CHAPTER 4Unit 2 Quadratic, Polnomial, and Radical Functions Comple Numbers, p. 28 f(z) 5 z 2 c Quadratic Functions and Factoring Prerequisite Skills... 234 4. Graph Quadratic Functions in Standard Form... 236 Graphing Calculator Activit Find Maimum and Minimum Values... 244 4.2 Graph Quadratic Functions in Verte or Intercept Form... 245 4.3 Solve 2 + b + c 5 0 b Factoring... 252 4.4 Solve a 2 + b + c 5 0 b Factoring... 259 4.5 Solve Quadratic Equations b Finding Square Roots... 266 Problem Solving Workshop... 272 Mied Review of Problem Solving... 274 4.6 Perform Operations with Comple Numbers... 275 4.7 Complete the Square... 284 Investigating Algebra: Using Algebra Tiles to Complete the Square... 283 4.8 Use the Quadratic Formula and the Discriminant... 292 4.9 Graph and Solve Quadratic Inequalities... 300 4.0 Write Quadratic Functions and Models... 309 Investigating Algebra: Modeling Data with a Quadratic Function... 308 Mied Review of Problem Solving... 36 ASSESSMENT Quizzes... 265, 29, 35 Chapter Summar and Review... 37 Chapter Test... 323 Standardized Test Preparation and Practice... 324 Activities... 235, 238, 247, 269, 279, 287, 300 Chapter 4 Highlights PROBLEM SOLVING Mied Review of Problem Solving, 274, 36 Multiple Representations, 242, 258, 272, 290, 306, 34 Multi-Step Problems, 250, 257, 274, 298, 307, 34, 36 Using Alternative Methods, 272 Real-World Problem Solving Eamples, 239, 246, 254, 262, 277, 287, 295, 303, 3 ASSESSMENT Standardized Test Practice Eamples, 254, 268, 286, Multiple Choice, 24, 249, 256, 263, 270, 280, 288, 296, 298, 305, 32, 35, 324 Short Response/Etended Response, 24, 242, 25, 256, 258, 264, 27, 274, 28, 282, 289, 290, 297, 299, 306, 33, 36 Writing/Open-Ended, 240, 250, 255, 257, 263, 269, 279, 289, 296, 304, 32, 36 TECHNOLOGY At classzone.com: Animated Algebra, 235, 238, 247, 269, 279, 287, 300 @Home Tutor, 234, 242, 244, 250, 257, 264, 270, 28, 290, 298, 306, 308, 34, 38, Online Quiz, 243, 25, 258, 265, 27, 282, 29, 299, 307, 35 Electronic Function Librar, 37 State Test Practice, 274, 36, 327 Contents i
4 Quadratic Functions and Factoring 4. Graph Quadratic Functions in Standard Form 4.2 Graph Quadratic Functions in Verte or Intercept Form 4.3 Solve 2 b c 5 0 b Factoring 4.4 Solve a 2 b c 5 0 b Factoring 4.5 Solve Quadratic Equations b Finding Square Roots 4.6 Perform Operations with Comple Numbers 4.7 Complete the Square 4.8 Use the Quadratic Formula and the Discriminant 4.9 Graph and Solve Quadratic Inequalities 4.0 Write Quadratic Functions and Models Before In previous chapters, ou learned the following skills, which ou ll use in Chapter 4: evaluating epressions, graphing functions, and solving equations. Prerequisite Skills VOCABULARY CHECK Cop and complete the statement. (0, 2). The -intercept of the line shown is?. (3, 0) 2. The -intercept of the line shown is?. SKILLS CHECK Evaluate the epression when 5 23. (Review p. 0 for 4., 4.7.) 3. 25 2 4. 2 2 2 8 5. ( 4)2 6. 23( 2 7)2 2 Graph the function and label the verte. (Review p. 23 for 4.2.) 7. 5 2 8. 5 2 3 9. 5 22 0. 5 2 5 4 Solve the equation. (Review p. 8 for 4.3, 4.4.). 8 5 0 2. 3 2 5 5 0 3. 2 5 4. 4( 2 3) 5 9 SFSFRVJTJUF TLJMMT QSBDUJDF BU DMBTT[POF DPN 234 n2pe-0400.indd 234 0/25/05 :36:40 AM
Now In Chapter 4, ou will appl the big ideas listed below and reviewed in the Chapter Summar on page 37. You will also use the ke vocabular listed below. Big Ideas Graphing and writing quadratic functions in several forms 2 Solving quadratic equations using a variet of methods 3 Performing operations with square roots and comple numbers KEY VOCABULARY standard form of a quadratic function, p. 236 root of an equation, p. 253 zero of a function, p. 254 completing the square, p. 284 parabola, p. 236 square root, p. 266 quadratic formula, p. 292 verte form, p. 245 comple number, p. 276 discriminant, p. 294 intercept form, p. 246 imaginar number, p. 276 best-fitting quadratic model, p. 3 quadratic equation, p. 253 Wh? You can use quadratic functions to model the heights of projectiles. For eample, the height of a baseball hit b a batter can be modeled b a quadratic function. Algebra The animation illustrated below for Eample 7 on page 287 helps ou answer this question: How does changing the ball speed and hitting angle affect the maimum height of a baseball? 4HE FUNCTION IS NOW IN VERTEX FORM Y n T n 2EMEMBER THAT THE VERTEX OF THE PARABOLA IS AT H K AND THAT THE MAXIMUM HEIGHT OF THE BASEBALL IN FLIGHT IS K 7HAT IS THE MAXIMUM HEIGHT OF THE BASEBALL IN FEET -AXIMUM HEIGHT FEET 3TART A quadratic function models the height of a baseball in flight. #HECK!NSWER Rewrite the function in verte form to find the maimum height of the ball. Algebra at classzone.com Other animations for Chapter 4: pages 238, 247, 269, 279, 300, and 37 235 n2pe-0400.indd 235 0/25/05 :36:5 AM
4. Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in Eample 5. Ke Vocabular quadratic function parabola verte ais of smmetr minimum value maimum value A quadratic function is a function that can be written in the standard form 5 a 2 b c where a Þ 0. The graph of a quadratic function is a parabola. KEY CONCEPT Parent Function for Quadratic Functions For Your Notebook The parent function for the famil of all quadratic functions is f() 5 2. The graph of f() 5 2 is the parabola shown below. The lowest or highest point on a parabola is the verte. The verte for f() 5 2 is (0, 0). 5 2 The ais of smmetr divides the parabola into mirror images and passes through the verte. For f() 5 2, and for an quadratic function g() 5 a 2 b c where b 5 0, the verte lies on the -ais and the ais of smmetr is 5 0. E XAMPLE Graph a function of the form 5 a 2 Graph 5 2 2. Compare the graph with the graph of 5 2. Solution STEP Make a table of values for 5 2 2. SKETCH A GRAPH Choose values of on both sides of the ais of smmetr 5 0. STEP 2 STEP 3 22 2 0 2 8 2 0 2 8 Plot the points from the table. Draw a smooth curve through the points. 3 5 2 5 2 2 STEP 4 Compare the graphs of 5 2 2 and 5 2. Both open up and have the same verte and ais of smmetr. The graph of 5 2 2 is narrower than the graph of 5 2. 236 Chapter 4 Quadratic Functions and Factoring
E XAMPLE 2 Graph a function of the form 5 a 2 c Graph 5 2 2 2 3. Compare the graph with the graph of 5 2. SKETCH A GRAPH Choose values of that are multiples of 2 so that the values of will be integers. Solution STEP Make a table of values for 5 2 2 2 3. STEP 2 STEP 3 24 22 0 2 4 25 3 25 Plot the points from the table. Draw a smooth curve through the points. 5 2 5 2 2 3 2 STEP 4 Compare the graphs of 5 2 2 2 3 and 5 2. Both graphs have the same ais of smmetr. However, the graph of 5 2 2 2 3 opens down and is wider than the graph of 5 2. Also, its verte is 3 units higher. GUIDED PRACTICE for Eamples and 2 Graph the function. Compare the graph with the graph of 5 2.. 5 24 2 2. 5 2 2 2 5 3. f() 5 4 2 2 GRAPHING ANY QUADRATIC FUNCTION You can use the following properties to graph an quadratic function 5 a 2 b c, including a function where b Þ 0. KEY CONCEPT For Your Notebook Properties of the Graph of 5 a 2 b c 5 a 2 b c, a > 0 5 a 2 b c, a < 0 5 2 b 2a (0, c) (0, c) 5 2 b 2a Characteristics of the graph of 5 a 2 b c: The graph opens up if a > 0 and opens down if a < 0. The graph is narrower than the graph of 5 2 if a > and wider if a <. The ais of smmetr is 5 2 b 2a and the verte has -coordinate 2 b 2a. The -intercept is c. So, the point (0, c) is on the parabola. 4. Graph Quadratic Functions in Standard Form 237
E XAMPLE 3 Graph a function of the form 5 a 2 b c Graph 5 2 2 2 8 6. AVOID ERRORS Be sure to include the negative sign before the fraction when calculating the -coordinate of the verte. Solution STEP Identif the coefficients of the function. The coefficients are a 5 2, b 5 28, and c 5 6. Because a > 0, the parabola opens up. STEP 2 Find the verte. Calculate the -coordinate. 5 2 b 2a 5 2 (28) 2(2) 5 2 Then find the -coordinate of the verte. 5 2(2) 2 2 8(2) 6 5 22 So, the verte is (2, 22). Plot this point. STEP 3 Draw the ais of smmetr 5 2. STEP 4 Identif the -intercept c, which is 6. Plot the point (0, 6). Then reflect this point in the ais of smmetr to plot another point, (4, 6). STEP 5 Evaluate the function for another value of, such as 5. ais of smmetr 5 2 verte (2, 22) 5 2() 2 2 8() 6 5 0 STEP 6 Plot the point (, 0) and its reflection (3, 0) in the ais of smmetr. Draw a parabola through the plotted points. (2, 22) 5 at classzone.com GUIDED PRACTICE for Eample 3 Graph the function. Label the verte and ais of smmetr. 4. 5 2 2 2 2 5. 5 2 2 6 3 6. f() 5 2 3 2 2 5 2 KEY CONCEPT For Your Notebook Minimum and Maimum Values Words For 5 a 2 b c, the verte s -coordinate is the minimum value of the function if a > 0 and the maimum value if a < 0. Graphs maimum minimum a is positive a is negative 238 Chapter 4 Quadratic Functions and Factoring
E XAMPLE 4 Find the minimum or maimum value Tell whether the function 5 3 2 2 8 20 has a minimum value or a maimum value. Then find the minimum or maimum value. Solution Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the verte. 5 2 b 2a 5 2 (28) 5 3 2(3) 5 3(3) 2 2 8(3) 20 5 27 c The minimum value is 5 27. You can check the answer on a graphing calculator. Minimum X=3 Y=-7 E XAMPLE 5 Solve a multi-step problem GO-CARTS A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for ever $ reduction in the price per racer. How can the owner of the go-cart track maimize weekl revenue? Solution STEP STEP 2 Define the variables. Let represent the price reduction and R() represent the weekl revenue. Write a verbal model. Then write and simplif a quadratic function. Revenue (dollars) 5 Price (dollars/racer) p Attendance (racers) INTERPRET FUNCTIONS Notice that a 5 220 < 0, so the revenue function has a maimum value. STEP 3 R() 5 (35 2 ) p (380 20) R() 5 3,300 700 2 380 2 20 2 R() 5 220 2 320 3,300 Find the coordinates (, R()) of the verte. 5 2 b 2a 5 2 320 5 8 2(220) Find -coordinate. R(8) 5 220(8) 2 320(8) 3,300 5 4,580 Evaluate R(8). c The verte is (8, 4,580), which means the owner should reduce the price per racer b $8 to increase the weekl revenue to $4,580. GUIDED PRACTICE for Eamples 4 and 5 7. Find the minimum value of 5 4 2 6 2 3. 8. WHAT IF? In Eample 5, suppose each $ reduction in the price per racer brings in 40 more racers per week. How can weekl revenue be maimized? 4. Graph Quadratic Functions in Standard Form 239
4. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 37, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 39, 40, 43, 53, 58, and 60 5 MULTIPLE REPRESENTATIONS E. 59. VOCABULARY Cop and complete: The graph of a quadratic function is called a(n)?. 2. WRITING Describe how to determine whether a quadratic function has a minimum value or a maimum value. EXAMPLE on p. 236 for Es. 3 2 USING A TABLE Cop and complete the table of values for the function. 3. 5 4 2 4. 5 23 2 22 2 0 2????? 22 2 0 2????? 5. 5 2 2 6. 5 2 3 2 24 22 0 2 4????? 26 23 0 3 6????? MAKING A GRAPH Graph the function. Compare the graph with the graph of 5 2. 7. 5 3 2 8. 5 5 2 9. 5 22 2 0. 5 2 2. f() 5 3 2 2. g() 5 2 4 2 EXAMPLE 2 on p. 237 for Es. 3 8 3. 5 5 2 4. 5 4 2 5. f() 5 2 2 2 6. g() 5 22 2 2 5 7. f() 5 3 4 2 2 5 8. g() 5 2 5 2 2 2 ERROR ANALYSIS Describe and correct the error in analzing the graph of 5 4 2 24 2 7. 9. The -coordinate of the verte is: 5 b 2a 5 24 2(4) 5 3 20. The -intercept of the graph is the value of c, which is 7. EXAMPLE 3 on p. 238 for Es. 2 32 MAKING A GRAPH Graph the function. Label the verte and ais of smmetr. 2. 5 2 2 22. 5 3 2 2 6 4 23. 5 24 2 8 2 24. 5 22 2 2 6 3 25. g() 5 2 2 2 2 2 26. f() 5 26 2 2 4 2 5 27. 5 2 3 2 2 3 6 28. 5 2 3 4 2 2 4 2 29. g() 5 2 3 5 2 2 2 30. f() 5 2 2 2 3 3. 5 8 5 2 2 4 5 32. 5 2 5 3 2 2 2 4 240 Chapter 4 Quadratic Functions and Factoring
EXAMPLE 4 on p. 239 for Es. 33 38 MINIMUMS OR MAXIMUMS Tell whether the function has a minimum value or a maimum value. Then find the minimum or maimum value. 33. 5 26 2 2 34. 5 9 2 7 35. f() 5 2 2 8 7 36. g() 5 23 2 8 2 5 37. f() 5 3 2 2 6 4 38. 5 2 4 2 2 7 2 39. MULTIPLE CHOICE What is the effect on the graph of the function 5 2 2 when it is changed to 5 2 2 3? A The graph widens. C The graph opens down. B The graph narrows. D The verte moves down the -ais. 40. MULTIPLE CHOICE Which function has the widest graph? A 5 2 2 B 5 2 C 5 0.5 2 D 5 2 2 IDENTIFYING COEFFICIENTS In Eercises 4 and 42, identif the values of a, b, and c for the quadratic function. 4. The path of a basketball thrown at an angle of 458 can be modeled b 5 20.02 2 6. 42. The path of a shot put released at an angle of 358 can be modeled b 5 20.0 2 0.7 6. 358 43. OPEN-ENDED MATH Write three different quadratic functions whose graphs have the line 5 4 as an ais of smmetr but have different -intercepts. MATCHING In Eercises 44 46, match the equation with its graph. 44. 5 0.5 2 2 2 45. 5 0.5 2 3 46. 5 0.5 2 2 2 3 A. (2, 5) (0, 3) B. (0, 3) (2, ) C. 2 (0, 0) (2, 22) MAKING A GRAPH Graph the function. Label the verte and ais of smmetr. 47. f() 5 0. 2 2 48. g() 5 20.5 2 2 5 49. 5 0.3 2 3 2 50. 5 0.25 2 2.5 3 5. f() 5 4.2 2 6 2 52. g() 5.75 2 2 2.5 53. SHORT RESPONSE The points (2, 3) and (24, 3) lie on the graph of a quadratic function. Eplain how these points can be used to find an equation of the ais of smmetr. Then write an equation of the ais of smmetr. 54. CHALLENGE For the graph of 5 a 2 b c, show that the -coordinate of the verte is 2 b2 4a c. 4. Graph Quadratic Functions in Standard Form 24
PROBLEM SOLVING EXAMPLE 5 on p. 239 for Es. 55 58 55. ONLINE MUSIC An online music store sells about 4000 songs each da when it charges $ per song. For each $.05 increase in price, about 80 fewer songs per da are sold. Use the verbal model and quadratic function to find how the store can maimize dail revenue. Revenue (dollars) 5 Price (dollars/song) p Sales (songs) R() 5 ( 0.05) p (4000 2 80) 56. DIGITAL CAMERAS An electronics store sells about 70 of a new model of digital camera per month at a price of $320 each. For each $20 decrease in price, about 5 more cameras per month are sold. Write a function that models the situation. Then tell how the store can maimize monthl revenue from sales of the camera. 57. GOLDEN GATE BRIDGE Each cable joining the two towers on the Golden Gate Bridge can be modeled b the function 5 9000 2 2 7 5 500 where and are measured in feet. What is the height h above the road of a cable at its lowest point? 58. SHORT RESPONSE A woodland jumping mouse hops along a parabolic path given b 5 20.2 2.3 where is the mouse s horizontal position (in feet) and is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Eplain. 59. MULTIPLE REPRESENTATIONS A communit theater sells about 50 tickets to a pla each week when it charges $20 per ticket. For each $ decrease in price, about 0 more tickets per week are sold. The theater has fied epenses of $500 per week. a. Writing a Model Write a verbal model and a quadratic function to represent the theater s weekl profit. b. Making a Table Make a table of values for the quadratic function. c. Drawing a Graph Use the table to graph the quadratic function. Then use the graph to find how the theater can maimize weekl profit. 5 WORKED-OUT SOLUTIONS 242 Chapter 4 Quadratic on p. WSFunctions and Factoring 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS
60. EXTENDED RESPONSE In 97, astronaut Alan Shepard hit a golf ball on the moon. The path of a golf ball hit at an angle of 458 and with a speed of 00 feet per second can be modeled b g 5 2 0,000 2 where is the ball s horizontal position (in feet), is the corresponding height (in feet), and g is the acceleration due to gravit (in feet per second squared). a. Model Use the information in the diagram to write functions for the paths of a golf ball hit on Earth and a golf ball hit on the moon. GRAPHING CALCULATOR In part (b), use the calculator s zero feature to answer the questions. b. Graphing Calculator Graph the functions from part (a) on a graphing calculator. How far does the golf ball travel on Earth? on the moon? c. Interpret Compare the distances traveled b a golf ball on Earth and on the moon. Your answer should include the following: a calculation of the ratio of the distances traveled a discussion of how the distances and values of g are related 6. CHALLENGE Lifeguards at a beach want to rope off a rectangular swimming section. The have P feet of rope with buos. In terms of P, what is the maimum area that the swimming section can have? MIXED REVIEW Solve the equation. (p. 8) 62. 2 3 5 0 63. 3 4 5 0 64. 29 7 5 24 2 5 65. 5 2 2 5 22 2 66. 0.7 3 5 0.2 2 2 67. 0.4 5 20.5 2 5 PREVIEW Prepare for Lesson 4.2 in Es. 68 73. Graph the function. (p. 23) 68. 5 2 5 69. 5 2 2 70. 5 3 2 7. 5 24 72. f() 5 2 2 3 6 73. g() 5 25 4 2 74. AVERAGE SPEED You are driving on a road trip. At 9:00 A.M., ou are 340 miles west of Nashville. At 2:00 P.M., ou are 70 miles west of Nashville. Find our average speed. (p. 82) EXTRA PRACTICE for Lesson 4., p. 03 4. Graph ONLINE Quadratic QUIZ Functions at classzone.com in Standard Form 243
4. Find Maimum and Minimum Values Use after Lesson 4. classzone.com Kestrokes QUESTION How can ou use a graphing calculator to find the maimum or minimum value of a function? EXAMPLE Find the maimum value of a function Find the maimum value of 522 2 2 0 2 5 and the value of where it occurs. STEP Graph function Graph the given function and select the maimum feature. STEP 2 Choose left bound Move the cursor to the left of the maimum point. Press. CALCULATE :value 2:zero 3:minimum 4:maimum 5:intersect 6:d/d Left Bound? X=-3.4042 Y=5.8646 STEP 3 Choose right bound Move the cursor to the right of the maimum point. Press. STEP 4 Find maimum Put the cursor approimatel on the maimum point. Press. Right Bound? X=-.4893 Y=5.4572 Maimum X=-2.5 Y=7.5 c The maimum value of the function is 5 7.5 and occurs at 522.5. P RACTICE Tell whether the function has a maimum value or a minimum value. Then find the maimum or minimum value and the value of where it occurs.. 5 2 2 6 4 2. f() 5 2 2 3 3 3. 523 2 9 2 4. 5 0.5 2 0.8 2 2 5. h() 5 2 2 2 3 2 6. 52 3 8 2 6 2 5 244 Chapter 4 Quadratic Functions and Factoring
4.2 Graph Quadratic Functions in Verte or Intercept Form Before You graphed quadratic functions in standard form. Now You will graph quadratic functions in verte form or intercept form. Wh? So ou can find the height of a jump, as in E. 5. Ke Vocabular verte form intercept form In Lesson 4., ou learned that the standard form of a quadratic function is 5 a 2 b c where a Þ 0. Another useful form of a quadratic function is the verte form, 5 a( 2 h) 2 k. KEY CONCEPT For Your Notebook Graph of Verte Form 5 a( 2 h) 2 k The graph of 5 a( 2 h) 2 k is the parabola 5 a 2 translated horizontall h units and verticall k units. Characteristics of the graph of 5 a( 2 h) 2 k: The verte is (h, k). The ais of smmetr is 5 h. 5 a 2 The graph opens up if a > 0 and down if a < 0. (0, 0) h (h, k) k 5 a( 2 h) 2 k E XAMPLE Graph a quadratic function in verte form Graph 5 2 4 ( 2)2 5. Solution STEP Identif the constants a 5 2, h 5 22, and 4 k 5 5. Because a < 0, the parabola opens down. verte (22, 5) STEP 2 Plot the verte (h, k) 5 (22, 5) and draw the ais of smmetr 5 22. STEP 3 Evaluate the function for two values of. ais of smmetr 5 22 5 0: 5 2 4 (0 2)2 5 5 4 5 2: 5 2 4 (2 2)2 5 5 (22, 5) STEP 4 Plot the points (0, 4) and (2, ) and their reflections in the ais of smmetr. Draw a parabola through the plotted points. 2 4.2 Graph Quadratic Functions in Verte or Intercept Form 245
E XAMPLE 2 Use a quadratic model in verte form CIVIL ENGINEERING The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadwa and are connected b suspension cables as shown. Each cable can be modeled b the function 5 7000 ( 2 400)2 27 where and are measured in feet. What is the distance d between the two towers? Solution The verte of the parabola is (400, 27). So, a cable s lowest point is 400 feet from the left tower shown above. Because the heights of the two towers are the same, the smmetr of the parabola implies that the verte is also 400 feet from the right tower. So, the distance between the two towers is d 5 2(400) 5 2800 feet. GUIDED PRACTICE for Eamples and 2 Graph the function. Label the verte and ais of smmetr.. 5 ( 2) 2 2 3 2. 5 2( 2 ) 2 5 3. f() 5 2 ( 2 3)2 2 4 4. WHAT IF? Suppose an architect designs a bridge with cables that can be modeled b 5 6500 ( 2 400)2 27 where and are measured in feet. Compare this function s graph to the graph of the function in Eample 2. INTERCEPT FORM If the graph of a quadratic function has at least one -intercept, then the function can be represented in intercept form, 5 a( 2 p)( 2 q). KEY CONCEPT For Your Notebook Graph of Intercept Form 5 a( 2 p)( 2 q) Characteristics of the graph of 5 a( 2 p)( 2 q): The -intercepts are p and q. The ais of smmetr is halfwa between (p, 0) and (q, 0). It has equation 5 p q. 2 The graph opens up if a > 0 and opens down if a < 0. (p, 0) 5 pq 2 5 a( 2 p)( 2 q) (q, 0) 246 Chapter 4 Quadratic Functions and Factoring
E XAMPLE 3 Graph a quadratic function in intercept form Graph 5 2( 3)( 2 ). AVOID ERRORS Remember that the -intercepts for a quadratic function written in the form 5 a( 2 p)( 2 q) are p and q, not 2p and 2q. Solution STEP Identif the -intercepts. Because p 5 23 and q 5, the -intercepts occur at the points (23, 0) and (, 0). STEP 2 Find the coordinates of the verte. 5 p q 5 23 5 2 2 2 (23, 0) 2 (, 0) 5 2(2 3)(2 2 ) 5 28 STEP 3 So, the verte is (2, 28). Draw a parabola through the verte and the points where the -intercepts occur. (2, 28) at classzone.com E XAMPLE 4 Use a quadratic function in intercept form FOOTBALL The path of a placekicked football can be modeled b the function 5 20.026( 2 46) where is the horizontal distance (in ards) and is the corresponding height (in ards). a. How far is the football kicked? b. What is the football s maimum height? Solution a. Rewrite the function as 5 20.026( 2 0)( 2 46). Because p 5 0 and q 5 46, ou know the -intercepts are 0 and 46. So, ou can conclude that the football is kicked a distance of 46 ards. b. To find the football s maimum height, calculate the coordinates of the verte. 5 p q 5 0 46 5 23 2 2 5 20.026(23)(23 2 46) ø 3.8 The maimum height is the -coordinate of the verte, or about 3.8 ards. GUIDED PRACTICE for Eamples 3 and 4 Graph the function. Label the verte, ais of smmetr, and -intercepts. 5. 5 ( 2 3)( 2 7) 6. f() 5 2( 2 4)( ) 7. 5 2( )( 2 5) 8. WHAT IF? In Eample 4, what is the maimum height of the football if the football s path can be modeled b the function 5 20.025( 2 50)? 4.2 Graph Quadratic Functions in Verte or Intercept Form 247
FOIL METHOD You can change quadratic functions from intercept form or verte form to standard form b multipling algebraic epressions. One method for multipling two epressions each containing two terms is FOIL. KEY CONCEPT For Your Notebook FOIL Method Words To multipl two epressions that each contain two terms, add the products of the First terms, the Outer terms, the Inner terms, and the Last terms. Eample F O I L ( 4)( 7) 5 2 7 4 28 5 2 28 E XAMPLE 5 Change from intercept form to standard form REVIEW FOIL For help with using the FOIL method, see p. 985. Write 5 22( 5)( 2 8) in standard form. 5 22( 5)( 2 8) Write original function. 5 22( 2 2 8 5 2 40) Multipl using FOIL. 5 22( 2 2 3 2 40) Combine like terms. 5 22 2 6 80 Distributive propert E XAMPLE 6 Change from verte form to standard form Write f() 5 4( 2 ) 2 9 in standard form. f() 5 4( 2 ) 2 9 Write original function. 5 4( 2 )( 2 ) 9 Rewrite ( 2 ) 2. 5 4( 2 2 2 ) 9 Multipl using FOIL. 5 4( 2 2 2 ) 9 Combine like terms. 5 4 2 2 8 4 9 Distributive propert 5 4 2 2 8 3 Combine like terms. GUIDED PRACTICE for Eamples 5 and 6 Write the quadratic function in standard form. 9. 5 2( 2 2)( 2 7) 0. 5 24( 2 )( 3). f() 5 2( 5)( 4) 2. 5 27( 2 6)( ) 3. 5 23( 5) 2 2 4. g() 5 6( 2 4) 2 2 0 5. f() 5 2( 2) 2 4 6. 5 2( 2 3) 2 9 248 Chapter 4 Quadratic Functions and Factoring
4.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 9, 29, and 53 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 22, 49, 54, and 55. VOCABULARY Cop and complete: A quadratic function in the form 5 a( 2 h) 2 k is in? form. 2. WRITING Eplain how to find a quadratic function s maimum value or minimum value when the function is given in intercept form. EXAMPLE on p. 245 for Es. 3 2 GRAPHING WITH VERTEX FORM Graph the function. Label the verte and ais of smmetr. 3. 5 ( 2 3) 2 4. 5 ( 4) 2 5. f() 5 2( 3) 2 5 6. 5 3( 2 7) 2 2 7. g() 5 24( 2 2) 2 4 8. 5 2( ) 2 2 3 9. f() 5 22( 2 ) 2 2 5 0. 5 2 4 ( 2)2. 5 2 ( 2 3)2 2 2. MULTIPLE CHOICE What is the verte of the graph of the function 5 3( 2) 2 2 5? A (2, 25) B (22, 25) C (25, 2) D (5, 22) EXAMPLE 3 on p. 247 for Es. 3 23 GRAPHING WITH INTERCEPT FORM Graph the function. Label the verte, ais of smmetr, and -intercepts. 3. 5 ( 3)( 2 3) 4. 5 ( )( 2 3) 5. 5 3( 2)( 6) 6. f() 5 2( 2 5)( 2 ) 7. 5 2( 2 4)( 6) 8. g() 5 24( 3)( 7) 9. 5 ( )( 2) 20. f() 5 22( 2 3)( 4) 2. 5 4( 2 7)( 2) 22. MULTIPLE CHOICE What is the verte of the graph of the function 5 2( 2 6)( 4)? A (, 25) B (2, 2) C (26, 4) D (6, 24) 23. ERROR ANALYSIS Describe and correct the error in analzing the graph of the function 5 5( 2 2)( 3). The -intercepts of the graph are 22 and 3. EXAMPLES 5 and 6 on p. 248 for Es. 24 32 WRITING IN STANDARD FORM Write the quadratic function in standard form. 24. 5 ( 4)( 3) 25. 5 ( 2 5)( 3) 26. h() 5 4( )( 2 6) 27. 5 23( 2 2)( 2 4) 28. f() 5 ( 5) 2 2 2 29. 5 ( 2 3) 2 6 30. g() 5 2( 6) 2 0 3. 5 5( 3) 2 2 4 32. f() 5 2( 2 ) 2 4 MINIMUM OR MAXIMUM VALUES Find the minimum value or the maimum value of the function. 33. 5 3( 2 3) 2 2 4 34. g() 5 24( 6) 2 2 2 35. 5 5( 2 25) 2 30 36. f() 5 3( 0)( 2 8) 37. 5 2( 2 36)( 8) 38. 5 22( 2 9) 39. 5 8( 5) 40. 5 2( 2 3)( 2 6) 4. g() 5 25( 9)( 2 4) 4.2 Graph Quadratic Functions in Verte or Intercept Form 249
42. GRAPHING CALCULATOR Consider the function 5 a( 2 h) 2 k where a 5, h 5 3, and k 5 22. Predict the effect of each change in a, h, or k described in parts (a) (c). Use a graphing calculator to check our prediction b graphing the original and revised functions in the same coordinate plane. a. a changes to 23 b. h changes to 2 c. k changes to 2 MAKING A GRAPH Graph the function. Label the verte and ais of smmetr. 43. 5 5( 2 2.25) 2 2 2.75 44. g() 5 28( 3.2) 2 6.4 45. 5 20.25( 2 5.2) 2 8.5 46. 5 2 2 3 2 2 2 2 4 5 47. f() 5 2 3 4 ( 5)( 8) 48. g() 5 5 2 2 4 3 2 2 2 5 2 49. OPEN-ENDED MATH Write two different quadratic functions in intercept form whose graphs have ais of smmetr 5 3. 50. CHALLENGE Write 5 a( 2 h) 2 k and 5 a( 2 p)( 2 q) in standard form. Knowing the verte of the graph of 5 a 2 b c occurs at 5 2 b 2a, show that the verte of the graph of 5 a( 2 h) 2 k occurs at 5 h and that the verte of the graph of 5 a( 2 p)( 2 q) occurs at 5 p q. 2 PROBLEM SOLVING EXAMPLES 2 and 4 on pp. 246 247 for Es. 5 54 5. BIOLOGY The function 5 20.03( 2 4) 2 6 models the jump of a red kangaroo where is the horizontal distance (in feet) and is the corresponding height (in feet). What is the kangaroo s maimum height? How long is the kangaroo s jump? 52. CIVIL ENGINEERING The arch of the Gateshead Millennium Bridge forms a parabola with equation 5 20.06( 2 52.5) 2 45 where is the horizontal distance (in meters) from the arch s left end and is the distance (in meters) from the base of the arch. What is the width of the arch? 53. MULTI-STEP PROBLEM Although a football field appears to be flat, its surface is actuall shaped like a parabola so that rain runs off to both sides. The cross section of a field with snthetic turf can be modeled b 5 20.000234( 2 60) where and are measured in feet. a. What is the field s width? b. What is the maimum height of the field s surface? surface of football field Not drawn to scale 5 WORKED-OUT SOLUTIONS 250 Chapter 4 Quadratic on p. WSFunctions and Factoring 5 STANDARDIZED TEST PRACTICE
54. SHORT RESPONSE A jump on a pogo stick with a conventional spring can be modeled b 5 20.5( 2 6) 2 8, and a jump on a pogo stick with a bow spring can be modeled b 5 2.7( 2 6) 2 42, where and are measured in inches. Compare the maimum heights of the jumps on the two pogo sticks. Which constants in the functions affect the maimum heights of the jumps? Which do not? Vertical position (in.) 40 30 20 0 0 0 bow spring conventional spring 2 4 6 8 0 2 Horizontal position (in.) 55. EXTENDED RESPONSE A kernel of popcorn contains water that epands when the kernel is heated, causing it to pop. The equations below give the popping volume (in cubic centimeters per gram) of popcorn with moisture content (as a percent of the popcorn s weight). Hot-air popping: 5 20.76( 2 5.52)( 2 22.6) Hot-oil popping: 5 20.652( 2 5.35)( 2 2.8) a. Interpret For hot-air popping, what moisture content maimizes popping volume? What is the maimum volume? b. Interpret For hot-oil popping, what moisture content maimizes popping volume? What is the maimum volume? c. Graphing Calculator Graph the functions in the same coordinate plane. What are the domain and range of each function in this situation? Eplain how ou determined the domain and range. 56. CHALLENGE Fling fish use their pectoral fins like airplane wings to glide through the air. Suppose a fling fish reaches a maimum height of 5 feet after fling a horizontal distance of 33 feet. Write a quadratic function 5 a( 2 h) 2 k that models the flight path, assuming the fish leaves the water at (0, 0). Describe how changing the value of a, h, or k affects the flight path. MIXED REVIEW PREVIEW Prepare for Lesson 4.3 in Es. 57 64. Solve the equation. 57. 2 5 5 0 (p. 8) 58. 2 3 5 0 (p. 8) 59. 23 2 4 5 25 2 7 (p. 8) 60. 25(3 4) 5 7 2 (p. 8) 6. 2 9 5 6 (p. 5) 62. 4 9 5 27 (p. 5) 63. 7 2 2 5 (p. 5) 64. 3 2 5 5 7 (p. 5) Use the given matrices to perform the indicated operation, if possible. If not possible, state the reason. A 5 F 2 3 B 5 2 25G, F 2 26 C 5 3 8G, F 2 4 D 5 22 3G, F 3 0 2 E 5 6 4G, F 22 4 3 2 5G 65. 2A B (p. 87) 66. 3(B C) (p. 87) 67. D 2 4E (p. 87) 68. AB (p. 95) 69. A(B 2 C) (p. 95) 70. 4(CD) (p. 95) EXTRA PRACTICE for Lesson 4.2, p. 03 Graph Quadratic ONLINE Functions QUIZ in Verte at classzone.com or Intercept Form 25
4.3 Solve 2 b c 5 0 b Factoring Before You graphed quadratic functions. Now You will solve quadratic equations. Wh? So ou can double the area of a picnic site, as in E. 42. Ke Vocabular monomial binomial trinomial quadratic equation root of an equation zero of a function A monomial is an epression that is either a number, a variable, or the product of a number and one or more variables. A binomial, such as 4, is the sum of two monomials. A trinomial, such as 2 28, is the sum of three monomials. You know how to use FOIL to write ( 4)( 7) as 2 28. You can use factoring to write a trinomial as a product of binomials. To factor 2 b c, find integers m and n such that: 2 b c 5 ( m)( n) 5 2 (m n) mn So, the sum of m and n must equal b and the product of m and n must equal c. E XAMPLE Factor trinomials of the form 2 b c Factor the epression. a. 2 2 9 20 b. 2 3 2 2 Solution a. You want 2 2 9 20 5 ( m)( n) where mn 5 20 and m n 5 29. AVOID ERRORS When factoring 2 b c where c > 0, ou must choose factors m and n such that m and n have the same sign. Factors of 20: m, n, 20 2, 220 2, 0 22, 20 4, 5 24, 25 Sum of factors: m n 2 22 2 22 9 29 c Notice that m 5 24 and n 5 25. So, 2 2 9 20 5 ( 2 4)( 2 5). b. You want 2 3 2 2 5 ( m)( n) where mn 5 22 and m n 5 3. Factors of 22: m, n 2, 2, 22 22, 6 2, 26 23, 4 3, 24 Sum of factors: m n 2 4 24 2 c Notice that there are no factors m and n such that m n 5 3. So, 2 3 2 2 cannot be factored. GUIDED PRACTICE for Eample Factor the epression. If the epression cannot be factored, sa so.. 2 2 3 2 8 2. n 2 2 3n 9 3. r 2 2r 2 63 252 Chapter 4 Quadratic Functions and Factoring
FACTORING SPECIAL PRODUCTS Factoring quadratic epressions often involves trial and error. However, some epressions are eas to factor because the follow special patterns. KEY CONCEPT For Your Notebook Special Factoring Patterns Pattern Name Pattern Eample Difference of Two Squares a 2 2 b 2 5 (a b)(a 2 b) 2 2 4 5 ( 2)( 2 2) Perfect Square Trinomial a 2 2ab b 2 5 (a b) 2 2 6 9 5 ( 3) 2 a 2 2 2ab b 2 5 (a 2 b) 2 2 2 4 4 5 ( 2 2) 2 E XAMPLE 2 Factor with special patterns Factor the epression. a. 2 2 49 5 2 2 7 2 Difference of two squares 5 ( 7)( 2 7) b. d 2 2d 36 5 d 2 2(d)(6) 6 2 Perfect square trinomial 5 (d 6) 2 c. z 2 2 26z 69 5 z 2 2 2(z)(3) 3 2 Perfect square trinomial 5 (z 2 3) 2 GUIDED PRACTICE for Eample 2 Factor the epression. 4. 2 2 9 5. q 2 2 00 6. 2 6 64 7. w 2 2 8w 8 SOLVING QUADRATIC EQUA TIONS You can use factoring to solve certain quadratic equations. A qu adratic equation in one variable can be written in the form a 2 b c 5 0 where a? 0. This is called the standard form of the equation. The solutions of a quadratic equation are called the roots of the equation. If the left side of a 2 b c 5 0 can be factored, th en the equation can be solved using the zero product propert. KEY CONCEPT For Your Notebook Zero Product Propert Words If the product of two epressions is zero, then one or both of the epressions equal zero. Algebra If A and B are epressions and AB 5 0, then A 5 0 or B 5 0. Eample If ( 5)( 2) 5 0, then 5 5 0 or 2 5 0. That is, 5 25 or 5 22. 4.3 Solve 2 b c 5 0 b Factoring 253
E XAMPLE 3 Standardized Test Practice UNDERSTAND ANSWER CHOICES Sometimes a standardized test question ma ask for the solution set of an equation. The answer choices will be given in the format {a, b. What are the roots of the equation 2 2 5 2 36 5 0? A 24, 29 B 4, 29 C 24, 9 D 4, 9 Solution 2 2 5 2 36 5 0 Write original equation. ( 2 9)( 4) 5 0 Factor. 2 9 5 0 or 4 5 0 Zero product propert 5 9 or 5 24 Solve for. c The correct answer is C. A B C D E XAMPLE 4 Use a quadratic equation as a model NATURE PRESERVE A town has a nature preserve with a rectangular field that measures 600 meters b 400 meters. The town wants to double the area of the field b adding land as shown. Find the new dimensions of the field. Solution New area (square meters) 5 New length (meters) p New width (meters) 2(600)(400) 5 (600 ) p (400 ) 480,000 5 240,000 000 2 Multipl using FOIL. 0 5 2 000 2 240,000 Write in standard form. 0 5 ( 2 200)( 200) Factor. 2 200 5 0 or 200 5 0 Zero product propert 5 200 or 5 2200 Solve for. c Reject the negative value, 2200. The field s length and width should each be increased b 200 meters. The new dimensions are 800 meters b 600 meters. GUIDED PRACTICE for Eamples 3 and 4 8. Solve the equation 2 2 2 42 5 0. 9. WHAT IF? In Eample 4, suppose the field initiall measures 000 meters b 300 meters. Find the new dimensions of the field. ZEROS OF A FUNCTION In Lesson 4.2, ou learned that the -intercepts of the graph of 5 a( 2 p)( 2 q) are p and q. Because the function s value is zero when 5 p and when 5 q, the numbers p and q are also called zeros of the function. 254 Chapter 4 Quadratic Functions and Factoring
E XAMPLE 5 Find the zeros of quadratic functions UNDERSTAND REPRESENTATIONS If a real number k is a zero of the function 5 a 2 b c, then k is an -intercept of this function s graph and k is also a root of the equation a 2 b c 5 0. Find the zeros of the function b rewriting the function in intercept form. a. 5 2 2 2 2 b. 5 2 2 36 Solution a. 5 2 2 2 2 Write original function. 5 ( 3)( 2 4) Factor. The zeros of the function are 23 and 4. CHECK Graph 5 2 2 2 2. The graph passes through (23, 0) and (4, 0). Zero X=-3 Y=0 b. 5 2 2 36 Write original function. 5 ( 6)( 6) Factor. The zero of the function is 26. CHECK Graph 5 2 2 36. The graph passes through (26, 0). Zero X=-6 Y=0 GUIDED PRACTICE for Eample 5 Find the zeros of the function b rewriting the function in intercept form. 0. 5 2 5 2 4. 5 2 2 7 2 30 2. f() 5 2 2 0 25 4.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 33, 47, and 67 5 STANDARDIZED TEST PRACTICE Es. 2, 4, 56, 58, 63, and 7 5 MULTIPLE REPRESENTATIONS E. 68. VOCABULARY What is a zero of a function 5 f()? 2. WRITING Eplain the difference between a monomial, a binomial, and a trinomial. Give an eample of each tpe of epression. EXAMPLE on p. 252 for Es. 3 4 FACTORING Factor the epression. If the epression cannot be factored, sa so. 3. 2 6 5 4. 2 2 7 0 5. a 2 2 3a 22 6. r 2 5r 56 7. p 2 2p 4 8. q 2 2 q 28 9. b 2 3b 2 40 0. 2 2 4 2 2. 2 2 7 2 8 2. c 2 2 9c 2 8 3. 2 9 2 36 4. m 2 8m 2 65 4.3 Solve 2 b c 5 0 b Factoring 255
EXAMPLE 2 on p. 253 for Es. 5 23 EXAMPLE 3 on p. 254 for Es. 24 4 FACTORING WITH SPECIAL PATTERNS Factor the epression. 5. 2 2 36 6. b 2 2 8 7. 2 2 24 44 8. t 2 2 6t 64 9. 2 8 6 20. c 2 28c 96 2. n 2 4n 49 22. s 2 2 26s 69 23. z 2 2 2 SOLVING EQUATIONS Solve the equation. 24. 2 2 8 2 5 0 25. 2 2 30 5 0 26. 2 2 2 35 5 0 27. a 2 2 49 5 0 28. b 2 2 6b 9 5 0 29. c 2 5c 4 5 0 30. n 2 2 6n 5 0 3. t 2 0t 25 5 0 32. w 2 2 6w 48 5 0 33. z 2 2 3z 5 54 34. r 2 2r 5 80 35. u 2 5 29u 36. m 2 5 7m 37. 4 2 49 5 2 38. 23 28 5 2 ERROR ANALYSIS Describe and correct the error in solving the equation. 39. 40. 2 2 2 6 5 0 ( 2 2)( 3) 5 0 2 2 5 0 or 3 5 0 5 2 or 5 23 2 7 6 5 4 ( 6)( ) 5 4 6 5 4 or 5 4 5 8 or 5 3 4. MULTIPLE CHOICE What are the roots of the equation 2 2 2 63 5 0? A 7, 29 B 27, 29 C 27, 9 D 7, 9 EXAMPLE 4 on p. 254 for Es. 42 43 EXAMPLE 5 on p. 255 for Es. 44 55 WRITING EQUATIONS Write an equation that ou can solve to find the value of. 42. A rectangular picnic site measures 24 feet b 0 feet. You want to double the site s area b adding the same distance to the length and the width. 43. A rectangular performing platform in a park measures 0 feet b 2 feet. You want to triple the platform s area b adding the same distance to the length and the width. FINDING ZEROS Find the zeros of the function b rewriting the function in intercept form. 44. 5 2 6 8 45. 5 2 2 8 6 46. 5 2 2 4 2 32 47. 5 2 7 2 30 48. f() 5 2 49. g() 5 2 2 8 50. 5 2 2 64 5. 5 2 2 25 52. f() 5 2 2 2 2 45 53. g() 5 2 9 84 54. 5 2 22 2 55. 5 2 2 56. MULTIPLE CHOICE What are the zeros of f() 5 2 6 2 55? A 2, 25 B 2, 5 C 25, D 5, 57. REASONING Write a quadratic equation of the form 2 b c 5 0 that has roots 8 and. 58. SHORT RESPONSE For what integers b can the epression 2 b 7 be factored? Eplain. 5 WORKED-OUT SOLUTIONS 256 Chapter 4 Quadratic on p. WSFunctions and Factoring 5 STANDARDIZED TEST PRACTICE
GEOMETRY Find the value of. 59. Area of rectangle 5 36 60. Area of rectangle 5 84 2 5 7 6. Area of triangle 5 42 62. Area of trapezoid 5 32 6 3 2 8 2 63. OPEN-ENDED MATH Write a quadratic function with zeros that are equidistant from 0 on a number line. 64. CHALLENGE Is there a formula for factoring the sum of two squares? You will investigate this question in parts (a) and (b). a. Consider the sum of two squares 2 6. If this sum can be factored, then there are integers m and n such that 2 6 5 ( m)( n). Write two equations that m and n must satisf. b. Show that there are no integers m and n that satisf both equations ou wrote in part (a). What can ou conclude? PROBLEM SOLVING EXAMPLE 4 on p. 254 for Es. 65 67 65. SKATE PARK A cit s skate park is a rectangle 00 feet long b 50 feet wide. The cit wants to triple the area of the skate park b adding the same distance to the length and the width. Write and solve an equation to find the value of. What are the new dimensions of the skate park? 66. ZOO A rectangular enclosure at a zoo is 35 feet long b 8 feet wide. The zoo wants to double the area of the enclosure b adding the same distance to the length and the width. Write and solve an equation to find the value of. What are the new dimensions of the enclosure? 67. MULTI-STEP PROBLEM A museum has a café with a rectangular patio. The museum wants to add 464 square feet to the area of the patio b epanding the eisting patio as shown. a. Find the area of the eisting patio. b. Write a verbal model and an equation that ou can use to find the value of. c. Solve our equation. B what distance should the length and the width of the patio be epanded? 4.3 Solve 2 b c 5 0 b Factoring 257
68. MULTIPLE REPRESENTATIONS Use the diagram shown. a. Writing an Epression Write a quadratic trinomial that represents the area of the diagram. b. Describing a Model Factor the epression from part (a). Eplain how the diagram models the factorization. c. Drawing a Diagram Draw a diagram that models the factorization 2 8 5 5 ( 5)( 3). 69. SCHOOL FAIR At last ear s school fair, an 8 foot b 5 foot rectangular section of land was roped off for a dunking booth. The length and width of the section will each be increased b feet for this ear s fair in order to triple the original area. Write and solve an equation to find the value of. What is the length of rope needed to enclose the new section? 70. RECREATION CENTER A rectangular deck for a recreation center is 2 feet long b 20 feet wide. Its area is to be halved b subtracting the same distance from the length and the width. Write and solve an equation to find the value of. What are the deck s new dimensions? 7. SHORT RESPONSE A square garden has sides that are 0 feet long. A gardener wants to double the area of the garden b adding the same distance to the length and the width. Write an equation that must satisf. Can ou solve the equation ou wrote b factoring? Eplain wh or wh not. 72. CHALLENGE A grocer store wants to double the area of its parking lot b epanding the eisting lot as shown. B what distance should the lot be epanded? 75 ft 65 ft Old lot Epanded part of lot 300 ft Grocer store 75 ft MIXED REVIEW PREVIEW Prepare for Lesson 4.4 in Es. 73 8. Solve the equation. 73. 2 2 5 0 (p. 8) 74. 3 4 5 0 (p. 8) 75. 28 7 5 0 (p. 8) 76. 6 5 5 0 (p. 8) 77. 4 2 5 5 0 (p. 8) 78. 3 5 0 (p. 8) 79. 2 6 5 7 (p. 5) 80. 2 2 5 5 0 (p. 5) 8. 4 2 3 5 8 (p. 5) Graph the function. 82. 5 3 2 (p. 89) 83. f() 5 2 2 (p. 89) 4 84. 5 2 4 2 4 (p. 23) 85. 5 2 2 (p. 23) 86. 5 22 2 8 7 (p. 236) 87. g() 5 22( ) 2 2 4 (p. 245) 88. f() 5 ( 4)( 2 2) (p. 245) 89. 5 2( 2 3)( 2 7) (p. 245) 90. PARK DESIGN A cit plans to place a plaground in a triangular region of a park. The vertices of the triangle are (0, 0), (4, 3), and (6, 25) where the coordinates are given in feet. Find the area of the triangular region. (p. 203) 258 Chapter 4 EXTRA Quadratic PRACTICE Functions and for Factoring Lesson 4.3, p. 03 ONLINE QUIZ at classzone.com
4.4 Solve a2 b c 5 0 b Factoring Before You used factoring to solve equations of the form 2 b c 5 0. Now You will use factoring to solve equations of the form a 2 b c 5 0. Wh? So ou can maimize a shop s revenue, as in E. 64. Ke Vocabular monomial, p. 252 To factor a 2 b c when a?, find integers k, l, m, and n such that: a 2 b c 5 (k m)(l n) 5 kl 2 (kn lm) mn So, k and l must be factors of a, and m and n must be factors of c. E XAMPLE Factor a 2 b c where c > 0 Factor 5 2 2 7 6. FACTOR EXPRESSIONS When factoring a 2 b c where a > 0, it is customar to choose factors k m and l n such that k and l are positive. Solution You want 5 2 2 7 6 5 (k m)(l n) where k and l are factors of 5 and m and n are factors of 6. You can assume that k and l are positive and k l. Because mn > 0, m and n have the same sign. So, m and n must both be negative because the coefficient of, 27, is negative. k, l 5, 5, 5, 5, m, n 26, 2 2, 26 23, 22 22, 23 (k m)(l n) (5 2 6)( 2 ) (5 2 )( 2 6) (5 2 3)( 2 2) (5 2 2)( 2 3) a 2 b c 5 2 2 6 5 2 2 3 6 5 2 2 3 6 5 2 2 7 6 c The correct factorization is 5 2 2 7 6 5 (5 2 2)( 2 3). E XAMPLE 2 Factor a 2 b c where c < 0 Factor 3 2 20 2 7. Solution You want 3 2 20 2 7 5 (k m)(l n) where k and l are factors of 3 and m and n are factors of 27. Because mn < 0, m and n have opposite signs. k, l 3, 3, 3, 3, m, n 7, 2 2, 7 27,, 27 (k m)(l n) (3 7)( 2 ) (3 2 )( 7) (3 2 7)( ) (3 )( 2 7) a 2 b c 3 2 4 2 7 3 2 20 2 7 3 2 2 4 2 7 3 2 2 20 2 7 c The correct factorization is 3 2 20 2 7 5 (3 2 )( 7). 4.4 Solve a 2 b c 5 0 b Factoring 259
GUIDED PRACTICE for Eamples and 2 Factor the epression. If the epression cannot be factored, sa so.. 7 2 2 20 2 3 2. 5z 2 6z 3 3. 2w 2 w 3 4. 3 2 5 2 2 5. 4u 2 2u 5 6. 4 2 2 9 2 FACTORING SPECIAL PRODUCTS If the values of a and c in a 2 b c are perfect squares, check to see whether ou can use one of the special factoring patterns from Lesson 4.3 to factor the epression. E XAMPLE 3 Factor with special patterns Factor the epression. a. 9 2 2 64 5 (3) 2 2 8 2 Difference of two squares 5 (3 8)(3 2 8) b. 4 2 20 25 5 (2) 2 2(2)(5) 5 2 Perfect square trinomial 5 (2 5) 2 c. 36w 2 2 2w 5 (6w) 2 2 2(6w)() 2 Perfect square trinomial 5 (6w 2 ) 2 GUIDED PRACTICE for Eample 3 Factor the epression. 7. 6 2 2 8. 9 2 2 4 9. 4r 2 2 28r 49 0. 25s 2 2 80s 64. 49z 2 42z 9 2. 36n 2 2 9 FACTORING OUT MONOMIALS When factoring an epression, first check to see whether the terms have a common monomial factor. E XAMPLE 4 Factor out monomials first AVOID ERRORS Be sure to factor out the common monomial from all of the terms of the epression, not just the first term. Factor the epression. a. 5 2 2 45 5 5( 2 2 9) b. 6q 2 2 4q 8 5 2(3q 2 2 7q 4) 5 5( 3)( 2 3) 5 2(3q 2 4)(q 2 ) c. 25z 2 20z 5 25z(z 2 4) d. 2p 2 2 2p 3 5 3(4p 2 2 7p ) GUIDED PRACTICE for Eample 4 Factor the epression. 3. 3s 2 2 24 4. 8t 2 38t 2 0 5. 6 2 24 5 6. 2 2 2 28 2 24 7. 26n 2 2n 8. 6z 2 33z 36 260 Chapter 4 Quadratic Functions and Factoring
SOLVING QUADRATIC EQUATIONS As ou saw in Lesson 4.3, if the left side of the quadratic equation a 2 b c 5 0 can be factored, then the equation can be solved using the zero product propert. E XAMPLE 5 Solve quadratic equations Solve (a) 3 2 0 2 8 5 0 and (b) 5p 2 2 6p 5 5 4p 2 5. a. 3 2 0 2 8 5 0 Write original equation. (3 2 2)( 4) 5 0 Factor. 3 2 2 5 0 or 4 5 0 Zero product propert 5 2 3 or 5 24 Solve for. b. 5p 2 2 6p 5 5 4p 2 5 Write original equation. 5p 2 2 20p 20 5 0 Write in standard form. INTERPRET EQUATIONS If the square of an epression is zero, then the epression itself must be zero. p 2 2 4p 4 5 0 Divide each side b 5. (p 2 2) 2 5 0 Factor. p 2 2 5 0 Zero product propert p 5 2 Solve for p. E XAMPLE 6 Use a quadratic equation as a model QUILTS You have made a rectangular quilt that is 5 feet b 4 feet. You want to use the remaining 0 square feet of fabric to add a decorative border of uniform width to the quilt. What should the width of the quilt s border be? Solution Write a verbal model. Then write an equation. Area of border (square feet) 5 Area of quilt and border (square feet) 2 Area of quilt (square feet) 0 5 (5 2)(4 2) 2 (5)(4) 0 5 20 8 4 2 2 20 Multipl using FOIL. 0 5 4 2 8 2 0 Write in standard form. 0 5 2 2 9 2 5 Divide each side b 2. 0 5 (2 2 )( 5) Factor. 2 2 5 0 or 5 5 0 Zero product propert 5 2 or 5 25 Solve for. c Reject the negative value, 25. The border s width should be ft, or 6 in. 2 4.4 Solve a 2 b c 5 0 b Factoring 26
FACTORING AND ZEROS To find the maimum or minimum value of a quadratic function, ou can first use factoring to write the function in intercept form 5 a( 2 p)( 2 q). Because the function s verte lies on the ais of smmetr 5 p q, the maimum or minimum occurs at the average of the zeros p and q. 2 E XAMPLE 7 Solve a multi-step problem MAGAZINES A monthl teen magazine has 28,000 subscribers when it charges $0 per annual subscription. For each $ increase in price, the magazine loses about 2000 subscribers. How much should the magazine charge to maimize annual revenue? What is the maimum annual revenue? Solution STEP STEP 2 Define the variables. Let represent the price increase and R() represent the annual revenue. Write a verbal model. Then write and simplif a quadratic function. Annual revenue (dollars) 5 Number of subscribers (people) p Subscription price (dollars/person) STEP 3 STEP 4 R() 5 (28,000 2 2000) p (0 ) R() 5 (22000 28,000)( 0) R() 5 22000( 2 4)( 0) Identif the zeros and find their average. Find how much each subscription should cost to maimize annual revenue. The zeros of the revenue function are 4 and 20. The average of the 4 (20) zeros is 5 2. To maimize revenue, each subscription 2 should cost $0 $2 5 $2. Find the maimum annual revenue. R(2) 5 22000(2 2 4)(2 0) 5 $288,000 c The magazine should charge $2 per subscription to maimize annual revenue. The maimum annual revenue is $288,000. GUIDED PRACTICE for Eamples 5, 6, and 7 Solve the equation. 9. 6 2 2 3 2 63 5 0 20. 2 2 7 2 5 8 2. 7 2 70 75 5 0 22. WHAT IF? In Eample 7, suppose the magazine initiall charges $ per annual subscription. How much should the magazine charge to maimize annual revenue? What is the maimum annual revenue? 262 Chapter 4 Quadratic Functions and Factoring
4.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 27, 39, and 63 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 64, 65, and 67. VOCABULARY What is the greatest common monomial factor of the terms of the epression 2 2 8 20? 2. WRITING Eplain how the values of a and c in a 2 b c help ou determine whether ou can use a perfect square trinomial factoring pattern. EXAMPLES and 2 on p. 259 for Es. 3 2 FACTORING Factor the epression. If the epression cannot be factored, sa so. 3. 2 2 5 3 4. 3n 2 7n 4 5. 4r 2 5r 6. 6p 2 5p 7. z 2 2z 2 9 8. 5 2 2 2 2 8 9. 4 2 2 5 2 4 0. 4m 2 m 2 3. 9d 2 2 3d 2 0 2. MULTIPLE CHOICE Which factorization of 5 2 4 2 3 is correct? A (5 2 3)( ) B (5 )( 2 3) C 5( 2 )( 3) D (5 2 )( 3) EXAMPLE 3 on p. 260 for Es. 3 2 FACTORING WITH SPECIAL PATTERNS Factor the epression. 3. 9 2 2 4. 4r 2 2 25 5. 49n 2 2 6 6. 6s 2 8s 7. 49 2 70 25 8. 64w 2 44w 8 9. 9p 2 2 2p 4 20. 25t 2 2 30t 9 2. 36 2 2 84 49 EXAMPLE 4 on p. 260 for Es. 22 3 FACTORING MONOMIALS FIRST Factor the epression. 22. 2 2 2 4 2 40 23. 8z 2 36z 6 24. 32v 2 2 2 25. 6u 2 2 24u 26. 2m 2 2 36m 27 27. 20 2 24 24 28. 2 2 2 77 2 28 29. 236n 2 48n 2 5 30. 28 2 28 2 60 3. ERROR ANALYSIS Describe and correct the error in factoring the epression. 4 2 2 36 5 4( 2 2 36) 5 4( 6)( 2 6) EXAMPLE 5 on p. 26 for Es. 32 40 SOLVING EQUATIONS Solve the equation. 32. 6 2 2 5 0 33. q 2 2 44 5 0 34. 4s 2 2 2s 5 0 35. 45n 2 0n 5 0 36. 4 2 2 20 25 5 0 37. 4p 2 2p 9 5 0 38. 5 2 7 2 2 5 0 39. 6r 2 2 7r 2 5 5 0 40. 36z 2 96z 5 5 0 EXAMPLE 7 on p. 262 for Es. 4 49 FINDING ZEROS Find the zeros of the function b rewriting the function in intercept form. 4. 5 4 2 2 9 2 5 42. g() 5 3 2 2 8 5 43. 5 5 2 2 27 2 8 44. f() 5 3 2 2 3 45. 5 2 2 9 2 6 46. 5 6 2 2 2 2 5 47. 5 5 2 2 5 2 20 48. 5 8 2 2 6 2 4 49. g() 5 2 2 5 2 7 4.4 Solve a 2 b c 5 0 b Factoring 263