stem-and-leaf plot for the set of data shown below. (Lesson 2-) 72, 64, 68, 66, 70, 89, 9, 54, 59, 7, 7, 85 PREREQUISITE SKILL Simplif each epression. (To review multipling monomials, see Lesson 4-2.) 40. (3) 4. (2)(4) 42. (t 2 )(6t) 43. (4m)(m 2 ) 44. (w 2 )( 3w) 45. (2r 2 )(5r 3 ) P ractice Quiz Lessons 3- through 3-3 Find the degree of each polnomial. (Lesson 3-). cd 3 2. a 4a Find each sum or difference. (Lessons 3-2 and 3-3) 4. (2 8) ( 7) 5. (4 5) (2 3) 6. (5d 2 3) (2d 2 7) 7. (3r 6s) (5r 9s) 8. ( 2 4 2) ( ) 9. (9 4) (2 9) 0. GEMETRY The perimeter of the triangle is 8 3 centimeters. Find the length of the third side. (Lessons 3-2 and 3-3) 4 cm 2 cm Lesson 3-3 Subtracting Polnomials 68

17 A Preview of Lesson 3-4 Modeling Multiplication Recall that algebra tiles are named based on their area. The area of each tile is the product of the width and length. 2 These algebra tiles can be placed together to form a rectangle whose length and width each represent a polnomial. The area of the rectangle is the product of the polnomials. Use algebra tiles to find ( 2). Step Make a rectangle with a width of and a length of 2. Use algebra tiles to mark off the dimensions on a product mat. Step 2 Using the marks as a guide, fill in the rectangle with algebra tiles. Step 3 The area of the rectangle is 2. In simplest form, the area is 2 2. Therefore, ( 2) Model and Analze Use algebra tiles to determine whether each statement is true or false.. ( ) 2 2. (2 3) ( 2) (3 ) 6 2 Find each product using algebra tiles. 5. ( 5) 6. (2 ) 7. (2 4)2 8. 3(2 ) 9. There is a square garden plot that measures feet on a side. a. Suppose ou double the length of the plot and increase the width b 3 feet. Write two epressions for the area of the new plot. b. If the original plot was 0 feet on a side, what is the area of the new plot? Etend the Activit 0. Write a multiplication sentence that is represented b the model at the right. 682 Investigating Slope-Intercept Form 682 Chapter 3 Polnomials and Nonlinear Functions 2 2

18 Multipling a Polnomial b a Monomial Multipl a polnomial b a monomial. is the Distributive Propert used to multipl a polnomial b a monomial? The Grande Arche office building in Paris, France, looks like a hollowed-out prism, as shown in the photo at the right. a. Write an epression that represents the area of the rectangular region outlined on the photo. b. Recall that 2(4 ) 2(4) 2() b the Distributive Propert. Use this propert to simplif the epression ou wrote in part a. c. The Grande Arche is approimatel w feet deep. Eplain how ou can write a polnomial to represent the volume of the hollowed-out region of the building. Then write the polnomial. 2w 52 w MULTIPLY A PLYNMIAL AND A MNMIAL You can model the multiplication of a polnomial and a monomial b using algebra tiles. 3 This model has a length of 3 and a width of Stud Tip Look Back To review the Distributive Propert, see Lesson 3-. The model shows the product of 2 and 3. The rectangular arrangement contains 2 2 tiles and 6 tiles. So, the product of 2 and 3 is In general, the Distributive Propert can be used to multipl a polnomial and a monomial. Eample Products of a Monomial and a Polnomial Find each product. a. 4(5 ) 4(5 ) 4(5) 4() Distributive Propert 20 4 Simplif. b. (2 6)(3) (2 6)(3) 2(3) 6(3) Distributive Propert Simplif. Lesson 3-4 Multipling a Polnomial b a Monomial 683

19 Eample 2 Find 3a(a 2 2ab 4b 2 ). Product of a Monomial and a Polnomial 3a(a 2 2ab 4b 2 ) 3a(a 2 ) 3a(2ab) 3a(4b 2 ) 3a 3 6a 2 b 2ab 2 Distributive Propert Simplif. Concept Check What is the product of 2 and? Sometimes problems can be solved b simplifing polnomial epressions. Eample 3 Use a Polnomial to Solve a Problem PLS The world s largest swimming pool is the rthlieb Pool in Casablanca, Morocco. It is 30 meters longer than 6 times its width. If the perimeter of the pool is 0 meters, what are the dimensions of the pool? Eplore Plan You know the perimeter of the pool. You want to find the dimensions of the pool. Let w represent the width of the pool. Then 6w 30 represents the length. Write an equation. Stud Tip Look Back To review equations with grouping smbols, see Lesson 3-. Perimeter equals twice the sum of the length and width. P 2 ( w) Solve P 2( w) Write the equation. 0 2(6w 30 w) Replace P with 0 and with 6w (7w 30) Combine like terms. 0 4w 60 Distributive Propert 050 4w Subtract 60 from each side. 75 w Divide each side b 4. Eamine } } } The width is 75 meters, and the length is 6w 30 or 480 meters. Check the reasonableness of the results b estimating. P 2( w) Formula for perimeter of a rectangle P 2(500 80) Round 480 to 500 and 75 to 80. P 2(580) or about 60 Since 60 is close to 0, the answer is reasonable. Concept Check. Determine whether the following statement is true or false. If ou change the order in which ou multipl a polnomial and a monomial, the product will be different. Eplain our reasoning or give a countereample. 2. Eplain the steps ou would take to find the product of 3 7 and PEN ENDED Write a monomial and a polnomial, each having a degree no greater than. Then find their product. 684 Chapter 3 Polnomials and Nonlinear Functions

20 Guided Practice Application Find each product. 4. (5 4)3 5. a(a 4) 6. t(7t 8) 7. (3 7)4 8. a(2a b) 9. 5( ) 0. TENNIS The perimeter of a tennis court is 228 feet. The length of the court is 6 feet more than twice the width. What are the dimensions of the tennis court? Practice and Appl For Eercises See Eamples 26, Etra Practice See page 756. Find each product.. 7(2n 5) 2. ( 4b)6 3. t(t 9) 4. ( 5) 5. a(7a 6) 6. (3 2) 7. 4n(0 2n) 8. 3(6 4) 9. 3( 2 2) 20. ab(a 2 7) 2. 5( ) 22. 4m(m 2 m) 23. 7( ) 24. 3( ) 25. 4c(c 3 7c 0) ( ) Solve each equation ( 2w 3) 28. 3(2a 2) 3a BASKETBALL The dimensions of high school basketball courts are different than the dimensions of college basketball courts, as shown in the table. Use the information in the table to find the length and width of each court. Basketball Courts Measure High School (ft) College (ft) Perimeter Width w w Length 2w 6 (2w 6) 0 Basketball Basketball originated in Springfield, Massachusetts, in 89. The first game was plaed with a soccer ball and two peach baskets used as goals. Source: BXES A bo large enough to hold 43,000 liters of water was made from one large sheet of cardboard. a. Write a polnomial that represents the area of the cardboard used to make the bo. Assume the top and bottom of the bo are the same. (Hint: (2 2) (6 2) ) b. If is.2 meters and is 0. meter, what is the total amount of cardboard in square meters used to make the bo? CRITICAL THINKING You have seen how algebra tiles can be used to connect multipling a polnomial b a monomial and the Distributive Propert. Draw a model and write a sentence to show how to multipl two binomials:(a b)(c d). 2 Lesson 3-4 Multipling a Polnomial b a Monomial 685

21 32. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How is the Distributive Propert used to multipl a polnomial b a monomial? Include the following in our answer: a description of the Distributive Propert, and an eample showing the steps used to multipl a polnomial and a monomial. Standardized Test Practice 33. What is the product of 2 and 8? A 2 8 B C D The area of the rectangle is 252 square centimeters. Find the length of the longer side. A 8 cm B 6 cm C 4 cm D 0 cm 2 0 cm cm Maintain Your Skills Mied Review Find each sum or difference. (Lessons 3-2 and 3-3) 35. (2 ) (9a 3a 2 ) (a 4) 37. ( 2 6 2) ( ) 38. (4 7) (2 2) 39. (9 8) ( 3) 40. (3n 2 6n 5) (6n 2 5) 4. STATISTICS Describe two was that a graph of sales of several brands of cereal could be misleading. (Lesson 2-5) State whether each transformation of the triangles is a reflection, translation, or rotation. (Lesson 0-3) B L' M' L A M C N' N A' C' B' Getting Read for the Net Lesson PREREQUISITE SKILL Complete each table to find the coordinates of four points through which the graph of each function passes. (To review using tables to find ordered pair solutions, see Lesson 8-2.) (, ) (, ) (, ) Chapter 3 Polnomials and Nonlinear Functions

22 Linear and Nonlinear Functions Determine whether a function is linear or nonlinear. Vocabular nonlinear function quadratic function cubic function can ou determine whether a function is linear? The sum of the lengths of three sides of a new deck is 40 feet. Suppose represents the width of the deck. Then the length of the deck is a. Write an epression to represent the area of the deck. b. Find the area of the deck for widths of 6, 8, 0, 2, and 4 feet. c. Graph the points whose ordered pairs are (width, area). Do the points fall along a straight line? Eplain. NNLINEAR FUNCTINS In Lesson 8-2, ou learned that linear functions have graphs that are straight lines. These graphs represent constant rates of change. Nonlinear functions do not have constant rates of change. Therefore, their graphs are not straight lines. Eample Identif Functions Using Graphs Determine whether each graph represents a linear or nonlinear function. Eplain. a. b. 2 2 The graph is a curve, not a straight line, so it represents a nonlinear function. This graph is also a curve, so it represents a nonlinear function. Recall that the equation for a linear function can be written in the form m b, where m represents the constant rate of change. Therefore, ou can determine whether a function is linear b looking at its equation. Eample 2 Identif Functions Using Equations Determine whether each equation represents a linear or nonlinear function. a. 0 b. 3 This is linear because it can This is nonlinear because is in be written as 0 0. the denominator and the equation cannot be written in the form m b. Lesson 3-5 Linear and Nonlinear Functions 687

23 The tables represent the functions in Eample 2. Compare the rates of change. Linear Nonlinear The rate of change is constant. The rate of change is not constant. A nonlinear function does not increase or decrease at the same rate. You can check this b using a table. Eample 3 Identif Functions Using Tables Determine whether each table represents a linear or nonlinear function. a. b As increases b 5, decreases b 20. So this is a linear function. As increases b 2, increases b a greater amount each time. So this is a nonlinear function. Reading Math Cubic Cubic means three-dimensional. A cubic function has a variable to the third power. Some nonlinear functions are given special names. Quadratic and Cubic Functions A quadratic function is a function that can be described b an equation of the form a 2 b c, where a 0. A cubic function is a function that can be described b an equation of the form a 3 b 2 c d, where a 0. Eamples of these and other nonlinear functions are shown below. Nonlinear Functions Quadratic Cubic Eponential Inverse Variation Chapter 3 Polnomials and Nonlinear Functions

24 Standardized Test Practice Eample 4 Multiple-Choice Test Item Describe a Linear Function Which rule describes a linear function? A B ( )5 C D Read the Test Item A rule describes a relationship between variables. A rule that can be written in the form m b describes a relationship that is linear. Test-Taking Tip Find out if there is a penalt for incorrect answers. If there is no penalt, making an educated guess can onl increase our score, or at worst, leave our score the same. Solve the Test Item cubic equation The variable has an eponent of quadratic equation The variable has an eponent of 2. You can eliminate choices A and D. ( ) This is a quadratic equation. Eliminate choice B. The answer is C. CHECK This equation is in the form m b. Concept Check Guided Practice. Describe two methods for determining whether a function is linear. 2. Eplain whether a compan would prefer profits that showed linear growth or eponential growth. 3. PEN ENDED Use newspapers, magazines, or the Internet to find reallife eamples of nonlinear situations. Determine whether each graph, equation, or table represents a linear or nonlinear function. Eplain Standardized Test Practice Which rule describes a nonlinear function? A 00 B 8 C 9 D Lesson 3-5 Linear and Nonlinear Functions 689

25 Practice and Appl For Eercises See Eamples 6, , 29 3 Etra Practice See page 757. Determine whether each graph, equation, or table represents a linear or nonlinear function. Eplain TECHNLGY The graph shows the increase of trademark applications for internet-related products or services. Would ou describe this growth as linear or nonlinear? Eplain. nline Research Data Update Is the growth of the Internet itself linear or nonlinear? Visit www. pre-alg.com/data_update to learn more. USA TDAY Snapshots Web leads trademark surge Led b new online businesses, trademark applications jumped 32% last ear to 265,342. Applications for Internet-related products or services: 307 3,059 8,22 9, Source: Dechert Price & Rhoads ( Trends in Trademarks, ,235 33,73 B Anne R. Care and Marc E. Mullins, USA TDAY 690 Chapter 3 Polnomials and Nonlinear Functions 28. CRITICAL THINKING Are all graphs of straight lines linear functions? Eplain.

26 The trend in farm income can be modeled with a nonlinear function. Visit webquest to continue work on our WebQuest project. 29. PATENTS The table shows the ears in which the first si million patents were issued. Is the number of patents issued a linear function of time? Eplain. Year Number of Patents Issued 9 million million 96 3 million million 99 5 million million Source: New York Times Standardized Test Practice 30. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can ou determine whether a function is linear? Include the following in our answer: a list of was in which a function can be represented, and an eplanation of how each representation can be used to identif the function as linear or nonlinear. 3. Which equation represents a linear function? A 2 B 3 2 C 2 D ( 4) Maintain Your Skills Mied Review 32. Determine which general rule represents a nonlinear function if a. A a B a C a D a Find each product. (Lesson 3-4) 33. t(4 9t) 34. 5n( 3n) 35. (a 2b)ab Find each difference. (Lesson 3-3) 36. (2 7) ( ) 37. (4 ) (5 ) 38. (6a a 2 ) (8a 3) 39. GEMETRY Classif a 65 angle as acute, obtuse, right, or straight. (Lesson 9-3) Getting Read for the Net Lesson PREREQUISITE SKILL Use a table to graph each line. (To review graphing equations, see Lesson 8-3.) P ractice Quiz 2 Find each product. (Lesson 3-4). c(2c 2 8) 2. (4 2)3 3. a 2 (5 a 2a 2 ) Determine whether each equation represents a linear or nonlinear function. Eplain. (Lesson 3-5) Lessons 3-4 and Lesson 3-5 Linear and Nonlinear Functions 69

27 Graphing Quadratic and Cubic Functions Graph quadratic functions. Graph cubic functions. are functions, formulas, tables, and graphs related? You can find the area of a square A b squaring the length of a side s. This relationship can be represented in different was. Equation Table Graph s s length of a Area equals side squared. A s 2 s s 2 (s, A) (0, 0) 2 (, ) (2, 4) Area A A s 2 a. The volume of cube V equals the cube of the length of an edge a. Write a formula to represent the volume of a cube as a function of edge length. b. Graph the volume as a function of edge length. (Hint: Use values of a like 0, 0.5,,.5, 2, and so on.) a a Side a s QUADRATIC FUNCTINS In Lesson 3-5, ou saw that functions can be represented using graphs, equations, and tables. This allows ou to graph quadratic functions such as A s 2 using an equation or a table of values. Eample Graph Quadratic Functions Graph each function. a. 2 2 Stud Tip Graphing It is often helpful to substitute decimal values of in order to graph points that are closer together. Make a table of values, plot the ordered pairs, and connect the points with a curve. 2 2 (, ).5 2(.5) (.5, 4.5) 2( ) 2 2 (, 2) 0 2(0) 2 0 (0, 0) 2() 2 2 (, 2).5 2(.5) (.5, 4.5) Chapter 3 Polnomials and Nonlinear Functions

28 b. 2 2 (, ) 2 ( 2) 2 3 ( 2, 3) ( ) 2 0 (, 0) 0 (0) 2 (0, ) () 2 0 (, 0) 2 (2) 2 3 (2, 3) 2 c (, ) 2 ( 2) 2 3 ( 2, ) ( ) (, 2) 0 (0) (0, 3) () (, 2) 2 (2) 2 3 (2, ) 2 3 You can also write a rule from a verbal description of a function, and then graph. Eample 2 Use a Function to Solve a Problem SKYDIVING The distance in feet that a skdiver falls is equal to siteen times the time squared, with the time given in seconds. Graph this function and estimate how far he will fall in 4.5 seconds. Words Variables Distance is equal to siteen times the time squared. Let d the distance in feet and t the time in seconds. is equal the time Distance to siteen times squared. } } } } Equation d 6 t 2 } Skdiving Skdivers fall at a speed of 0 20 miles per hour. Source: The equation is d 6t 2. Since the variable t has an eponent of 2, this function is nonlinear. Now graph d 6t 2. Since time cannot be negative, use onl positive values of t. t d 6t 2 (t, d) 0 6(0) 2 0 (0, 0) 6() 2 6 (, 6) 2 6(2) 2 64 (2, 64) 3 6(3) 2 44 (3, 44) 4 6(4) (4, 256) 5 6(5) (5, 400) 6 6(6) (6, 576) Distance (ft) d t Time (s) B looking at the graph, we find that in 4.5 seconds, the skdiver will fall approimatel 320 feet. You could find the eact distance b substituting 4.5 for t in the equation d 6t 2. Lesson 3-6 Graphing Quadratic and Cubic Functions 693

29 CUBIC FUNCTINS You can also graph cubic functions such as the formula for the volume of a cube b making a table of values. Eample 3 Graph Cubic Functions Graph each function. a. 3 3 (, ).5 (.5) (.5, 3.4) ( ) 3 (, ) 0 (0) 3 0 (0, 0) () 3 (, ).5 (.5) (.5, 3.4) 3 b. 3 3 (, ).5 (.5) (.5, 4.4) ( ) 3 2 (, 2) 0 (0) 3 (0, ) () 3 0 (, 0).5 (.5) (.5, 2.4) 3 Concept Check Guided Practice Application. Describe one difference between the graph of n 2 and the graph of n 3 for an rational number n. 2. Eplain how to determine whether a function is quadratic. 3. PEN ENDED Write a quadratic function and eplain how to graph it. Graph each function GEMETRY A cube has edges measuring a units. a. Write a quadratic equation for the surface area S of the cube. b. Graph the surface area as a function of a. (Hint: Use values of a like 0, 0.5,,.5, 2, and so on.) Practice and Appl 694 Chapter 3 Polnomials and Nonlinear Functions Graph each function

30 For Eercises See Eamples 22, Etra Practice See page Graph 2 4 and 4 3. Are these equations functions? Eplain. 24. Graph 2 and 3 in the first quadrant on the same coordinate plane. Eplain which graph shows faster growth. The maimum point of a graph is the point with the greatest value coordinate. The minimum point is the point with the least value coordinate. Find the coordinates of each point. 25. the maimum point of the graph of the minimum point of the graph of 2 6 Graph each pair of equations on the same coordinate plane. Describe their similarities and differences CNSTRUCTIN For Eercises 3 33, use the information below and the figure at the right. A dog trainer is building a dog pen with a 00-foot ft roll of chain link fence. 3. Write an equation to represent the area A of the pen. 50 ft 32. Graph the equation ou wrote in Eercise What should the dimensions of the dog pen be to enclose the maimum area inside the fence? (Hint: Find the coordinates of the maimum point of the graph.) GEMETRY Write a function for each of the following. Then graph the function in the first quadrant. 34. the volume V of a rectangular prism as a function of a fied height of 2 units and a square base of varing lengths s 35. the volume V of a clinder as a function of a fied height of 0.2 unit and radius r 36. CRITICAL THINKING Describe how ou can find real number solutions of the quadratic equation a 2 b c 0 from the graph of the quadratic function a 2 b c. Standardized Test Practice 37. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are functions, formulas, tables, and graphs related? Include the following in our answer: an eplanation of how to make a graph b using a rule, and an eplanation of how to write a rule b using a graph. 38. Which equation represents the graph at the right? A 4 2 B C D Lesson 3-6 Graphing Quadratic and Cubic Functions 695

31 39. For a certain frozen pizza, as the cost goes from $2 to $4, the demand can be modeled b the formula , where represents the cost and represents the number of pizzas sold. Estimate the cost that will result in the greatest demand. Monthl Sales ($ thousands) A $0 B $ C $3 D $8 Cost ($) Etending the Lesson Just as ou can estimate the area of irregular figures, ou can also estimate the area under a curve that is graphed on the coordinate plane. Estimate the shaded area under each curve to the nearest square unit Maintain Your Skills Mied Review 42. SCIENCE The graph shows how water vapor pressure increases as the temperature increases. Is this relationship linear or nonlinear? Eplain. (Lesson 3-5) Water Vapor Pressure Temperature ( o C) Find each product. (Lesson 3-4) 43. (2 4)5 44. n(n 6) 45. 3(8 7) Vapor Pressure (kpa) Write an equation in slope-intercept form for the line passing through each pair of points. (Lesson 8-7) 46. (3, 6) and (0, 9) 47. (2, 5) and (, 7) 48. ( 4, 3) and (8, 6) Famil Farms It is time to complete our project. Use the information and data ou have gathered to prepare a Web page about farming or ranching in the United States. Be sure to include at least five graphs or tables that show statistics about farming or ranching and at least one scatter plot that shows a farming or ranching statistic over time, from which ou can make predictions Chapter 3 Polnomials and Nonlinear Functions

32 A Follow-Up of Lesson 3-6 Families of Quadratic Functions A quadratic function can be described b an equation of the form a 2 b c, where a 0. The graph of a quadratic function is called a parabola. Recall that families of linear graphs share the same slope or -intercept. Similarl, families of parabolas share the same maimum or minimum point, or have the same shape. Graph = 2 and = on the same screen and describe how the are related. Enter the function 2. Enter 2 as Y. X,T,,n KEYSTRKES: ENTER Enter the function 2 4. Enter 2 4 as Y2. X,T,,n KEYSTRKES: 4 ENTER Displa the graph. Graph both quadratic functions on the same screen. ZM KEYSTRKES: 6 The first function graphed is Y or 2. The second is Y2 or 2 4. Press TRACE and move along each function b using the right and left arrow kes. Move from one function to another b using the up and down arrow kes. The graphs are similar in that the are both parabolas. However, the graph of 2 has its verte at (0, 0), whereas the graph of 2 4 has its verte at (0, 4) Eercises. Graph 2, 2 5, and 2 3 on the same screen and draw the parabolas on grid paper. Compare and contrast the three parabolas. 2. Make a conjecture about how adding or subtracting a constant c affects the graph of a quadratic function. 3. The three parabolas at the right are graphed in the standard viewing window and have the same shape as the graph of 2. Write an equation for each, beginning with the lowest parabola. 4. Clear all functions from the menu. Enter as Y, 2 as Y2, and 3 2 as Y3. Graph the functions in the standard viewing window on the same screen. Then draw the graphs on the same coodinate grid. How does the shape of the parabola change as the coefficient of 2 increases? Graphing Calculator Investigation 697

33 Vocabular and Concept Check binomial (p. 669) cubic function (p. 688) degree (p. 670) nonlinear function (p. 687) polnomial (p. 669) quadratic function (p. 688) trinomial (p. 669) Choose the correct term to complete each sentence.. A (binomial, trinomial) is the sum or difference of three monomials. 2. Monomials that contain the same variables with the same ( power, sign) are like terms. 3. The function 2 3 is an eample of a ( cubic, quadratic) function. 4. The equation 2 5 is an eample of a (cubic, quadratic ) function and 4 2 are eamples of (binomials, like terms). 6. The equation is an eample of a (quadratic, cubic) function. 7. The graph of a quadratic function is a (straight line, curve). 8. To multipl a polnomial and a monomial, use the ( Distributive, Commutative) Propert. 3- See pages Eample Polnomials Concept Summar A polnomial is an algebraic epression that contains one or more monomials. A binomial has two terms and a trinomial has three terms. The degree of a monomial is the sum of the eponents of its variables. State whether 3 2 is a monomial, binomial, or trinomial. Then find the degree. The epression is the difference of two monomials. So it is a binomial. 3 has degree 3, and 2 has degree or 2. So, the degree of 3 2 is 3. Eercises Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial. See Eample on page c t b a n Find the degree of each polnomial. See Eamples 2 and 3 on page a 2 b 20. n z Chapter 3 Polnomials and Nonlinear Functions

34 Chapter 3 Stud Guide and Review 3-2 See pages Eample Adding Polnomials Concept Summar To add polnomials, add like terms. Find ( ) ( 2 6) ( ) 2 6 Align like terms Add. The sum is Eercises Find each sum. See Eample on pages 674 and b ( ) 5b 5 ( ) ( ) (9m 3n) (0m 4n) 29. ( 3 2 2) (4 2 3) 3-3 See pages Eample Subtracting Polnomials Concept Summar To subtract polnomials, subtract like terms or add the additive inverse. Find ( ) ( 2 2 ) ( ) 2 2 Align like terms Subtract. The difference is Eercises Find each difference. See Eamples and 2 on pages 678 and a ( )3a 2 0 ( ) ( ) ( 8) (2 7) 34. (3n 2 7) (n 2 n 4) 3-4 See pages Eample Multipling a Polnomial b a Monomial Concept Summar To multipl a polnomial and a monomial, use the Distributive Propert. Find 3( 8). 3( 8) 3() ( 3)(8) Distributive Propert Simplif. Eercises Find each product. See Eample on page (4t 2) 36. (2 3)7 37. k(6k 3) 38. 4d(2d 5) 39. 2a(9 a 2 ) 40. 6( ) Chapter 3 Stud Guide and Review 699

35 Etra Practice, see pages Mied Problem Solving, see page See pages Eample Linear and Nonlinear Functions Concept Summar Nonlinear functions do not have constant rates of change. Determine whether each graph, equation, or table represents a linear or nonlinear function. Eplain. a. b. 2 c. Linear; equation can be written as m b Nonlinear; graph is not a straight line. Linear; rate of change is constant. Eercises Determine whether each graph, equation, or table represents a linear or nonlinear function. Eplain. See Eamples 3 on pages 687 and See pages Eample Graphing Quadratic and Cubic Functions Concept Summar Quadratic and cubic functions can be graphed b plotting points. Graph (, ) 2 ( 2) 2 3 ( 2, ) ( ) (, 2) 0 (0) (0, 3) () (, 2) 2 (2) 2 3 (2, ) 2 3 Eercises Graph each function. See Eamples and 3 on pages Chapter 3 Polnomials and Nonlinear Functions

36 Vocabular and Concepts. Define polnomial. 2. Eplain how the degree of a monomial is found. 3. PEN ENDED Draw the graph of a linear and a nonlinear function. Skills and Applications Determine whether each epression is a polnomial. If it is, classif it as a monomial, binomial, or trinomial m 5 p4 Find the degree of each polnomial. 7. 5ab 3 8. w 5 3w 3 4 Find each sum or difference. 9. (5 8) ( 2 3) 0. (5a 2b) ( 4a 5b). ( 3m 3 5m 9) (7m 3 2m 2 4) 2. (6p 5) (3p 8) 3. (5w 3) (6w 4) 4. ( 2s 2 4s 7) (6s 2 7s 9) Find each product. 5. (3 5) 6. 5a(a 2 b 2 ) 7. 6p( 2p 2 3p 4) Determine whether each graph, equation, or table represents a linear or nonlinear function. Eplain Graph each function GEMETRY Refer to the rectangle. a. Write an epression for the perimeter of the rectangle. b. Find the value of if the perimeter is 4 inches. 2 2 in. 25. STANDARDIZED TEST PRACTICE The length of a garden is equal to 5 less than four times its width. The perimeter of the garden is 40 feet. Find the length of the garden. A ft B 5 ft C 0 ft D 5 ft 3 9 in. Chapter 3 Practice Test 70

37 Part Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. If n represents a positive number, which of these epressions is equivalent to n n n? (Lesson -3) A n 3 B 3n C n 3 D 3(n ) 2. Connor sold 4 fewer tickets to the band concert than Miguel sold. Klie sold 3 times as man tickets as Connor. If the number of tickets Miguel sold is represented b m, which of these epressions represents the number of tickets that Klie sold? (Lesson 3-6) A m 4 B C D 4 3m 3m 4 3(m 4) 3. Melissa s famil calculated that the drove an average of 400 miles per da during their three-da trip. The drove 460 miles on the first da and 360 miles on the second da. How man miles did the drive on the third da? A (Lesson 5-7) 340 B 380 C 40 D What is the ratio of the length of a side of a square to its perimeter? (Lesson 6-) A C 6 B 4 3 Multiple Choice D 2 5. The table shows values of and, where is proportional to. What are the missing values, S and T? (Lesson 6-3) A B C D S 36 and T 3 S 2 and T 5 S 5 and T 2 S 3 and T In the figure at the right, lines and m are parallel. Choose two angles whose measures have a sum of 80. (Lesson 0-2) A and 5 B C D 2 and 8 2 and 5 4 and 8 7. The point represented b coordinates (4, 6) is reflected across the -ais. What are the coordinates of the image? (Lesson 0-3) A ( 6, 4) B ( 4, 6) C ( 4, 6) D (4, 6) 8. If is subtracted from , what is the difference? (Lesson 3-3) A B C D S 5 T Which function includes all of the ordered pairs in the table? It ma help ou to sketch a graph of the points. (Lesson 3-6) m Test-Taking Tip Question 2 If ou have time at the end of a test, go back to check our calculations and answers. If the test allows ou to use a calculator, use it to check our calculations. 702 Chapter 3 Polnomials and Nonlinear Functions A C D 2 B