Quadratic Functions and Factoring

Transcription

1 Quadratic Functions and Factoring Lesson. CC.9-2.F.IF.7a*.2 CC.9-2.F.IF.7a*.3 CC.9-2.A.SSE.3a*.4 CC.9-2.A.SSE.3a*.5 CC.9-2.A.REI.4b.6 CC.9-2.N.CN.2.7 CC.9-2.A.REI.4a.8 CC.9-2.N.CN.7.9 CC.9-2.A.REI.4b. Graph Quadratic Functions in Standard Form.2 Graph Quadratic Functions in Verte or Intercept Form.3 Solve 2 b c 5 0 b Factoring.4 Solve a 2 b c 5 0 b Factoring.5 Solve Quadratic Equations b Finding Square Roots.6 Perform Operations with Comple Numbers.7 Complete the Square.8 Use the Quadratic Formula and the Discriminant.9 Graph and Solve Quadratic Inequalities Before Previousl, ou learned the following skills, which ou ll use in this chapter: evaluating epressions, simplifing epressions, and solving equations. Prerequisite Skills VOCABULARY CHECK Cop and complete the statement.. The -intercept of the line shown is?. 2. The -intercept of the line shown is?. (0, 2) (3, 0) SKILLS CHECK Evaluate the epression (25) Simplif. 7. ( 3)( 2) 8. ( 2 4)( 6) 9. (3h 7)(h 9) 0. (4n 2 0)(3n 2 ) Solve the equation ( 2 3) ( 2 5) 5 4(2 2 ) Gett Images

2 Now In this chapter, ou will appl the big ideas listed below and reviewed in the Chapter Summar. 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 parabola verte form intercept form quadratic equation root of an equation zero of a function square root comple number imaginar number completing the square quadratic formula discriminant 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 helps ou answer a question from this chapter: How does changing the ball speed and hitting angle affect the maimum height of a baseball? The function is now in verte form: = 6(t t 3) Remember that the verte of the parabola is at (h, k) and that the maimum height of the baseball in flight is k. What is the maimum height of the baseball in feet? Maimum height = feet Start A quadratic function models the height of a baseball in flight. Check Answer Rewrite the function in verte form to find the maimum height of the ball. Algebra at m.hrw.com

3 . 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 CC.9-2.F.IF.7 Graph functions epressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases.* a. Graph linear and quadratic functions and show intercepts, maima, and minima. 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 5 2 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). 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 Compare the graph with the graph of 5 2. Solution STEP Make a table of values for SKETCH A GRAPH Choose values of on both sides of the ais of smmetr 5 0. STEP 2 STEP Plot the points from the table. Draw a smooth curve through the points Chapter Quadratic Functions and Factoring STEP 4 Compare the graphs of and 5 2. Both open up and have the same verte and ais of smmetr. The graph of is narrower than the graph of 5 2. C B Knight/Gett Images

4 E XAMPLE 2 Graph a function of the form 5 a 2 c Graph 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 STEP 2 STEP Plot the points from the table. Draw a smooth curve through the points. STEP 4 Compare the graphs of and 5 2. Both graphs have the same ais of smmetr. However, the graph of opens down and is wider than the graph of 5 2. Also, its verte is 3 units higher. 5 2 GUIDED PRACTICE for Eamples and 2 Graph the function. Compare the graph with the graph of f() 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 (0, c) 5 2 b 2a 5 2 b 2a (0, c) 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.. Graph Quadratic Functions in Standard Form 3

5 E XAMPLE 3 Graph a function of the form 5 a 2 b c Graph 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) 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() 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 m.hrw.com GUIDED PRACTICE for Eample 3 Graph the function. Label the verte and ais of smmetr f() 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 4 Chapter Quadratic Functions and Factoring

6 E XAMPLE 4 Find the minimum or maimum value Tell whether the function 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) c The minimum value is You can check the answer on a graphing calculator. 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 < 0, so the revenue function has a maimum value. R() 5 (35 2 ) p (380 20) R() 5 3, R() ,300 STEP 3 Find the coordinates (, R()) of the verte. 5 2 b 2a Find -coordinate. 2(220) R(8) 5 220(8) 2 320(8) 3, ,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 Rick Friedman/Corbis 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?. Graph Quadratic Functions in Standard Form 5

7 . EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 5, 37, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 39, 40, 43, 53, 58, and 60 5 MULTIPLE REPRESENTATIONS E. 59 SKILL PRACTICE. 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 for Es. 3 2 USING A TABLE Cop and complete the table of values for the function ????? ????? ????? ????? MAKING A GRAPH Graph the function. Compare the graph with the graph of f() g() EXAMPLE 2 for Es f() g() f() g() ERROR ANALYSIS Describe and correct the error in analzing the graph of The -coordinate of the verte is: 5 b 2a (4) The -intercept of the graph is the value of c, which is 7. EXAMPLE 3 for Es MAKING A GRAPH Graph the function. Label the verte and ais of smmetr g() f() g() f() Chapter Quadratic Functions and Factoring

8 EXAMPLE 4 for Es MINIMUMS OR MAXIMUMS Tell whether the function has a minimum value or a maimum value. Then find the minimum or maimum value f() g() f() MULTIPLE CHOICE What is the effect on the graph of the function when it is changed to ? 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 B 5 2 C D 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 The path of a shot put released at an angle of 358 can be modeled b 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 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() g() f() g() 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.. Graph Quadratic Functions in Standard Form 7

9 PROBLEM SOLVING EXAMPLE 5 for Es 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 ( ) 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 where and are measured in feet. What is the height h above the road of a cable at its lowest point? 500 ft 220 ft h 58. SHORT RESPONSE A woodland jumping mouse hops along a parabolic path given b 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. 8 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

10 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? w w See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 9

11 Graphing Calculator ACTIVITY Find Maimum and Minimum Values Use after Graph Quadratic Functions in Standard Form Use appropriate tools strategicall. m.hrw.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 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. 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. c The maimum value of the function is and occurs at 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 f() h() Chapter Quadratic Functions and Factoring

12 .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 Previousl, 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 ANALYZE GRAPHS The graph of a quadratic function 5 a( 2 h) 2 k where a > 0 is decreasing on the interval (2`, h) and increasing on the interval (h, `). 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 (0, 0) h (h, k) k 5 a( 2 h) 2 k The graph opens up if a > 0 and down if a < 0. CC.9-2.F.IF.7 Graph functions epressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases.* a. Graph linear and quadratic functions and show intercepts, maima, and minima. E XAMPLE Graph ( 2)2 5. Solution Graph a quadratic function in verte form 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 STEP 3 Evaluate the function for two values of. ais of smmetr 5 22 Roalt Free/Corbis STEP 4 5 0: (0 2) : (2 2)2 5 5 Plot the points (0, 4) and (2, ) and their reflections in the ais of smmetr. Draw a parabola through the plotted points. (22, 5).2 Graph Quadratic Functions in Verte or Intercept Form 2

13 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 307 ft 307 ft ( 2 400)2 27 where and are measured in feet. What is the distance d between the two towers? Solution d Not drawn to scale 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) feet. GUIDED PRACTICE for Eamples and 2 Graph the function. Label the verte and ais of smmetr.. 5 ( 2) ( 2 ) f() 5 2 ( 2 3) WHAT IF? Suppose an architect designs a bridge with cables that can be modeled b ( 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) 2 Chapter Quadratic Functions and Factoring

14 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 (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. at m.hrw.com (2, 28) E XAMPLE 4 Use a quadratic function in intercept form FOOTBALL The path of a placekicked football can be modeled b the function ( 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 ( 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 (23)( ) 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 ( 2 3)( 2 7) 6. f() 5 2( 2 4)( ) ( )( 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 ( 2 50)?.2 Graph Quadratic Functions in Verte or Intercept Form 3

15 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) E XAMPLE 5 Change from intercept form to standard form Write 5 22( 5)( 2 8) in standard form. 5 22( 5)( 2 8) Write original function. 5 22( ) Multipl using FOIL. 5 22( ) Combine like terms 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 ) ( ) 9 Multipl using FOIL. 5 4( ) 9 Combine like terms Distributive propert Combine like terms. GUIDED PRACTICE for Eamples 5 and 6 Write the quadratic function in standard form ( 2 2)( 2 7) ( 2 )( 3). f() 5 2( 5)( 4) ( 2 6)( ) ( 5) g() 5 6( 2 4) f() 5 2( 2) ( 2 3) Chapter Quadratic Functions and Factoring

16 .2 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 9, 29, and 53 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 22, 54, and 55 SKILL PRACTICE. 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 for Es. 3 2 GRAPHING WITH VERTEX FORM Graph the function. Label the verte and ais of smmetr ( 2 3) ( 4) 2 5. f() 5 2( 3) ( 2 7) g() 5 24( 2 2) ( ) f() 5 22( 2 ) ( 2) ( 2 3) 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 for Es GRAPHING WITH INTERCEPT FORM Graph the function. Label the verte, ais of smmetr, and -intercepts ( 3)( 2 3) 4. 5 ( )( 2 3) ( 2)( 6) 6. f() 5 2( 2 5)( 2 ) ( 2 4)( 6) 8. g() 5 24( 3)( 7) 9. 5 ( )( 2) 20. f() 5 22( 2 3)( 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 for Es WRITING IN STANDARD FORM Write the quadratic function in standard form ( 4)( 3) ( 2 5)( 3) 26. h() 5 4( )( 2 6) ( 2 2)( 2 4) 28. f() 5 ( 5) ( 2 3) g() 5 2( 6) ( 3) f() 5 2( 2 ) 2 4 MINIMUM OR MAXIMUM VALUES Find the minimum value or the maimum value of the function ( 2 3) g() 5 24( 6) ( 2 25) f() 5 3( 0)( 2 8) ( 2 36)( 8) ( 2 9) ( 5) ( 2 3)( 2 6) 4. g() 5 25( 9)( 2 4).2 Graph Quadratic Functions in Verte or Intercept Form 5

17 42. GRAPHING CALCULATOR Consider the function 5 a( 2 h) 2 k where a 5, h 5 3, and k 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 ( ) g() 5 28( 3.2) ( 2 5.2) f() ( 5)( 8) 48. g() ANALYZING A GRAPH On what interval(s) are the values of f() 5 22( )( 2 5) positive? negative? 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 for Es BIOLOGY The function ( 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 ( ) 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 ( 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 6 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE

18 54. SHORT RESPONSE A jump on a pogo stick with a conventional spring can be modeled b ( 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.) bow spring conventional spring 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: ( )( ) Hot-oil popping: ( )( 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. David Hall/Nature Picture Librar See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 7

19 .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 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. CC.9-2.A.SSE.3a Factor a quadratic epression to reveal the zeros of the function it defines.* E XAMPLE Factor the epression. Factor trinomials of the form 2 b c a b Solution a. You want ( m)( n) where mn 5 20 and m n 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 c Notice that m 5 24 and n So, ( 2 4)( 2 5). b. You want ( 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 c Notice that there are no factors m and n such that m n 5 3. So, cannot be factored. GUIDED PRACTICE for Eample Factor the epression. If the epression cannot be factored, sa so n 2 2 3n 9 3. r 2 2r Chapter Quadratic Functions and Factoring Kate M. Deioma/PhotoEdit, Inc.

20 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 2) Perfect Square Trinomial a 2 2ab b 2 5 (a b) ( 3) 2 a 2 2 2ab b 2 5 (a 2 b) ( 2 2) 2 E XAMPLE 2 Factor with special patterns Factor the epression. a 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 z 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 q w 2 2 8w 8 SOLVING QUADRATIC EQUATIONS You can use factoring to solve certain quadratic equations. A quadratic 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, then 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 or That is, 5 25 or Solve 2 b c 5 0 b Factoring 9

21 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 ? A 24, 29 B 4, 29 C 24, 9 D 4, 9 Solution Write original equation. ( 2 9)( 4) 5 0 Factor or 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, , Multipl using FOIL ,000 Write in standard form. 0 5 ( 2 200)( 200) Factor or Zero product propert or Solve for. c Reject the negative value, 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 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 You know 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. 20 Chapter Quadratic Functions and Factoring

22 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 b Solution a Write original function. 5 ( 3)( 2 4) Factor. The zeros of the function are 23 and 4. CHECK Graph The graph passes through (23, 0) and (4, 0). b Write original function. 5 ( 6)( 6) Factor. The zero of the function is 26. CHECK Graph The graph passes through (26, 0). GUIDED PRACTICE for Eample 5 Find the zeros of the function b rewriting the function in intercept form f() EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 33, 47, and 67 5 STANDARDIZED TEST PRACTICE Es. 2, 4, 56, 58, 63, and 7 5 MULTIPLE REPRESENTATIONS E. 68 SKILL PRACTICE. 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 for Es. 3 4 FACTORING Factor the epression. If the epression cannot be factored, sa so a 2 2 3a r 2 5r p 2 2p 4 8. q 2 2 q b 2 3b c 2 2 9c m 2 8m Solve 2 b c 5 0 b Factoring 2

23 EXAMPLE 2 for Es EXAMPLE 3 for Es FACTORING WITH SPECIAL PATTERNS Factor the epression b t 2 2 6t c 2 28c n 2 4n s s z SOLVING EQUATIONS Solve the equation a b 2 2 6b c 2 5c n 2 2 6n t 2 0t w 2 2 6w z 2 2 3z r 2 2r u u 36. m 2 5 7m ERROR ANALYSIS Describe and correct the error in solving the equation ( 2 2)( 3) or or ( 6)( ) or or MULTIPLE CHOICE What are the roots of the equation ? A 7, 29 B 27, 29 C 27, 9 D 7, 9 EXAMPLE 4 for Es EXAMPLE 5 for Es 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 f() g() f() g() MULTIPLE CHOICE What are the zeros of f() ? 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 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE

24 GEOMETRY Find the value of. 59. Area of rectangle Area of rectangle Area of triangle Area of trapezoid 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 ( 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 for Es 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? Imagesource/Photolibrar 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? Eisting patio 30 ft 20 ft.3 Solve 2 b c 5 0 b Factoring 23

25 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 ( 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 300 ft Grocer store Old lot Epanded part of lot 75 ft 24 See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com

26 .4 Solve a 2 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 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. CC.9-2.A.SSE.3a Factor a quadratic epression to reveal the zeros of the function it defines.* 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. E XAMPLE Factor a 2 b c where c > 0 Factor Solution You want (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 c The correct factorization is (5 2 2)( 2 3). E XAMPLE 2 Factor a 2 b c where c < 0 Factor Solution You want (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) Greg Huglin/SuperStock a 2 b c c The correct factorization is (3 2 )( 7)..4 Solve a 2 b c 5 0 b Factoring 25

27 GUIDED PRACTICE for Eamples and 2 Factor the epression. If the epression cannot be factored, sa so z 2 6z w 2 w u 2 2u 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 (3) Difference of two squares 5 (3 8)(3 2 8) b (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 r r s s z 2 42z n 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 ( 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 t 2 38t n 2 2n 8. 6z 2 33z Chapter Quadratic Functions and Factoring

28 SOLVING QUADRATIC EQUATIONS As ou have seen, 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) and (b) 5p 2 2 6p 5 5 4p 2 5. a Write original equation. (3 2 2)( 4) 5 0 Factor or Zero product propert or 5 24 Solve for. b. 5p 2 2 6p 5 5 4p 2 5 Write original equation. 5p p 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 Divide each side b 5. (p 2 2) Factor. p 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? 4 2 Solution 5 2 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) Multipl using FOIL Write in standard form Divide each side b (2 2 )( 5) Factor or Zero product propert Jack Hollingsworth/Corbis 5 or 5 25 Solve for. 2 c Reject the negative value, 25. The border s width should be ft, or 6 in. 2.4 Solve a 2 b c 5 0 b Factoring 27

29 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, ) p (0 ) R() 5 ( ,000)( 0) R() ( 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) (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 Chapter Quadratic Functions and Factoring 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? Photodisc/Gett Images

30 .4 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 27, 39, and 63 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 64, 65, and 67 SKILL PRACTICE. VOCABULARY What is the greatest common monomial factor of the terms of the epression ? 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 for Es. 3 2 FACTORING Factor the epression. If the epression cannot be factored, sa so n 2 7n r 2 5r 6. 6p 2 5p 7. z 2 2z m 2 m d 2 2 3d MULTIPLE CHOICE Which factorization of is correct? A (5 2 3)( ) B (5 )( 2 3) C 5( 2 )( 3) D (5 2 )( 3) EXAMPLE 3 for Es. 3 2 FACTORING WITH SPECIAL PATTERNS Factor the epression r n s 2 8s w 2 44w p 2 2 2p t t EXAMPLE 4 for Es FACTORING MONOMIALS FIRST Factor the epression z 2 36z v u u 26. 2m m n 2 48n ERROR ANALYSIS Describe and correct the error in factoring the epression ( ) 5 4( 6)( 2 6) EXAMPLE 5 for Es SOLVING EQUATIONS Solve the equation q s 2 2 2s n 2 0n p 2 2p r 2 2 7r z 2 96z EXAMPLE 7 for Es FINDING ZEROS Find the zeros of the function b rewriting the function in intercept form g() f() g() Solve a 2 b c 5 0 b Factoring 29

31 GEOMETRY Find the value of. 50. Area of square Area of rectangle Area of triangle SOLVING EQUATIONS Solve the equation ( 8) CHALLENGE Factor the epression PROBLEM SOLVING EXAMPLE 6 for Es ARTS AND CRAFTS You have a rectangular stained glass window that measures 2 feet b foot. You have 4 square feet of glass with which to make a border of uniform width around the window. What should the width of the border be? 63. URBAN PLANNING You have just planted a rectangular flower bed of red roses in a cit park. You want to plant a border of ellow roses around the flower bed as shown. Because ou bought the same number of red and ellow roses, the areas of the border and flower bed will be equal. What should the width of the border of ellow roses be? 2 ft 8 ft EXAMPLE 7 for Es MULTIPLE CHOICE A surfboard shop sells 45 surfboards per month when it charges $500 per surfboard. For each $20 decrease in price, the store sells 5 more surfboards per month. How much should the shop charge per surfboard in order to maimize monthl revenue? A $340 B $492 C $508 D $ SHORT RESPONSE A restaurant sells about 330 sandwiches each da at a price of $6 each. For each $.25 decrease in price, 5 more sandwiches are sold per da. How much should the restaurant charge to maimize dail revenue? Eplain each step of our solution. What is the maimum dail revenue? PAINTINGS You place a mat around a 25 inch b 2 inch painting as shown. The mat is twice as wide at the left and right of the painting as it is at the top and bottom of the painting. The area of the mat is 74 square inches. How wide is the mat at the left and right of the painting? at the top and bottom of the painting? 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE 2 25 in. 2 2 in. Seth Thompson/Gett Images

32 67. EXTENDED RESPONSE A U.S. Postal Service guideline states that for a rectangular package like the one shown, the sum of the length and the girth cannot eceed 08 inches. Suppose that for one such package, the length is 36 inches and the girth is as large as possible. a. What is the girth of the package? b. Write an epression for the package s width w in terms of h. Write an equation giving the package s volume V in terms of h. c. What height and width maimize the volume of the package? What is the maimum volume? Eplain how ou found it. 68. CHALLENGE Recall from geometr the theorem about the products of the lengths of segments of two chords that intersect in the interior of a circle. Use this theorem to find the value of in the diagram QUIZ Graph the function. Label the verte and ais of smmetr f() Write the quadratic function in standard form ( 2 4)( 2 8) 5. g() 5 22( 3)( 2 7) ( 6) Solve the equation n 2 2 n z 2 2 3z s 2 2 4s a a DVD PLAYERS A store sells about 50 of a new model of DVD plaer per month at a price of $40 each. For each $0 decrease in price, about 5 more DVD plaers per month are sold. How much should the store charge in order to maimize monthl revenue? What is the maimum monthl revenue? See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 3

33 .5 Solve Quadratic Equations b Finding Square Roots Before You solved quadratic equations b factoring. Now You will solve quadratic equations b finding square roots. Wh? So ou can solve problems about astronom, as in E. 39. Ke Vocabular square root radical radicand rationalizing the denominator conjugates A number r is a square root of a number s if r 2 5 s. A positive number s has two square roots, written as Ï s and 2 Ï s. For eample, because and (23) 2 5 9, the two square roots of 9 are Ï and 2 Ï The positive square root of a number is also called the principal square root. The epression Ï s is called a radical. The smbol Ï is a radical sign, and the number s beneath the radical sign is the radicand of the epression. KEY CONCEPT For Your Notebook Properties of Square Roots (a > 0, b > 0) CC.9-2.A.REI.4b Solve quadratic equations b inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a 6 bi for real numbers a and b. Product Propert Ï ab 5 Ï a p Ï b Eample Ï 8 5 Ï 9 p Ï Ï 2 Quotient Propert Î a b 5 Ï a Ï b Eample Î Ï 2 Ï 25 5 Ï 2 5 SIMPLIFYING SQUARE ROOTS You can use the properties above to simplif epressions containing square roots. A square-root epression is simplified if: no radicand has a perfect-square factor other than, and there is no radical in a denominator E XAMPLE Use properties of square roots USE A CALCULATOR You can use a calculator to approimate Ï s when s is not a perfect square. For eample, Ï 80 ø Simplif the epression. a. Ï 80 5 Ï 6 p Ï Ï 5 b. Ï 6 p Ï 2 5 Ï 26 5 Ï 9 p Ï Ï 4 c. Î Ï 4 Ï d. Î Ï 7 Ï 6 5 Ï 7 4 GUIDED PRACTICE for Eample Simplif the epression.. Ï Ï Ï 0 p Ï 5 4. Ï 8 p Ï Î Î Î Î Chapter Quadratic Functions and Factoring NASA

34 RATIONALIZING THE DENOMINATOR Suppose the denominator of a fraction has the form Ï b, a Ï b, or a 2 Ï b where a and b are rational numbers. The table shows how to eliminate the radical from the denominator. This is called rationalizing the denominator. The epressions a Ï b and a 2 Ï b are called conjugates of each other. Their product is alwas a rational number. Form of the denominator Ï b a Ï b a 2 Ï b Multipl numerator and denominator b: Ï b a 2 Ï b a Ï b E XAMPLE 2 Rationalize denominators of fractions Simplif (a) Î 5 2 and (b) 3 7 Ï 2. Solution a. Î Ï 5 Ï 2 5 Ï 5 Ï 2 5 Ï 0 2 p Ï 2 Ï 2 b. 3 7 Ï Ï 2 p 7 2 Ï Ï Ï Ï 2 7 Ï Ï 2 47 SOLVING QUADRATIC EQUATIONS You can use square roots to solve some tpes of quadratic equations. For eample, if s > 0, then the equation 2 5 s has two real-number solutions: 5 Ï s and 5 2 Ï s. These solutions are often written in condensed form as 5 6 Ï s (read as plus or minus the square root of s ). E XAMPLE 3 Solve a quadratic equation Solve Write original equation. Subtract 5 from each side. AVOID ERRORS When solving an equation of the form 2 5 s where s > 0, make sure to find both the positive and negative solutions Divide each side b Ï 2 Take square roots of each side. 5 6 Ï 4 p Ï 3 Product propert 5 62 Ï 3 Simplif. c The solutions are 2 Ï 3 and 22 Ï 3. CHECK Check the solutions b substituting them into the original equation (2 Ï 3 ) (22 Ï 3 ) (2) (2) Solve Quadratic Equations b Finding Square Roots 33

35 E XAMPLE 4 Standardized Test Practice What are the solutions of the equation 5 (z 3)2 5 7? A 238, 32 B Ï 7, 23 5 Ï 7 C 23 2 Ï 35, 23 Ï 35 D 23 2 Ï 35 5, 23 Ï 35 5 Solution 5 (z 3)2 5 7 Write original equation. (z 3) Multipl each side b 5. z Ï 35 z Ï 35 Take square roots of each side. Subtract 3 from each side. The solutions are 23 Ï 35 and 23 2 Ï 35. c The correct answer is C. A B C D GUIDED PRACTICE for Eamples 2, 3, and 4 Simplif the epression. 9. Î Î 9 8. Î Î Ï Ï Ï Ï 3 Solve the equation z ( 2 2) MODELING DROPPED OBJECTS When an object is dropped, its height h (in feet) above the ground after t seconds can be modeled b the function h 5 26t 2 h 0 where h 0 is the object s initial height (in feet). The graph of h 5 26t 2 200, representing the height of an object dropped from an initial height of 200 feet, is shown at the right. The model h 5 26t 2 h 0 assumes that the force of air resistance on the object is negligible. Also, this model works onl on Earth. For planets with stronger or weaker gravit, different models are used (see Eercise 39). 34 Chapter Quadratic Functions and Factoring

36 E XAMPLE 5 Model a dropped object with a quadratic function SCIENCE COMPETITION For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground? ANOTHER WAY For alternative methods for solving the problem in Eample 5, see the Problem Solving Workshop. Solution h 5 26t 2 h 0 Write height function t 2 50 Substitute 0 for h and 50 for h t 2 Subtract 50 from each side t 2 Divide each side b Î 50 5 t Take square roots of each side ø t Use a calculator. After a successful egg drop c Reject the negative solution, 2.8, because time must be positive. The container will fall for about.8 seconds before it hits the ground. at m.hrw.com GUIDED PRACTICE for Eample WHAT IF? In Eample 5, suppose the egg container is dropped from a height of 30 feet. How long does the container take to hit the ground?.5 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7, 27, and 4 5 STANDARDIZED TEST PRACTICE Es. 2, 9, 34, 35, 36, 40, and 4 SKILL PRACTICE. VOCABULARY In the epression Ï 72, what is 72 called? 2. WRITING Eplain what it means to rationalize the denominator of a quotient containing square roots. EXAMPLES and 2 for Es SIMPLIFYING RADICAL EXPRESSIONS Simplif the epression. 3. Ï Ï Ï Ï 3 p Ï Ï 6 p Ï Ï 24 p 3 Ï 0 9. Î Î AP Images/Mike Mergen Ï Ï Ï Ï 6 3. Î 8 7. Ï 2 4 Ï 5 4. Î Ï Ï 0.5 Solve Quadratic Equations b Finding Square Roots 35

37 9. MULTIPLE CHOICE What is a completel simplified epression for Ï 08? A 2 Ï 27 B 3 Ï 2 C 6 Ï 3 D 0 Ï 8 ERROR ANALYSIS Describe and correct the error in simplifing the epression or solving the equation. 20. Ï 96 5 Ï 4 p Ï Ï EXAMPLES 3 and 4 for Es SOLVING QUADRATIC EQUATIONS Solve the equation. 22. s a z p w r t ( 2 ) ( 2 4) ( 2) SOLVING EQUATIONS AND INEQUALITIES Solve the equation or inequalit. Simplif an radical epressions. a. Ï 2 2 Ï 2 Ï 3 b. Ï 3 > 2 c. 2 Ï Ï SHORT RESPONSE Describe two different methods for solving the equation Include the steps for each method. 36. OPEN-ENDED MATH Write an equation of the form 2 5 s that has (a) two real solutions, (b) eactl one real solution, and (c) no real solutions. 37. CHALLENGE Solve the equation a( b) 2 5 c in terms of a, b, and c. PROBLEM SOLVING EXAMPLE 5 for Es CLIFF DIVING A cliff diver dives off a cliff 40 feet above water. Write an equation giving the diver s height h (in feet) above the water after t seconds. How long is the diver in the air? ASTRONOMY On an planet, the height h (in feet) of a falling object t seconds after it is dropped can be modeled b h 5 2 g 2 t 2 h 0 where h 0 is the object s initial height (in feet) and g is the acceleration (in feet per second squared) due to gravit. For each planet (or dwarf planet) in the table, find the time it takes for a rock dropped from a height of 50 feet to hit the surface. Planet Earth Mars Jupiter Saturn Pluto (dwarf planet) g (ft/sec 2 ) See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE Steve Allen/Brand X Pictures/PictureQuest/Jupiter Images

38 40. SHORT RESPONSE The equation h s 2 gives the height h (in feet) of the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5 foot waves and 20 foot waves. 4. EXTENDED RESPONSE You want to transform a square gravel parking lot with 0 foot sides into a circular lot. You want the circle to have the same area as the square so that ou do not have to bu an additional gravel. a. Model Write an equation ou can use to find the radius r of the circular lot. b. Solve What should the radius of the circular lot be? c. Generalize In general, if a square has sides of length s, what is the radius r of a circle with the same area? Justif our answer algebraicall. 42. BICYCLING The air resistance R (in pounds) on a racing cclist is given b the equation R s 2 where s is the biccle s speed (in miles per hour). a. What is the speed of a racing cclist who eperiences 5 pounds of air resistance? b. What happens to the air resistance if the cclist s speed doubles? Justif our answer algebraicall. 43. CHALLENGE For a swimming pool with a rectangular base, Torricelli s law implies that the height h of water in the pool t seconds after it begins draining is given b h 5 Ï h 0 2 2πd2 Ï 3 lw t 2 2 where l and w are the pool s length and width, d is the diameter of the drain, and h 0 is the water s initial height. (All measurements are in inches.) In terms of l, w, d, and h 0, what is the time required to drain the pool when it is completel filled? 0 ft r 0 ft Duomo/Corbis See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 37

39 Using ALTERNATIVE METHODS LESSON.5 Another Wa to Solve Eample 5 Make sense of problems and persevere in solving them. MULTIPLE REPRESENTATIONS In Eample 5, ou solved a quadratic equation b finding square roots. You can also solve a quadratic equation using a table or a graph. P ROBLEM SCIENCE COMPETITION For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground? M ETHOD Using a Table One alternative approach is to write a quadratic equation and then use a table of values to solve the equation. You can use a graphing calculator to make the table. STEP Write an equation that models the situation using the height function h 5 26t 2 h 0. h 5 26t 2 h 0 Write height function t 2 50 Substitute 0 for h and 50 for h 0. STEP 2 Enter the function into a graphing calculator. Note that time is now represented b and height is now represented b. STEP 3 Make a table of values for the function. Set the table so that the -values start at 0 and increase in increments of 0.. STEP 4 Scroll through the table to find the time at which the height of the container is 0 feet. The table shows that 5 0 between 5.7 and 5.8 because has a change of sign. c The container hits the ground between.7 and.8 seconds after it is dropped. 38 Chapter Quadratic Functions and Factoring

40 M ETHOD 2 Using a Graph Another approach is to write a quadratic equation and then use a graph to solve the equation. You can use a graphing calculator to make the graph. STEP Write an equation that models the situation using the height function h 5 26t 2 h 0. h 5 26t 2 h 0 Write height function t 2 50 Substitute 0 for h and 50 for h 0. STEP 2 Enter the function into a graphing calculator. Note that time is now represented b and height is now represented b. STEP 3 Graph the height function. Adjust the viewing window so that ou can see the point where the graph crosses the positive -ais. Find the positive -value for which 5 0 using the zero feature. The graph shows that 5 0 when ø.8. c The container hits the ground about.8 seconds after it is dropped. P RACTICE SOLVING EQUATIONS Solve the quadratic equation using a table and using a graph WHAT IF? How long does it take for an egg container to hit the ground when dropped from a height of 00 feet? Find the answer using a table and using a graph. 7. WIND PRESSURE The pressure P (in pounds per square foot) from wind blowing at s miles per hour is given b P s 2. What wind speed produces a pressure of 30 lb/ft 2? Solve this problem using a table and using a graph. 8. BIRDS A bird fling at a height of 30 feet carries a shellfish. The bird drops the shellfish to break it and get the food inside. How long does it take for the shellfish to hit the ground? Find the answer using a table and using a graph. 9. DROPPED OBJECT You are dropping a ball from a window 29 feet above the ground to our friend who will catch it 4 feet above the ground. How long is the ball in the air before our friend catches it? Solve this problem using a table and using a graph. 0. REASONING Eplain how to use the table feature of a graphing calculator to approimate the solution of the problem on the preceding page to the nearest hundredth of a second. Use this procedure to find the approimate solution. Using Alternative Methods 39

41 MIXED REVIEW of Problem Solving Make sense of problems and persevere in solving them.. MULTI-STEP PROBLEM A pinecone falls from a tree branch that is 20 feet above the ground. a. Write a function that models the height of the pinecone as it falls. b. Graph the function. c. After how man seconds does the pinecone hit the ground? d. What are the domain and range of the function? 2. MULTI-STEP PROBLEM Some harbor police departments have fire-fighting boats with water cannons. The boats are used to fight fires that occur within the harbor. 4. SHORT RESPONSE You are creating a metal border of uniform width for a wall mirror that is 20 inches b 24 inches. You have 46 square inches of metal to use. Write and solve an equation to find the border s width. Does doubling the width require eactl twice as much metal? Eplain. 24 in. 20 in. a. The function ( ) models the path of water shot b a water cannon where is the horizontal distance (in feet) and is the corresponding height (in feet). What are the domain and range of this function? b. How far does the water cannon shoot? c. What is the maimum height of the water? 3. EXTENDED RESPONSE The diagonal of the screen on a laptop computer measures 5 inches. The ratio of the screen s width w to its height h is 4 : 3. a. Write an epression for w in terms of h. b. Use the Pthagorean theorem and the result from part (a) to write an equation that ou can use to find h. c. Solve the equation from part (b). Eplain wh ou must reject one of the solutions. d. What are the height, width, and area of the laptop screen? 40 Chapter Quadratic Functions and Factoring 5. EXTENDED RESPONSE A pizza shop sells about 80 slices of pizza each da during lunch when it charges $2 per slice. For each $.25 increase in price, about 5 fewer slices are sold each da during lunch. a. Write a function that gives the pizza shop s revenue R if there are price increases. b. What value of maimizes R? Eplain the meaning of our answer in this situation. c. Repeat parts (a) and (b) using the equivalent assumption that each $.25 decrease in price results in 5 more slices being sold. Wh do ou obtain a negative -value in part (b)? 6. GRIDDED ANSWER You have a rectangular vegetable garden that measures 42 feet b 8 feet. You want to double the area of the garden b epanding the length and width as shown. Find the value of. 42 ft 7. OPEN-ENDED Write three different quadratic functions in standard form whose graphs have a verte of (23, 2). 8 ft AP Images/Cherl Hatch

42 .6 Perform Operations with Comple Numbers Before Now You performed operations with real numbers. You will perform operations with comple numbers. Wh? So ou can solve problems involving fractals, as in E. 76. Ke Vocabular imaginar unit i comple number imaginar number comple conjugates comple plane absolute value of a comple number Not all quadratic equations have real-number solutions. For eample, has no real-number solutions because the square of an real number is never a negative number. To overcome this problem, mathematicians created an epanded sstem of numbers using the imaginar unit i, defined as i 5 Ï 2. Note that i The imaginar unit i can be used to write the square root of an negative number. KEY CONCEPT For Your Notebook The Square Root of a Negative Number CC.9-2. N.CN.2 Use the relation i and the commutative, associative, and distributive properties to add, subtract, and multipl comple numbers. Propert Eample. If r is a positive real number, then Ï 2r 5 i Ï r. Ï 23 5 i Ï 3 2. B Propert (), it follows that (i Ï r ) 2 5 2r. (i Ï 3 ) 2 5 i 2 p E XAMPLE Solve a quadratic equation Solve Write original equation Subtract from each side Divide each side b Ï 224 Take square roots of each side. 5 6i Ï 24 Write in terms of i. 5 62i Ï 6 Simplif radical. c The solutions are 2i Ï 6 and 22i Ï 6. GUIDED PRACTICE for Eample Stephen Johnson/Stone/Gett Images Solve the equation Perform Operations with Comple Numbers 4

43 COMPLEX NUMBERS A comple number written in standard form is a number a bi where a and b are real numbers. The number a is the real part of the comple number, and the number bi is the imaginar part. If b ± 0, then a bi is an imaginar number. If a 5 0 and b ± 0, then a bi is a pure imaginar number. The diagram shows how different tpes of comple numbers are related. Two comple numbers a bi and c di are equal if and onl if a 5 c and b 5 d. For eample, if i i, then 5 5 and Real Numbers (a 0i ) 2 p Imaginar Numbers (a bi, b Þ 0) 2 3i 24i 5 2 5i Pure Imaginar Numbers (0 bi, b Þ 0) 6i KEY CONCEPT For Your Notebook Sums and Differences of Comple Numbers To add (or subtract) two comple numbers, add (or subtract) their real parts and their imaginar parts separatel. Sum of comple numbers: Difference of comple numbers: (a bi) (c di) 5 (a c) (b d)i (a bi) 2 (c di) 5 (a 2 c) (b 2 d)i E XAMPLE 2 Add and subtract comple numbers Write the epression as a comple number in standard form. a. (8 2 i) (5 4i) b. (7 2 6i) 2 (3 2 6i) c. 0 2 (6 7i) 4i Solution a. (8 2 i) (5 4i) 5 (8 5) (2 4)i Definition of comple addition 5 3 3i Write in standard form. b. (7 2 6i) 2 (3 2 6i) 5 (7 2 3) (26 6)i Definition of comple subtraction 5 4 0i Simplif. 5 4 Write in standard form. c. 0 2 (6 7i) 4i 5 [(0 2 6) 2 7i] 4i Definition of comple subtraction 5 (4 2 7i) 4i Simplif. 5 4 (27 4)i Definition of comple addition i Write in standard form. GUIDED PRACTICE for Eample 2 Write the epression as a comple number in standard form. 7. (9 2 i) (26 7i) 8. (3 7i) 2 (8 2 2i) ( i) 2 (5 9i) 42 Chapter Quadratic Functions and Factoring

44 E XAMPLE 3 Use addition of comple numbers in real life ELECTRICITY Circuit components such as resistors, inductors, and capacitors all oppose the flow of current. This opposition is called resistance for resistors and reactance for inductors and capacitors. A circuit s total opposition to current flow is impedance. All of these quantities are measured in ohms (Ω). READING Note that while a component s resistance or reactance is a real number, its impedance is a comple number. Component and smbol Resistor Inductor Capacitor Resistance or reactance R L C Impedance R Li 2Ci Alternating current source The table shows the relationship between a component s resistance or reactance and its contribution to impedance. A series circuit is also shown with the resistance or reactance of each component labeled. The impedance for a series circuit is the sum of the impedances for the individual components. Find the impedance of the circuit shown above. Solution The resistor has a resistance of 5 ohms, so its impedance is 5 ohms. The inductor has a reactance of 3 ohms, so its impedance is 3i ohms. The capacitor has a reactance of 4 ohms, so its impedance is 24i ohms. Impedance of circuit 5 5 3i (24i) Add the individual impedances i Simplif. c The impedance of the circuit is 5 2 i ohms. MULTIPLYING COMPLEX NUMBERS To multipl two comple numbers, use the distributive propert or the FOIL method just as ou do when multipling real numbers or algebraic epressions. E XAMPLE 4 Multipl comple numbers Write the epression as a comple number in standard form. a. 4i(26 i) b. (9 2 2i)(24 7i) Solution AVOID ERRORS When simplifing an epression that involves comple numbers, be sure to simplif i 2 to 2. a. 4i(26 i) 5 224i 4i 2 Distributive propert 5 224i 4(2) Use i i 2 4 Simplif i Write in standard form. b. (9 2 2i)(24 7i) i 8i 2 4i 2 Multipl using FOIL i 2 4(2) Simplif and use i i 4 Simplif i Write in standard form..6 Perform Operations with Comple Numbers 43

45 COMPLEX CONJUGATES Two comple numbers of the form a bi and a 2 bi are called comple conjugates. The product of comple conjugates is alwas a real number. For eample, (2 4i)(2 2 4i) i 8i You can use this fact to write the quotient of two comple numbers in standard form. E XAMPLE 5 Divide comple numbers REWRITE QUOTIENTS When a quotient has an imaginar number in the denominator, rewrite the denominator as a real number so ou can epress the quotient in standard form. Write the quotient 7 5i in standard form. 2 4i 7 5i 2 4i 5 7 5i 2 4i p 4i 4i Multipl numerator and denominator b 4i, the comple conjugate of 2 4i. 7 28i 5i 20i 2 4i 2 4i 2 6i 2 Multipl using FOIL. 7 33i 20(2) 2 6(2) 23 33i 7 Simplif and use i 2 5. Simplif i Write in standard form. 7 GUIDED PRACTICE for Eamples 3, 4, and 5 0. WHAT IF? In Eample 3, what is the impedance of the circuit if the given capacitor is replaced with one having a reactance of 7 ohms? Write the epression as a comple number in standard form.. i(9 2 i) 2. (3 i)(5 2 i) 3. 5 i i 3 2 2i COMPLEX PLANE Just as ever real number corresponds to a point on the real number line, ever comple number corresponds to a point in the comple plane. As shown in the net eample, the comple plane has a horizontal ais called the real ais and a vertical ais called the imaginar ais. E XAMPLE 6 Plot comple numbers Plot the comple numbers in the same comple plane. a i b. 22 4i c. 3i d i Solution a. To plot 3 2 2i, start at the origin, move 3 units to the right, and then move 2 units down. b. To plot 22 4i, start at the origin, move 2 units to the left, and then move 4 units up. c. To plot 3i, start at the origin and move 3 units up. d. To plot i, start at the origin, move 4 units to the left, and then move 3 units down. 22 4i i i imaginar 3i real 3 2 2i 44 Chapter Quadratic Functions and Factoring

46 KEY CONCEPT For Your Notebook Absolute Value of a Comple Number The absolute value of a comple number z 5 a bi, denoted z, is a nonnegative real number defined as z 5 Ï a 2 b 2. This is the distance between z and the the origin in the comple plane. imaginar u z u 5 a 2 b 2 a z 5 a bi bi real E XAMPLE 7 Find absolute values of comple numbers Find the absolute value of (a) 24 3i and (b) 23i. a. 24 3i 5 Ï (24) Ï b. 23i 5 0 (23i) 5 Ï 0 2 (23) 2 5 Ï at m.hrw.com GUIDED PRACTICE for Eamples 6 and 7 Plot the comple numbers in the same comple plane. Then find the absolute value of each comple number i i i 8. 24i.6 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es., 29, and 67 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 50, 60, 69, and 74 SKILL PRACTICE. VOCABULARY What is the comple conjugate of a 2 bi? 2. WRITING Is ever comple number an imaginar number? Eplain. EXAMPLE for Es. 3 SOLVING QUADRATIC EQUATIONS Solve the equation r z s t t p p (n 2 3) EXAMPLE 2 for Es. 2 2 ADDING AND SUBTRACTING Write the epression as a comple number in standard form. 2. (6 2 3i) (5 4i) 3. (9 8i) (8 2 9i) 4. (22 2 6i) 2 (4 2 6i) 5. (2 i) 2 (7 2 5i) 6. (8 20i) 2 (28 2i) 7. (8 2 5i) 2 (2 4i) 8. (0 2 2i) (2 2 7i) 9. (4 3i) (7 6i) 20. (2 4i) (29 2 2i).6 Perform Operations with Comple Numbers 45

47 2. MULTIPLE CHOICE What is the standard form of the epression (2 3i) 2 (7 4i)? A 24 B 25 7i C 25 2 i D 5 i EXAMPLES 4 and 5 for Es MULTIPLYING AND DIVIDING Write the epression as a comple number in standard form i(3 2i) 23. 2i(4 2 8i) 24. ( 2 i) (22 5i)(2 4i) 26. (22i) (8 2 3i)(8 3i) 28. 7i 8 i 29. 6i 3 2 i i 3i i 2i i 2 2 3i i 5 9i EXAMPLE 6 for Es PLOTTING COMPLEX NUMBERS Plot the numbers in the same comple plane i i i 37. 4i i i EXAMPLE 7 for Es FINDING ABSOLUTE VALUE Find the absolute value of the comple number i i i i i 47. 4i i i 50. MULTIPLE CHOICE What is the absolute value of 9 2i? A 7 B 5 C 08 D 225 STANDARD FORM Write the epression as a comple number in standard form (3 2i) 2 (9 2 4i) 52. (3 2i) (5 2 i) 6i 53. 5i(3 2i)(8 3i) 54. ( 2 9i)( 2 4i)(4 2 3i) 55. (5 2 2i ) (5 3i ) ( i ) 2 (2 2 4i ) 56. (0 4i ) 2 (3 2 2i ) (6 2 7i )( 2 2i ) ERROR ANALYSIS Describe and correct the error in simplifing the epression. 57. ( 2i)(4 2 i) i 5 Ï i 8i 2 2i 2 5 Ï i 2 7i 4 5 i Ï ADDITIVE AND MULTIPLICATIVE INVERSES The additive inverse of a comple number z is a comple number z a such that z z a 5 0. The multiplicative inverse of z is a comple number z m such that z p z m 5. Find the additive and multiplicative inverses of each comple number. a. z 5 2 i b. z i c. z 5 2 3i 60. OPEN-ENDED MATH Find two imaginar numbers whose sum is a real number. How are the imaginar numbers related? CHALLENGE Write the epression as a comple number in standard form. 6. a bi c di 62. a 2 bi c 2 di 63. a bi c 2 di 64. a 2 bi c di 46 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE

48 PROBLEM SOLVING EXAMPLE 3 for Es CIRCUITS In Eercises 65 67, each component of the circuit has been labeled with its resistance or reactance. Find the impedance of the circuit VISUAL THINKING The graph shows how ou can geometricall add two comple numbers (in this case, 4 i and 2 5i) to find their sum (in this case, 6 6i). Find each of the following sums b drawing a graph. imaginar 6 6i 5i a. (5 i) ( 4i) b. (27 3i) (2 2 2i) c. (3 2 2i) (2 2 i) d. (4 2i) (25 2 3i) i 4 2 i real 69. SHORT RESPONSE Make a table that shows the powers of i from i to i 8 in the first row and the simplified forms of these powers in the second row. Describe the pattern ou observe in the table. Verif that the pattern continues b evaluating the net four powers of i. In Eercises 70 73, use the given numbers to verif that the given propert etends to the comple numbers. Refer to the eample. E XAMPLE Etend real number properties The associative propert of addition for real numbers can be etended to the comple numbers. Verif this propert for the numbers 2 i, 2 2i, and 3i. Solution The associative propert of addition for real numbers states that for real numbers a, b, and c, (a b) c 5 a (b c). [(2 i) ( 2 2i)] 3i 5 [(2 ) ( 2 2)i] 3i 5 (3 2 i) 3i 5 3 (2 3)i 5 3 2i (2 i) [( 2 2i) 3i] 5 (2 i) [ (22 3)i] 5 (2 i) ( i) 5 (2 ) ( )i 5 3 2i c Because the sums are equal, the propert is verified for the given numbers i, 23 i ; commutative propert of multiplication i, 3 2i, 5 i ; distributive propert i, i, 2 i ; associative propert of multiplication 73. a bi, c di ; commutative propert of addition.6 Perform Operations with Comple Numbers 47

49 74. SHORT RESPONSE Evaluate Ï 24 p Ï 225 and Ï 00. Does the rule Ï a p Ï b 5 Ï ab hold when a and b are negative numbers? 75. PARALLEL CIRCUITS In a parallel circuit, there is more than one pathwa through which current can flow. To find the impedance Z of a parallel circuit with two pathwas, first calculate the impedances Z and Z 2 of the pathwas separatel b treating each pathwa as a series circuit. Then appl this formula: Z Z 5 Z 2 Z Z 2 What is the impedance of each parallel circuit shown below? a. 4 7 b. 6 0 c Z 3 Z 2 8 Z Z 2 Z 6 Z CHALLENGE Julia sets are fractals defined on the comple plane. For ever comple number c, there is an associated Julia set determined b the function f(z) 5 z 2 c. For eample, the Julia set corresponding to c 5 i is determined b the function f(z) 5 z 2 i. A number z 0 is a member of this Julia set if the absolute values of the numbers z 5 f(z 0 ), z 2 5 f(z ), z 3 5 f(z 2 ),... are all less than some fied number N, and z 0 is not a member if these absolute values grow infinitel large. Tell whether the given number z 0 belongs to the Julia set associated with the function f(z) 5 z 2 i. A Julia set a. z 0 5 i b. z 0 5 c. z 0 5 2i d. z i 48 See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com Alfred Pasieka/SPL/Photo Researchers, Inc.

50 Investigating Algebra ACTIVITY Use before Complete the Square Using Algebra Tiles to Complete the Square Use appropriate tools strategicall. MATERIALS algebra tiles QUESTION How can ou use algebra tiles to complete the square for a quadratic epression? If ou are given an epression of the form 2 b, ou can add a constant c to the epression so that the result 2 b c is a perfect square trinomial. This process is called completing the square. EXPLORE Complete the square for the epression 2 6 STEP STEP 2 STEP 3 Model the epression Use algebra tiles to model the epression 2 6. You will need to use one 2 -tile and si -tiles for this epression. Make a square Arrange the tiles in a square. You want the length and width of the square to be equal. Your arrangement will be incomplete in one of the corners. Complete the square Find the number of -tiles needed to complete the square. B adding nine -tiles, ou can see that is equal to ( 3) 2. DRAW CONCLUSIONS Use our observations to complete these eercises. Cop and complete the table at the right b following the steps above. 2. Look for patterns in the last column of our table. Consider the general statement 2 b c 5 ( d) 2. a. How is d related to b in each case? b. How is c related to d in each case? c. How can ou obtain the numbers in the table s second column directl from the coefficients of in the epressions from the first column? Epression Completing the Square Number of -tiles needed to complete the square Epression written as a square 2 2??? 2 4??? 2 6? ( 3) 2 2 8??? 2 0??? HMH Photo.7 Complete the Square 49

51 .7 Complete the Square Before You solved quadratic equations b finding square roots. Now You will solve quadratic equations b completing the square. Wh? So ou can find a baseball s maimum height, as in Eample 7. Ke Vocabular completing the square Previousl, ou have solved equations of the form 2 5 k b finding square roots. This method also works if one side of an equation is a perfect square trinomial. E XAMPLE Solve a quadratic equation b finding square roots Solve Write original equation. ANOTHER WAY You can also find the solutions b writing the given equation as and solving this equation b factoring. CC.9-2.A.REI.4a Use the method of completing the square to transform an quadratic equation in into an equation of the form ( 2 p) 2 5 q that has the same solutions. Derive the quadratic formula from this form. ( 2 4) Write left side as a binomial squared Solve for. Take square roots of each side. c The solutions are and PERFECT SQUARES In Eample, the trinomial is a perfect square because it equals ( 2 4) 2. Sometimes ou need to add a term to an epression 2 b to make it a square. This process is called completing the square. KEY CONCEPT Completing the Square For Your Notebook Words To complete the square for the epression 2 b, add b Diagrams In each diagram, the combined area of the shaded regions is 2 b. Adding b completes the square in the second diagram. b b 2 2 b 2 b s 2d b 2 b s 2 d b s d Chapter Quadratic Functions and Factoring Algebra 2 b b b 2 2 b b Jim Cummins/Tai/Gett Images

52 E XAMPLE 2 Make a perfect square trinomial Find the value of c that makes 2 6 c a perfect square trinomial. Then write the epression as the square of a binomial. Solution STEP Find half the coefficient of STEP 2 Square the result of Step STEP 3 Replace c with the result of Step c The trinomial 2 6 c is a perfect square when c Then ( 8)( 8) 5 ( 8) GUIDED PRACTICE for Eamples and 2 Solve the equation b finding square roots Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial c c c SOLVING EQUATIONS The method of completing the square can be used to solve an quadratic equation. When ou complete a square as part of solving an equation, ou must add the same number to both sides of the equation. E XAMPLE 3 Solve a 2 b c 5 0 when a 5 Solve b completing the square Write original equation. Write left side in the form 2 b Add (26) to each side. ( 2 6) Write left side as a binomial squared Ï 32 Take square roots of each side Ï 32 Solve for Ï 2 Simplif: Ï 32 5 Ï 6 p Ï Ï 2 c The solutions are 6 4 Ï 2 and Ï 2. CHECK You can use algebra or a graph. Algebra Substitute each solution in the original equation to verif that it is correct. Graph Use a graphing calculator to graph The -intercepts are about 0.34 ø Ï 2 and.66 ø 6 4 Ï 2..7 Complete the Square 5

53 E XAMPLE 4 Solve a 2 b c 5 0 when a Þ Solve b completing the square Write original equation Divide each side b the coefficient of Write left side in the form 2 b Add to each side. ( 2) Write left side as a binomial squared Ï 23 Take square roots of each side Ï 23 Solve for i Ï 3 Write in terms of the imaginar unit i. c The solutions are 22 i Ï 3 and 22 2 i Ï 3. E XAMPLE 5 Standardized Test Practice The area of the rectangle shown is 72 square units. What is the value of? 2 ELIMINATE CHOICES You can eliminate choices A and D because the side lengths are negative when A 26 B 4 3 C 8.48 D 26 or 4 Solution Use the formula for the area of a rectangle to write an equation. 3( 2) Length 3 Width 5 Area Distributive propert Divide each side b the coefficient of Add to each side. ( ) Write left side as a binomial squared Take square roots of each side Solve for. So, or You can reject 5 26 because the side lengths would be 28 and 24, and side lengths cannot be negative. c The value of is 4. The correct answer is B. A B C D GUIDED PRACTICE for Eamples 3, 4, and 5 Solve the equation b completing the square n 2 2 4n ( 8) p(p 2 2) Chapter Quadratic Functions and Factoring

54 VERTEX FORM Recall that the verte form of a quadratic function is 5 a( 2 h) 2 k where (h, k) is the verte of the function s graph. To write a quadratic function in verte form, use completing the square. E XAMPLE 6 Write a quadratic function in verte form Write in verte form. Then identif the verte ? 5 ( 2 2 0? ) 22 Write original function. Prepare to complete the square ( ) 22 Add (25) to each side ( 2 5) 2 22 Write as a binomial squared. 5 ( 2 5) Solve for. c The verte form of the function is 5 ( 2 5) The verte is (5, 23). E XAMPLE 7 Find the maimum value of a quadratic function BASEBALL The height (in feet) of a baseball t seconds after it is hit is given b this function: 5 26t 2 96t 3 Find the maimum height of the baseball. Solution The maimum height of the baseball is the -coordinate of the verte of the parabola with the given equation. 5 26t 2 96t (t 2 2 6t) 3 Write original function. Factor 26 from first two terms. AVOID ERRORS When ou complete the square, be sure to add (26)(9) to each side, not just 9. (26)(? ) 5 26(t 2 2 6t? ) 3 Prepare to complete the square. (26)(9) 5 26(t 2 2 6t 9) 3 Add (26)(9) to each side (t 2 3) 2 3 Write t 2 2 6t 9 as a binomial squared. 5 26(t 2 3) 2 47 Solve for. c The verte is (3, 47), so the maimum height of the baseball is 47 feet. at m.hrw.com GUIDED PRACTICE for Eamples 6 and 7 David Madison/Stone/Gett Images Write the quadratic function in verte form. Then identif the verte f() WHAT IF? In Eample 7, suppose the height of the baseball is given b 5 26t 2 80t 2. Find the maimum height of the baseball..7 Complete the Square 53

55 .7 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 27, 45, and 65 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 34, 58, 59, and 67 5 MULTIPLE REPRESENTATIONS E. 66 SKILL PRACTICE. VOCABULARY What is the difference between a binomial and a trinomial? 2. WRITING Describe what completing the square means for an epression of the form 2 b. EXAMPLE for Es. 3 2 SOLVING BY SQUARE ROOTS Solve the equation b finding square roots n 2 6n m 2 2 2m t 2 8t u 2 4u MULTIPLE CHOICE What are the solutions of ? A 2 6 i B 22 6 i C 23, 2 D, 3 EXAMPLE 2 for Es. 3 2 FINDING C Find the value of c that makes the epression a perfect square trinomial. Then write the epression as the square of a binomial c c c c c c c c c EXAMPLES 3 and 4 for Es COMPLETING THE SQUARE Solve the equation b completing the square k 2 6k s 2 6s t 2 28t r 2 6r MULTIPLE CHOICE What are the solutions of ? A Ï 3 B Ï 3 C Ï 3 D Ï 3 EXAMPLE 5 for Es GEOMETRY Find the value of. 35. Area of rectangle Area of parallelogram Area of triangle Area of trapezoid Chapter Quadratic Functions and Factoring

56 FINDING THE VERTEX In Eercises 39 and 40, use completing the square to find the verte of the given function s graph. Then tell what the verte represents. 39. At Buckingham Fountain in Chicago, the water s height h (in feet) above the main nozzle can be modeled b h 5 26t t where t is the time (in seconds) since the water has left the nozzle. 25 ft EXAMPLES 6 and 7 for Es When ou walk meters per minute, our rate of energ use (in calories per minute) can be modeled b WRITING IN VERTEX FORM Write the quadratic function in verte form. Then identif the verte Buckingham Fountain f() g() ERROR ANALYSIS Describe and correct the error in solving the equation ( 2 6) ( 2 6 9) 5 9 ( 5) ( 3) Ï 2 ( 3) Ï Ï Ï Ï 5 COMPLETING THE SQUARE Solve the equation b completing the square q 2 0q 5 2q v 2 0.7v 5 0.3v OPEN-ENDED MATH Write a quadratic equation with real-number solutions that can be solved b completing the square but not b factoring. 59. SHORT RESPONSE In this eercise, ou will investigate the graphical effect of completing the square. a. Graph each pair of functions in the same coordinate plane ( ) 2 5 ( 2) 2 5 ( 2 3) 2 b. Compare the graphs of 5 2 b and 5 b What happens to the graph of 5 2 b when ou complete the square? 60. REASONING For what value(s) of k does 2 b b k have eactl real solution? 2 real solutions? 2 imaginar solutions? Ron Watts/Corbis 6. CHALLENGE Solve 2 b c 5 0 b completing the square. Your answer will be an epression for in terms of b and c..7 Complete the Square 55

57 PROBLEM SOLVING EXAMPLE 7 for Es DRUM MAJOR While marching, a drum major tosses a baton into the air and catches it. The height h (in feet) of the baton after t seconds can be modeled b h 5 26t 2 32t 6. Find the maimum height of the baton. 63. VOLLEYBALL The height h (in feet) of a volleball t seconds after it is hit can be modeled b h 5 26t 2 48t 4. Find the volleball s maimum height. 64. SKATEBOARD REVENUE A skateboard shop sells about 50 skateboards per week for the price advertised. For each $ decrease in price, about more skateboard per week is sold. The shop s revenue can be modeled b 5 (70 2 )(50 ). Use verte form to find how the shop can maimize weekl revenue. 65. VIDEO GAME REVENUE A store sells about 40 video game sstems each month when it charges $200 per sstem. For each $0 increase in price, about less sstem per month is sold. The store s revenue can be modeled b 5 (200 0)(40 2 ). Use verte form to find how the store can maimize monthl revenue. 66. MULTIPLE REPRESENTATIONS The path of a ball thrown b a softball plaer can be modeled b the function where is the softball s horizontal position (in feet) and is the corresponding height (in feet). a. Rewriting a Function Write the given function in verte form. b. Making a Table Make a table of values for the function. Include values of from 0 to 20 in increments of 0. c. Drawing a Graph Use our table to graph the function. What is the maimum height of the softball? How far does it travel? 67. EXTENDED RESPONSE Your school is adding a rectangular outdoor eating section along part of a 70 foot side of the school. The eating section will be enclosed b a fence along its three open sides. The school has 20 feet of fencing and plans to use 500 square feet of land for the eating section. 70 ft a. Write an equation for the area of the eating section. b. Solve the equation. Eplain wh ou must reject one of the solutions. c. What are the dimensions of the eating section? 20 2 Eating section 56 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

58 GEOMETRY REVIEW The volume of cla equals the difference of the volumes of two clinders. 68. CHALLENGE In our potter class, ou are given a lump of cla with a volume of 200 cubic centimeters and are asked to make a clindrical pencil holder. The pencil holder should be 9 centimeters high and have an inner radius of 3 centimeters. What thickness should our pencil holder have if ou want to use all of the cla? 3 cm cm 3 cm cm cm 9 cm Top view cm Side view QUIZ Solve the equation (p 2 ) (m 5) z s Write the epression as a comple number in standard form. 7. (5 2 3i) (22 5i) 8. (22 9i) 2 (7 8i) 9. 3i(7 2 9i) 0. (8 2 3i)(26 2 0i). 4i 26 2 i i 28 5i Write the quadratic function in verte form. Then identif the verte f() g() FALLING OBJECT A student drops a ball from a school roof 45 feet above ground. How long is the ball in the air? Digital Vision/Alam See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 57

59 .8 Use the Quadratic Formula and the Discriminant Before You solved quadratic equations b completing the square. Now You will solve quadratic equations using the quadratic formula. Wh? So ou can model the heights of thrown objects, as in Eample 5. Ke Vocabular quadratic formula discriminant CC.9-2.N.CN.7 Solve quadratic equations with real coefficients that have comple solutions. Previousl, ou solved quadratic equations b completing the square for each equation separatel. B completing the square once for the general equation a 2 b c 5 0, ou can develop a formula that gives the solutions of an quadratic equation. The formula for the solutions is called the quadratic formula. KEY CONCEPT The Quadratic Formula For Your Notebook Let a, b, and c be real numbers such that a Þ 0. The solutions of the quadratic equation a 2 b c 5 0 are 5 2b 6 Ï b 2 2 4ac. 2a E XAMPLE Solve an equation with two real solutions AVOID ERRORS Remember to write the quadratic equation in standard form before appling the quadratic formula. Solve b 6 Ï b 2 2 4ac 2a 23 6 Ï ()(22) 2() Ï 7 2 Write original equation. Write in standard form. Quadratic formula a 5, b 5 3, c 5 22 Simplif. c The solutions are 5 23 Ï 7 2 CHECK Graph and note that the -intercepts are about 0.56 and about ø 0.56 and Ï 7 ø Chapter Quadratic Functions and Factoring Mike Yamashita/Corbis

60 E XAMPLE 2 Solve an equation with one real solution ANOTHER WAY You can also use factoring to solve this equation because the left side factors as (5 2 3) 2. Solve Ï (230) (25)(9) 2(25) Ï Write original equation. Write in standard form. a 5 25, b 5 230, c 5 9 Simplif. Simplif. c The solution is 3 5. CHECK Graph and note that the onl -intercept is E XAMPLE 3 Solve an equation with imaginar solutions Solve Ï (2)(25) 2(2) Ï Write original equation. Write in standard form. a 5 2, b 5 4, c 5 25 Simplif i i c The solutions are 2 i and 2 2 i. CHECK Rewrite using the imaginar unit i. Simplif. Graph There are no -intercepts. So, the original equation has no real solutions. The algebraic check for the imaginar solution 2 i is shown. 2(2 i) 2 4(2 i) i 8 4i GUIDED PRACTICE for Eamples, 2, and 3 Use the quadratic formula to solve the equation Use the Quadratic Formula and the Discriminant 59

61 DISCRIMINANT In the quadratic formula, the epression b 2 2 4ac is called the discriminant of the associated equation a 2 b c b 6 Ï b 2 2 4ac 2a discriminant You can use the discriminant of a quadratic equation to determine the equation s number and tpe of solutions. KEY CONCEPT For Your Notebook Using the Discriminant of a 2 b c 5 0 Value of discriminant b 2 2 4ac > 0 b 2 2 4ac 5 0 b 2 2 4ac < 0 Number and tpe of solutions Two real solutions One real solution Two imaginar solutions Graph of 5 a 2 b c Two -intercepts One -intercept No -intercept E XAMPLE 4 Use the discriminant Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation. a b c Solution Equation Discriminant Solution(s) a 2 b c 5 0 b 2 2 4ac 5 2b 6 Ï b 2 2 4ac 2a a (28) 2 2 4()(7) 5 24 Two imaginar: 4 6 i b (28) 2 2 4()(6) 5 0 One real: 4 c (28) 2 2 4()(5) 5 4 Two real: 3, 5 GUIDED PRACTICE for Eample 4 Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation Chapter Quadratic Functions and Factoring

62 MODELING LAUNCHED OBJECTS The function h 5 26t 2 h 0 is used to model the height of a dropped object. For an object that is launched or thrown, an etra term v 0 t must be added to the model to account for the object s initial vertical velocit v 0 (in feet per second). Recall that h is the height (in feet), t is the time in motion (in seconds), and h 0 is the initial height (in feet). h 5 26t 2 h 0 h 5 26t 2 v 0 t h 0 Object is dropped. Object is launched or thrown. As shown below, the value of v 0 can be positive, negative, or zero depending on whether the object is launched upward, downward, or parallel to the ground. v 0 > 0 v 0 < 0 v E XAMPLE 5 Solve a vertical motion problem JUGGLING A juggler tosses a ball into the air. The ball leaves the juggler s hand 4 feet above the ground and has an initial vertical velocit of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air? Solution Because the ball is thrown, use the model h 5 26t 2 v 0 t h 0. To find how long the ball is in the air, solve for t when h 5 3. h 5 26t 2 v 0 t h 0 Write height model t 2 40t 4 Substitute 3 for h, 40 for v 0, and 4 for h t 2 40t Write in standard form Ï 40 t (26)() 2(26) Quadratic formula (tl) David Madison/Gett Images; (tc) Ro Morsch/Corbis; (tr) Amwell/Gett Images t Ï t ø or t ø 2.5 Simplif. Use a calculator. c Reject the solution because the ball s time in the air cannot be negative. So, the ball is in the air for about 2.5 seconds. GUIDED PRACTICE for Eample 5 0. WHAT IF? In Eample 5, suppose the ball leaves the juggler s hand with an initial vertical velocit of 50 feet per second. How long is the ball in the air?.8 Use the Quadratic Formula and the Discriminant 6

63 .8 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 9, 39, and 7 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 5, 55, 62, 69, 72, and 73 SKILL PRACTICE. VOCABULARY Cop and complete: You can use the? of a quadratic equation to determine the equation s number and tpe of solutions. 2. WRITING Describe a real-life situation in which ou can use the model h 5 26t 2 v 0 t h 0 but not the model h 5 26t 2 h 0. EXAMPLES, 2, and 3 for Es EQUATIONS IN STANDARD FORM Use the quadratic formula to solve the equation t 2 8t w 2 2 8w p 2 2 0p u 2 4u r 2 2 8r MULTIPLE CHOICE What are the comple solutions of the equation ? A 4 3i, 4 2 3i C 6 3i, 6 2 3i B 4 2i, 4 2 2i D 6 2i, 6 2 2i EQUATIONS NOT IN STANDARD FORM Use the quadratic formula to solve the equation. 3. 3w 2 2 2w s s v 2 5v 2 5 2v t 2 5 9t n 2 3n n SOLVING USING TWO METHODS Solve the equation using the quadratic formula. Then solve the equation b factoring to check our solution(s). 22. z 2 5z m 2 5m m 25. s 2 2 s s 26. r 2 2 4r 8 5 5r p 2 40p n n n EXAMPLE 4 for Es USING THE DISCRIMINANT Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation s 2 7s p 2 8p w 2 w z z 2 2 7z n 2 2 4n 2 5 5n r r 9r 2 SOLVING QUADRATIC EQUATIONS Solve the equation using an method t 2 2 7t 5 7t ( 2 ) v 2 5 2v(v 5) ( ) (2r 2.75) ( s) 2 62 Chapter Quadratic Functions and Factoring

64 ERROR ANALYSIS Describe and correct the error in solving the equation Ï (3)(5) 2(3) Ï ()(8) 2() Ï Ï or or SHORT RESPONSE For a quadratic equation a 2 b c 5 0 with two real solutions, show that the mean of the solutions is 2 b. How is this fact 2a related to the smmetr of the graph of 5 a 2 b c? VISUAL THINKING In Eercises 52 54, the graph of a quadratic function 5 a 2 b c is shown. Tell whether the discriminant of a 2 b c 5 0 is positive, negative, or zero MULTIPLE CHOICE What is the value of c if the discriminant of c 5 0 is 223? A 223 B 26 C 6 D 4 THE CONSTANT TERM Use the discriminant to find all values of c for which the equation has (a) two real solutions, (b) one real solution, and (c) two imaginar solutions c c c c c c OPEN-ENDED MATH Write a quadratic equation in standard form that has a discriminant of 20. WRITING EQUATIONS Write a quadratic equation in the form a 2 b c 5 0 such that c 5 4 and the equation has the given solutions and and i and 2 2 i REASONING Show that there is no quadratic equation a 2 b c 5 0 such that a, b, and c are real numbers and 3i and 22i are solutions. 67. CHALLENGE Solve the equation. a b Use the Quadratic Formula and the Discriminant 63

65 PROBLEM SOLVING EXAMPLE 5 for Es FOOTBALL In a football game, a defensive plaer jumps up to block a pass b the opposing team s quarterback. The plaer bats the ball downward with his hand at an initial vertical velocit of 250 feet per second when the ball is 7 feet above the ground. How long do the defensive plaer s teammates have to intercept the ball before it hits the ground? 69. MULTIPLE CHOICE For the period , the number S (in thousands) of cellular telephone subscribers in the United States can be modeled b S 5 858t 2 42t 4982 where t is the number of ears since 990. In what ear did the number of subscribers reach 50 million? A 99 B 992 C 996 D MULTI-STEP PROBLEM A stunt motorcclist makes a jump from one ramp 20 feet off the ground to another ramp 20 feet off the ground. The jump between the ramps can be modeled b where is the 4 horizontal distance (in feet) and is the height above the ground (in feet). a. What is the motorccle s height r when it lands on the ramp? b. What is the distance d between the ramps? c. What is the horizontal distance h the motorccle has traveled when it reaches its maimum height? d. What is the motorccle s maimum height k above the ground? 7. BIOLOGY The number S of ant species in Kle Canon, Nevada, can be modeled b the function S E E 2 2 where E is the elevation (in meters). Predict the elevation(s) at which ou would epect to find 0 species of ants. 72. SHORT RESPONSE A cit planner wants to create adjacent sections for athletics and picnics in the ard of a outh center. The sections will be rectangular and will be surrounded b fencing as shown. There is 900 feet of fencing available. Each section should have an area of 2,000 square feet. Athletics section Picnic section w a. Show that w l. b. Find the possible dimensions of each section See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE

66 73. EXTENDED RESPONSE You can model the position (, ) of a moving object using a pair of parametric equations. Such equations give and in terms of a third variable t that represents time. For eample, suppose that when a basketball plaer attempts a free throw, the path of the basketball can be modeled b the parametric equations 5 20t 5 26t 2 2t 6 where and are measured in feet, t is measured in seconds, and the plaer s feet are at (0, 0). a. Evaluate Make a table of values giving the position (, ) of the basketball after 0, 0.25, 0.5, 0.75, and second. b. Graph Use our table from part (a) to graph the parametric equations. c. Solve The position of the basketball rim is (5, 0). The top of the backboard is (5, 2). Does the plaer make the free throw? Eplain. 74. CHALLENGE The Stratosphere Tower in Las Vegas is 92 feet tall and has a needle at its top that etends even higher into the air. A thrill ride called the Big Shot catapults riders 60 feet up the needle and then lets them fall back to the launching pad. a. The height h (in feet) of a rider on the Big Shot can be modeled b h 5 26t 2 v 0 t 92 where t is the elapsed time (in seconds) after launch and v 0 is the initial vertical velocit (in feet per second). Find v 0 using the fact that the maimum value of h is feet. b. A brochure for the Big Shot states that the ride up the needle takes two seconds. Compare this time with the time given b the model h 5 26t 2 v 0 t 92 where v 0 is the value ou found in part (a). Discuss the model s accurac. Big Shot ride Ben Mangor/SuperStock See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 65

67 .9 Graph and Solve Quadratic Inequalities Before You graphed and solved linear inequalities. Now You will graph and solve quadratic inequalities. Wh? So ou can model the strength of a rope, as in Eample 2. Ke Vocabular quadratic inequalit in two variables quadratic inequalit in one variable A quadratic inequalit in two variables can be written in one of the following forms: < a 2 b c a 2 b c > a 2 b c a 2 b c The graph of an such inequalit consists of all solutions (, ) of the inequalit. KEY CONCEPT For Your Notebook Graphing a Quadratic Inequalit in Two Variables CC.9-2.A.REI.4b Solve quadratic equations b inspection (e.g., for ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a 6 bi for real numbers a and b. To graph a quadratic inequalit in one of the forms above, follow these steps: STEP STEP 2 Graph the parabola with equation 5 a 2 b c. Make the parabola dashed for inequalities with < or > and solid for inequalities with or. Test a point (, ) inside the parabola to determine whether the point is a solution of the inequalit. STEP 3 Shade the region inside the parabola if the point from Step 2 is a solution. Shade the region outside the parabola if it is not a solution. E XAMPLE Graph a quadratic inequalit Graph > AVOID ERRORS Be sure to use a dashed parabola if the smbol is > or < and a solid parabola if the smbol is or. Solution STEP Graph Because the inequalit smbol is >, make the parabola dashed. STEP 2 Test a point inside the parabola, such as (0, 0). > (0, 0) 2 0 >? 0 2 3(0) > 24 So, (0, 0) is a solution of the inequalit. STEP 3 Shade the region inside the parabola. at m.hrw.com 66 Chapter Quadratic Functions and Factoring Photodisc/Gett Images

68 E XAMPLE 2 Use a quadratic inequalit in real life RAPPELLING A manila rope used for rappelling down a cliff can safel support a weight W (in pounds) provided W 480d 2 where d is the rope s diameter (in inches). Graph the inequalit. Solution Graph W 5 480d 2 for nonnegative values of d. Because the inequalit smbol is, make the parabola solid. Test a point inside the parabola, such as (, 2000). W 480d ? 480() Because (, 2000) is not a solution, shade the region below the parabola. Weight (lb) W (, 2000) W 480d d Diameter (in.) SYSTEMS OF QUADRATIC INEQUALITIES Graphing a sstem of quadratic inequalities is similar to graphing a sstem of linear inequalities. First graph each inequalit in the sstem. Then identif the region in the coordinate plane common to all of the graphs. This region is called the graph of the sstem. E XAMPLE 3 Graph a sstem of quadratic inequalities Graph the sstem of quadratic inequalities Inequalit > Inequalit 2 Solution STEP Graph The graph is the red region inside and including the parabola STEP 2 Graph > The graph is the blue region inside (but not including) the parabola > STEP 3 Identif the purple region where the two graphs overlap. This region is the graph of the sstem GUIDED PRACTICE for Eamples, 2, and 3 Graph the inequalit.. > < Graph the sstem of inequalities consisting of 2 and < Graph and Solve Quadratic Inequalities 67

69 ONE-VARIABLE INEQUALITIES A quadratic inequalit in one variable can be written in one of the following forms: a 2 b c < 0 a 2 b c 0 a 2 b c > 0 a 2 b c 0 You can solve quadratic inequalities using tables, graphs, or algebraic methods. E XAMPLE 4 Solve a quadratic inequalit using a table Solve 2 6 using a table. Solution Rewrite the inequalit as Then make a table of values. MAKE A TABLE To give the eact solution, our table needs to include the -values for which the value of the quadratic epression is Notice that when the values of are between 23 and 2, inclusive. c The solution of the inequalit is GRAPHING TO SOLVE INEQUALITIES Another wa to solve a 2 b c < 0 is to first graph the related function 5 a 2 b c. Then, because the inequalit smbol is <, identif the -values for which the graph lies below the -ais. You can use a similar procedure to solve quadratic inequalities that involve, >, or. E XAMPLE 5 Solve a quadratic inequalit b graphing Solve b graphing. Solution The solution consists of the -values for which the graph of lies on or above the -ais. Find the graph s -intercepts b letting 5 0 and using the quadratic formula to solve for ± Ï (2)(24) 2(2) ± Ï 33 4 ø.9 or ø Sketch a parabola that opens up and has.9 and 2.69 as -intercepts. The graph lies on or above the -ais to the left of (and including) and to the right of (and including) 5.9. c The solution of the inequalit is approimatel 2.69 or.9. GUIDED PRACTICE for Eamples 4 and 5 5. Solve the inequalit using a table and using a graph. 68 Chapter Quadratic Functions and Factoring

70 E XAMPLE 6 Use a quadratic inequalit as a model ROBOTICS The number T of teams that have participated in a robot-building competition for high school students can be modeled b T() , 0 9 where is the number of ears since 992. For what ears was the number of teams greater than 00? Solution You want to find the values of for which: T() > > > 0 Graph on the domain 0 9. The graph s -intercept is about 4.2. The graph lies above the -ais when 4.2 < 9. c There were more than 00 teams participating in the ears E XAMPLE 7 Solve a quadratic inequalit algebraicall Solve > 5 algebraicall. Solution First, write and solve the equation obtained b replacing > with Write equation that corresponds to original inequalit. Write in standard form. ( 3)( 2 5) 5 0 Factor or 5 5 Zero product propert The numbers 23 and 5 are the critical -values of the inequalit > 5. Plot 23 and 5 on a number line, using open dots because the values do not satisf the inequalit. The critical -values partition the number line into three intervals. Test an -value in each interval to see if it satisfies the inequalit Test 5 24: Test 5 : Test 5 6: (24) 2 2 2(24) 5 24 > () (6) 5 24 > 5 c The solution is < 23 or > 5. AP Images/Marcio Jose Sanchez GUIDED PRACTICE for Eamples 6 and 7 6. ROBOTICS Use the information in Eample 6 to determine in what ears at least 200 teams participated in the robot-building competition. 7. Solve the inequalit > 4 algebraicall..9 Graph and Solve Quadratic Inequalities 69

71 .9 EXERCISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7, 39, and 73 5 STANDARDIZED TEST PRACTICE Es. 2, 44, 45, 68, and 73 5 MULTIPLE REPRESENTATIONS E. 74 SKILL PRACTICE. VOCABULARY Give an eample of a quadratic inequalit in one variable and an eample of a quadratic inequalit in two variables. 2. WRITING Eplain how to solve < 0 using a table, b graphing, and algebraicall. EXAMPLE for Es. 3 9 MATCHING INEQUALITIES WITH GRAPHS Match the inequalit with its graph > < A. B. C GRAPHING QUADRATIC INEQUALITIES Graph the inequalit. 6. < > < > < > ERROR ANALYSIS Describe and correct the error in graphing EXAMPLE 3 for Es EXAMPLE 4 for Es GRAPHING SYSTEMS Graph the sstem of inequalities > < 2 2 > > < < SOLVING USING A TABLE Solve the inequalit using a table < > > < > Chapter Quadratic Functions and Factoring

72 EXAMPLE 5 for Es SOLVING BY GRAPHING Solve the inequalit b graphing < > > < < MULTIPLE CHOICE What is the solution of > 0? A < 2 or > 4 3 B 2 < < 4 3 C < or > D < < MULTIPLE CHOICE What is the solution of ? A 28 or 3.5 B C 0 or 4.5 D EXAMPLE 7 for Es SOLVING ALGEBRAICALLY Solve the inequalit algebraicall < < > > < > GRAPHING CALCULATOR In this eercise, ou will use a different graphical method to solve Eample 6. a. Enter the equations and 5 00 into a graphing calculator. b. Graph the equations from part (a) for 0 9 and c. Use the intersect feature to find the point where the graphs intersect. d. During what ears was the number of participating teams greater than 00? Eplain our reasoning. CHOOSING A METHOD Solve the inequalit using an method < > > < < OPEN-ENDED MATH Write a quadratic inequalit in one variable that has a solution of < 22 or > CHALLENGE The area A of the region bounded b a parabola and a horizontal line is given b A bh where b and h are as defined in the diagram. Find the area of the region determined b each pair of inequalities. a b b h.9 Graph and Solve Quadratic Inequalities 7

73 PROBLEM SOLVING EXAMPLE 2 for Es ENGINEERING A wire rope can safel support a weight W (in pounds) provided W 8000d 2 where d is the rope s diameter (in inches). Graph the inequalit. 7. WOODWORKING A hardwood shelf in a wooden bookcase can safel support a weight W (in pounds) provided W 5 2 where is the shelf s thickness (in inches). Graph the inequalit. EXAMPLE 6 for Es ARCHITECTURE The arch of the Sdne Harbor Bridge in Sdne, Australia, can be modeled b where is the distance (in meters) from the left plons and is the height (in meters) of the arch above the water. For what distances is the arch above the road? plon 52 m 73. SHORT RESPONSE The length L (in millimeters) of the larvae of the black porg fish can be modeled b L() , 0 40 where is the age (in das) of the larvae. Write and solve an inequalit to find at what ages a larvae s length tends to be greater than 0 millimeters. Eplain how the given domain affects the solution MULTIPLE REPRESENTATIONS A stud found that a driver s reaction time A() to audio stimuli and his or her reaction time V() to visual stimuli (both in milliseconds) can be modeled b A() , 6 70 V() , 6 70 where is the driver s age (in ears). a. Writing an Inequalit Write an inequalit that ou can use to find the -values for which A() is less than V(). b. Making a Table Use a table to find the solution of the inequalit from part (a). Your table should contain -values from 6 to 70 in increments of 6. c. Drawing a Graph Check the solution ou found in part (b) b using a graphing calculator to solve the inequalit A() < V() graphicall. Describe how ou used the domain 6 70 to determine a reasonable solution. d. Interpret Based on our results from parts (b) and (c), do ou think a driver would react more quickl to a traffic light changing from green to ellow or to the siren of an approaching ambulance? Eplain. 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS Paul Steel/The Stock Market/Corbis

74 75. SOCCER The path of a soccer ball kicked from the ground can be modeled b where is the horizontal distance (in feet) from where the ball was kicked and is the corresponding height (in feet). a. A soccer goal is 8 feet high. Write and solve an inequalit to find at what values of the ball is low enough to go into the goal. b. A soccer plaer kicks the ball toward the goal from a distance of 5 feet awa. No one is blocking the goal. Will the plaer score a goal? Eplain our reasoning. 76. MULTI-STEP PROBLEM A truck that is feet tall and 7 feet wide is traveling under an arch. The arch can be modeled b where and are measured in feet. a. Will the truck fit under the arch? Eplain our reasoning. b. What is the maimum width that a truck feet tall can have and still make it under the arch? c. What is the maimum height that a truck 7 feet wide can have and still make it under the arch? ENTRANCE 77. CHALLENGE For clear blue ice on lakes and ponds, the maimum weight w (in tons) that the ice can support is given b w() where is the thickness of the ice (in inches). a. Calculate What thicknesses of ice can support a weight of 20 tons? b. Interpret Eplain how ou can use the graph of w() to determine the minimum -value in the domain for which the function gives meaningful results. QUIZ Use the quadratic formula to solve the equation Graph the inequalit. 4. < > Solve the inequalit > 2 5 Find the discriminant of the quadratic equation and give the number and tpe of solutions of the equation SPORTS A person throws a baseball into the air with an initial vertical velocit of 30 feet per second and then lets the ball hit the ground. The ball is released 5 feet above the ground. How long is the ball in the air? See EXTRA PRACTICE in Student Resources ONLINE QUIZ at m.hrw.com 73

75 MIXED REVIEW of Problem Solving Make sense of problems and persevere in solving them.. MULTI-STEP PROBLEM You are plaing a lawn version of tic-tac-toe in which ou toss bean bags onto a large board. One of our tosses can be modeled b where is the bean bag s horizontal position (in feet) and is the corresponding height (in feet). a. Write the given function in verte form. b. Graph the function. c. What is the bean bag s maimum height? 2. SHORT RESPONSE Given the functions f() 5 2 and g() 5 ( 2 ) 2 2, compare the graph of g() with the graph of f(). 5. GRIDDED ANSWER What is the product of 5 2 9i and its comple conjugate? 6. GRIDDED RESPONSE An interior designer is making a hanging glass lamp, modeled on the equation If each unit represents 2 inches, how wide is the base of the lamp, to the nearest tenth of an inch? (3, 3.8) 3. EXTENDED RESPONSE The table shows the average price of a car part from 2003 through (0, 0) Years since 2003, t Price (dollars), p a. Graph the price of the part versus time. b. Describe the shape the data appear to follow. What kind of equation would model this shape? c. Do ou think such a model will give a good estimate of the price of the part in 205? Eplain. 4. OPEN-ENDED Name three different comple numbers with an absolute value of 25. Then plot our answers in the same comple plane. 7. EXTENDED RESPONSE You throw a ball to our friend. The ball leaves our hand 5 feet above the ground and has an initial vertical velocit of 50 feet per second. Your friend catches the ball when it falls to a height of 3 feet. a. Write a function that gives the ball s height h (in feet) t seconds after ou throw it. b. How long is the ball in the air? c. Describe three methods ou could use to find the maimum height of the ball. Then find the maimum height using each method. 8. SHORT RESPONSE You are designing notepaper with a solid stripe along the paper s top and left sides as shown. The stripes will take up one third of the area of the paper. The paper measures 5 inches b 8 inches. What will the width of the stripes be? Eplain wh ou must reject one of the solutions. 8 in. 5 in. 9. GRIDDED ANSWER What is the discriminant of the equation ? 74 Chapter Quadratic Functions and Factoring Stockbte

76 CHAPTER SUMMARY Algebra m.hrw.com Electronic Function Librar Big Idea BIG IDEAS Graphing and Writing Quadratic Functions in Several Forms For Your Notebook You can graph or write a quadratic function in standard form, verte form, or intercept form. Form Equation Information about quadratic function Standard form 5 a 2 b c The -coordinate of the verte is 2 b 2a. The ais of smmetr is 5 2 b 2a. Verte form 5 a( 2 h) 2 k The verte is (h, k). The ais of smmetr is 5 h. Intercept form 5 a( 2 p)( 2 q) The -intercepts are p and q. The ais of the smmetr is 5 p q 2. Big Idea 2 Solving Quadratic Equations Using a Variet of Methods There are several different methods ou can use to solve a quadratic equation. Equation contains: Eample Method Binomial without -term Isolate the 2 -term. Then take square roots of each side. Factorable trinomial Factor the trinomial. Then use the zero product propert. Unfactorable trinomial Complete the square, or use the quadratic formula. Big Idea 3 Performing Operations with Square Roots and Comple Numbers You can use the following properties to simplif epressions involving square roots or comple numbers. Square roots Comple numbers If a > 0 and b > 0, then Ï ab 5 Ï a p Ï a b and Î b 5 Ï a Ï. b The imaginar unit i is defined as i 5 Ï 2, so that i If r is a positive real number, then Ï 2r 5 i Ï r and (i Ï r) 2 5 2r. (a bi) (c di) 5 (a c) (b d)i (a bi) 2 (c di) 5 (a 2 c) (b 2 d)i a bi 5 Ï a 2 b 2 Chapter Summar 75

77 CHAPTER REVIEW REVIEW KEY VOCABULARY m.hrw.com Multi-Language Glossar Vocabular practice quadratic function standard form of a quadratic function parabola verte ais of smmetr minimum, maimum value verte form intercept form monomial, binomial, trinomial quadratic equation standard form of a quadratic equation root of an equation zero of a function square root radical, radicand rationalizing the denominator conjugates imaginar unit i comple number standard form of a comple number imaginar number pure imaginar number comple conjugates comple plane absolute value of a comple number completing the square quadratic formula discriminant quadratic inequalit in two variables quadratic inequalit in one variable VOCABULARY EXERCISES. WRITING Given a quadratic function in standard form, eplain how to determine whether the function has a maimum value or a minimum value. 2. Cop and complete: A(n)? is a comple number a bi where a 5 0 and b Þ Cop and complete: A function of the form 5 a( 2 h) 2 k is written in?. 4. Give an eample of a quadratic equation that has a negative discriminant. REVIEW EXAMPLES AND EXERCISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of this chapter.. Graph Quadratic Functions in Standard Form E XAMPLE Graph Because a < 0, the parabola opens down. Find and plot the verte (22, 2). Draw the ais of smmetr Plot the -intercept at (0, 25), and plot its reflection (24, 25) in the ais of smmetr. Plot two other points: (2, 22) and its reflection (23, 22) in the ais of smmetr. Draw a parabola through the plotted points. (22, 2) EXAMPLE 3 for Es. 5 7 EXERCISES Graph the function. Label the verte and ais of smmetr f() Chapter Quadratic Functions and Factoring

78 m.hrw.com Chapter Review Practice.2 Graph Quadratic Functions in Verte or Intercept Form E XAMPLE Graph 5 ( 2 4)( 2). Identif the -intercepts. The quadratic function is in intercept form 5 a( 2 p)( 2 q) where a 5, p 5 4, and q Plot the -intercepts at (4, 0) and (22, 0). Find the coordinates of the verte. 5 p q 5 4 (22) ( 2 4)( 2) 5 29 Plot the verte at (, 29). Draw a parabola through the plotted points as shown. (22, 0) 2 (4, 0) 6 (, 29) EXAMPLES, 3, and 4 for Es. 8 4 EXERCISES Graph the function. Label the verte and ais of smmetr ( 2 )( 5) 9. g() 5 ( 3)( 2 2) ( )( 2 6). 5 ( 2 2) f() 5 ( 6) ( 8) BIOLOGY A flea s jump can be modeled b the function ( 2 33) where is the horizontal distance (in centimeters) and is the corresponding height (in centimeters). How far did the flea jump? What was the flea s maimum height?.3 Solve 2 b c 5 0 b Factoring E XAMPLE Solve Use factoring to solve for Write original equation. ( 2 6)( 3) 5 0 Factor or Zero product propert 5 6 or 5 23 Solve for. EXAMPLE 3 for Es. 5 2 EXERCISES Solve the equation z z 7. s 2 2 6s k 2 2k n 2 5n URBAN PLANNING A cit wants to double the area of a rectangular plaground that is 72 feet b 48 feet b adding the same distance to the length and the width. Write and solve an equation to find the value of. Chapter Review 77

79 CHAPTER REVIEW.4 Solve a 2 b c 5 0 b Factoring E XAMPLE Solve Write original equation Divide each side b 23. (5 2 4)(2 ) 5 0 Factor or Zero product propert or Solve for. EXAMPLE 5 for Es EXERCISES Solve the equation r 2 2r a 2 2 3a Solve Quadratic Equations b Finding Square Roots E XAMPLE Solve 4( 2 7) ( 2 7) Write original equation. ( 2 7) Divide each side b Ï Ï 5 Take square roots of each side. Add 7 to each side and simplif. EXAMPLES 3 and 4 for Es EXERCISES Solve the equation (p ) GEOGRAPHY The total surface area of Earth is 50,000,000 square kilometers. Use the formula S 5 4πr 2, which gives the surface area of a sphere with radius r, to find the radius of Earth..6 Perform Operations with Comple Numbers E XAMPLE Write (6 2 4i)( 2 3i) as a comple number in standard form. (6 2 4i)( 2 3i) i 2 4i 2i 2 Multipl using FOIL i 2(2) Simplif and use i i Write in standard form. 78 Chapter Quadratic Functions and Factoring

80 m.hrw.com Chapter Review Practice EXAMPLES 2, 4, and 5 for Es EXERCISES Write the epression as a comple number in standard form i(2 2 i) 30. (5 i)(4 2 2i) 3. (2 2 5i)(2 5i) 32. (8 2 6i) (7 4i) 33. (2 2 3i) 2 (6 2 5i) 34. 4i 23 6i.7 Complete the Square E XAMPLE Solve b completing the square Write original equation. Write left side in the form 2 b Add (24) to each side. ( 2 4) Write left side as a binomial squared Ï 3 Take square roots of each side Ï 3 Solve for. EXAMPLES 3 and 4 for Es EXERCISES Solve the equation b completing the square Use the Quadratic Formula and the Discriminant E XAMPLE Solve Ï (3)(2) 2(3) Ï 3 3 Write original equation. Write in standard form. Use a 5 3, b 5 6, and c 5 2 in quadratic formula. Simplif. EXAMPLES, 2, 3, and 5 for Es EXERCISES Use the quadratic formula to solve the equation VOLLEYBALL A person spikes a volleball over a net when the ball is 9 feet above the ground. The volleball has an initial vertical velocit of 240 feet per second. The volleball is allowed to fall to the ground. How long is the ball in the air after it is spiked? Chapter Review 79

81 .9 CHAPTER REVIEW Graph and Solve Quadratic Inequalities E XAMPLE Solve The solution consists of the -values for which the graph of lies on or below the -ais. Find the graph s -intercepts b letting 5 0 and using the quadratic formula to solve for Ï (22)(5) 2(22) Ï Ï 22 ø 2.6 or ø 2.6 Sketch a parabola that opens down and has 2.6 and 2.6 as -intercepts. The solution of the inequalit is approimatel 2.6 or EXAMPLE 5 for Es EXERCISES Solve the inequalit b graphing < > 0 80 Chapter Quadratic Functions and Factoring

82 CHAPTER TEST Graph the function. Label the verte and ais of smmetr ( 3) f() 5 2( 4)( 2 2) Factor the epression z 2 2z n s 2 7s t 2 23t 20 Solve the equation r 2 2 3r w 2 3w (m 2 7) ( 2) Write the epression as a comple number in standard form. 6. (3 4i) 2 (2 2 5i) 7. (2 2 7i)( 2i) 8. 3 i 2 2 3i Solve the equation b completing the square Use the quadratic formula to solve the equation Graph the inequalit < > Graph the sstem of quadratic inequalities < < ASPECT RATIO The aspect ratio of a widescreen TV is the ratio of the screen s width to its height, or 6 : 9. What are the width and the height of a 32 inch widescreen TV? (Hint: Use the Pthagorean theorem and the fact that TV sizes such as 32 inches refer to the length of the screen s diagonal.) 32 in. 6 9 Chapter Test 8

83 Standardized TEST PREPARATION CONTEXT-BASED MULTIPLE CHOICE QUESTIONS Some of the information ou need to solve a contet-based multiple choice question ma appear in a table, a diagram, or a graph. P ROBLEM The area of the shaded region is 56 square meters. What is the height of the trapezoid? A 3 meters B 4 meters C 6 meters D 7.5 meters m 8 m Plan INTERPRET THE DIAGRAM You know the area of the shaded region. Use the diagram to find the area of the rectangle, and write an epression for the area of the trapezoid. The difference of these two areas is the area of the shaded region. STEP Find epressions for the two areas. Solution Area of rectangle: Area of trapezoid: A 5 lw A 5 2 ( b b 2 )h 5 3(8) 5 [( 2) (3 2)](2) m 2 5 (2)(4 4) 2 5 (4 4) STEP 2 Write an equation for the area of the shaded region and solve b factoring. Area of shaded region 5 Area of rectangle 2 Area of trapezoid (4 2 4) Substitute Subtract 04 from each side Write in standard form Divide each side b 4. ( 2 3)( 4) 5 0 Factor. STEP 3 Find the possible heights. STEP 4 Reject the negative height. 5 3 or 5 24 Zero product propert The height of the trapezoid is given b the epression 2. Therefore, the possible heights are 2(3) 5 6 meters and 2(24) 5 28 meters. Because height cannot be negative, the height of the trapezoid is 6 meters. c The correct answer is C. A B C D 82 Chapter Quadratic Functions and Factoring

84 P ROBLEM 2 The height h (in feet) of a lobbed tennis ball after t seconds is shown b the graph. What is the initial vertical velocit of the tennis ball? A 3 feet/second B 6 feet/second C 37.5 feet/second D 47 feet/second h 5 (0, 3) (3, 0) 2 3 t Plan INTERPRET THE GRAPH The graph is a parabola passing through the points (0, 3) and (3, 0). In order to find the initial vertical velocit of the tennis ball, ou must write an equation of the parabola. STEP Write the model for an object that is launched. STEP 2 Use the initial height in our model. STEP 3 Find the initial vertical velocit b substituting a point on the parabola. Solution Because the tennis ball is launched, the parabola has an equation of the form h 5 26t 2 v 0 t h 0 where v 0 is the initial vertical velocit and h 0 is the initial height of the tennis ball. The graph passes through (0, 3), so the initial height of the tennis ball is 3 feet. When ou substitute 3 for h 0 in the model, ou obtain h 5 26t 2 v 0 t 3. Use the fact that the graph of h 5 26t 2 v 0 t 3 passes through (3, 0) to find the initial vertical velocit v (3) 2 v 0 (3) 3 Substitute 0 for h and 3 for t v 0 Simplif v 0 Solve for v 0. The initial vertical velocit is 47 feet per second. c The correct answer is D. A B C D PRACTICE In Eercises and 2, use the graph in Problem 2.. What is the maimum height of the tennis ball to the nearest tenth of a foot? A 36.3 feet B 36.8 feet C 37.5 feet D 38.0 feet 2. What does the -coordinate of the verte of the graph represent? A The maimum height of the tennis ball B The number of seconds the ball is in the air C The number of seconds it takes the ball to reach its maimum height D The initial height of the tennis ball Test Preparation 83

85 Standardized TEST PRACTICE CONTEXT-BASED MULTIPLE CHOICE In Eercises and 2, use the parabola below. verte 5. The graph of which inequalit is shown?. Which statement is not true about the parabola? A The -intercepts are 22 and 4. B The -intercept is 22. C The maimum value is 3. D The ais of smmetr is What is an equation of the parabola? A 5 ( 2 2)( 4) B 5 2 ( 2)( 2 4) 3 C 5 2( 2)( 2 4) D 5 23( 2)( 2 4) 3. You are using glass tiles to make a picture frame for a square photograph with sides 0 inches long. You want the frame to form a uniform border around the photograph. You have enough tiles to cover 300 square inches. What is the largest possible frame width? A 3.6 inches B 5 inches C 7.3 inches D 5 inches 0 in. 0 in. 4. At a flea market held each weekend, an artist sells handmade earrings. The table below shows the average number of pairs of earrings sold for several prices. Given the pattern in the table, how much should the artist charge to maimize revenue? Price $5 $4 $3 $2 Pairs sold A B C > D In Eercises 6 and 7, use the information below. The graph shows the height h (in feet) after t seconds of a horseshoe tossed during a game of horseshoes. The initial vertical velocit of the horseshoe is 30 feet per second. 4 h (0, 2) 2 6. To the nearest tenth of a second, how long is the horseshoe in the air? A 0. seconds C 2. seconds B.9 seconds D 3.9 seconds 7. To the nearest tenth of a foot, what is the maimum height of the horseshoe? A 5.9 feet C 29.2 feet B 6. feet D 32.2 feet 8. The diagram shows a circle inscribed in a square. The area of the shaded region is 2.5 square inches. To the nearest tenth of an inch, how long is a side of the square? A 4.6 inches B 8.7 inches C 9.7 inches D 0.0 inches r t A $5 B $7.50 C $0 D $5 84 Chapter Quadratic Functions and Factoring

86 GRIDDED ANSWER 9. What is the value of k in the equation (3 2)(2 2 k)? 0. What is the real part of the standard form of the epression (5 i)(0 2 i)?. For what value of c is c a perfect square trinomial? 2. What is the maimum value of the function 5 23( 2 2) 2 6? 3. What is the greatest zero of the function ? 4. What is the absolute value of 25 2i? 5. What is the -coordinate of the verte of the parabola that passes through the points (0, 222), (2, 26), and (5, 22)? 6. What is the minimum value of the function f() ? SHORT RESPONSE 7. Solve b completing the square. Eplain all of our steps. 8. The surface area of a cube is given b the function where is an edge length. Graph the function. Compare this graph to the graph of The height h (in feet) of an object after it is launched is given b the function h 5 26t 2 v 0 t h 0 where v 0 is the initial vertical velocit, h 0 is the initial height of the object, and t is the time (in seconds) after the object is launched. Eplain how this function is related to the general function for a dropped object. 20. At what two points do the graphs of and intersect? Eplain our reasoning. EXTENDED RESPONSE 2. The table below shows the New York Yankees paroll (in millions of dollars) from 989 through Years since Paroll Years since Paroll a. Make a scatter plot of the data. b. Draw the parabola that ou think best fits the data. c. Estimate the coordinates of three points on the parabola from part (b). Then use a sstem of equations to write a quadratic model for the data. d. Use our model from part (c) to make a table of data for the ears listed in the original table. Compare the numbers given b our model with the actual data. Assess the accurac of our model. 22. A volleball is hit upward b a plaer in a game. The height h (in feet) of the volleball after t seconds is given b the function h 5 26t 2 30t 6. a. What is the maimum height of the volleball? Eplain our reasoning. b. After how man seconds does the volleball reach its maimum height? c. After how man seconds does the volleball hit the ground? Test Practice 85