Write Quadratic Functions and Models

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4.0 A..B, A.6.B, A.6.C, A.8.A TEKS Write Quadratic Functions and Models Before You wrote linear functions and models. Now You will write quadratic functions and models. Wh? So ou can model the cross section of parabolic dishes, as in E. 46. Ke Vocabular best-fitting quadratic model In Lessons 4. and 4., ou learned how to graph quadratic functions. In this lesson, ou will write quadratic functions given information about their graphs. E XAMPLE Write a quadratic function in verte form Write a quadratic function for the parabola shown. Use verte form because the verte is given. 5 a( h) k Verte form 5 a( ) Substitute for h and for k. (3, ) verte (, ) Use the other given point, (3, ), to find a. 5 a(3 ) Substitute 3 for and for. 5 4a Simplif coefficient of a. 5 a Solve for a. c A quadratic function for the parabola is 5 ( ). E XAMPLE Write a quadratic function in intercept form Write a quadratic function for the parabola shown. 3 (3, ) Use intercept form because the -intercepts are given. 5 a( p)( q) Intercept form 5 a( )( 4) Substitute for p and 4 for q. 4 AVOID ERRORS Be sure to substitute the -intercepts and the coordinates of the given point for the correct letters in 5 a( p)( q). Use the other given point, (3, ), to find a. 5 a(3 )(3 4) Substitute 3 for and for. 54a Simplif coefficient of a. } 5 a Solve for a. c A quadratic function for the parabola is 5} ( )( 4). 4.0 Write Quadratic Functions and Models 309

E XAMPLE 3 Write a quadratic function in standard form Write a quadratic function in standard form for the parabola that passes through the points (, 3), (0, 4), and (, 6). STEP Substitute the coordinates of each point into 5 a b c to obtain the sstem of three linear equations shown below. 3 5 a() b() c Substitute for and 3 for. 3 5 a b c Equation 4 5 a(0) b(0) c Substitute 0 for and 4 for. 4 5 c Equation 6 5 a() b() c Substitute for and 6 for. 65 4a b c Equation 3 REVIEW SYSTEMS OF EQUATIONS For help with solving sstems of linear equations in three variables, see p. 78. STEP Rewrite the sstem of three equations in Step as a sstem of two equations b substituting 4 for c in Equations and 3. a b c 53 Equation a b 4 53 Substitute 4 for c. a b 5 Revised Equation STEP 3 4a b c 5 6 Equation 3 4a b 4 5 6 Substitute 4 for c. 4a b 5 0 Revised Equation 3 Solve the sstem consisting of revised Equations and 3. Use the elimination method. a b 5 3 a b 5 4 a b 5 0 4a b 5 0 6a 5 a 5 So b 5, which means b 5. The solution is a 5, b 5, and c 54. c A quadratic function for the parabola is 5 4. GUIDED PRACTICE for Eamples,, and 3 Write a quadratic function whose graph has the given characteristics.. verte: (4, 5). verte: (3, ) 3. -intercepts:, 5 passes through: (, ) passes through: (0, 8) passes through: (6, ) Write a quadratic function in standard form for the parabola that passes through the given points. 4. (, 5), (0, ), (, ) 5. (, ), (0, 3), (4, ) 6. (, 0), (, ), (, 5) 30 Chapter 4 Quadratic Functions and Factoring

QUADRATIC REGRESSION In Chapter, ou used a graphing calculator to perform linear regression on a data set in order to find a linear model for the data. A graphing calculator can also be used to perform quadratic regression. The model given b quadratic regression is called the best-fitting quadratic model. E XAMPLE 4 TAKS REASONING: Multi-Step Problem i PUMPKIN TOSSING A pumpkin tossing contest is held each ear in Morton, Illinois, where people compete to see whose catapult will send pumpkins the farthest. One catapult launches pumpkins from 5 feet above the ground at a speed of 5 feet per second. The table shows the horizontal distances (in feet) the pumpkins travel when launched at different angles. Use a graphing calculator to find the best-fitting quadratic model for the data. Angle (degrees) 0 30 40 50 60 70 Distance (feet) 37 46 509 50 437 33 STEP Enter the data into two lists of a graphing calculator. STEP Make a scatter plot of the data. Note that the points show a parabolic trend. L L 30 46 40 509 50 50 60 437 70 33 L(6)=33 STEP 3 Use the quadratic regression feature to find the bestfitting quadratic model for the data. STEP 4 Check how well the model fits the data b graphing the model and the data in the same viewing window. QuadReg =a+b+c a=-.648574 b=.594857 c=3.085743 c The best-fitting quadratic model is 50.6.6 3.0. GUIDED PRACTICE for Eample 4 7. PUMPKIN TOSSING In Eample 4, at what angle does the pumpkin travel the farthest? Eplain how ou found our answer. 4.0 Write Quadratic Functions and Models 3

4.0 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 9, 35, and 49 5 TAKS PRACTICE AND REASONING Es. 5, 6, 43, 44, 5, 53, and 54 5 MULTIPLE REPRESENTATIONS E. 50. VOCABULARY Cop and complete: When ou perform quadratic regression on a set of data, the quadratic model obtained is called the?.. WRITING Describe how to write an equation of a parabola if ou know three points on the parabola that are not the verte or -intercepts. EXAMPLE on p. 309 for Es. 3 5 WRITING IN VERTEX FORM Write a quadratic function in verte form for the parabola shown. 3. 4. verte 5. (5, 6) (, ) verte (3, ) (, ) verte (, 3) (, ) WRITING IN VERTEX FORM Write a quadratic function in verte form whose graph has the given verte and passes through the given point. 6. verte: (4, ) 7. verte: (, 6) 8. verte: (5, 4) point: (, 5) point: (, ) point: (, 0) 9. verte: (3, 3) 0. verte: (5, 0). verte: (4, ) point: (, ) point: (, 7) point: (0, 30). verte: (, ) 3. verte: (, 4) 4. verte: (3, 5) point: (4, ) point: (, ) point: (7, 3) 5. TAKS REASONING The verte of a parabola is (5, 3) and another point on the parabola is (, 5). Which point is also on the parabola? A (0, 3) B (, 9) C (, 5) D (7, 7) EXAMPLE on p. 309 for Es. 6 6 6. TAKS REASONING The -intercepts of a parabola are 4 and 7 and another point on the parabola is (, 0). Which point is also on the parabola? A (, ) B (8,4) C (5, 40) D (5, 4) WRITING IN INTERCEPT FORM Write a quadratic function in intercept form for the parabola shown. 7. (0, 6) 8. (3, 3) 9. 3 3 3 6 4 (, 4) 3 Chapter 4 Quadratic Functions and Factoring

WRITING IN INTERCEPT FORM Write a quadratic function in intercept form whose graph has the given -intercepts and passes through the given point. 0. -intercepts:, 5. -intercepts: 3, 0. -intercepts:, 4 point: (4, ) point: (, 0) point: (, 4) 3. -intercepts: 3, 7 4. -intercepts: 5, 5. -intercepts: 6, 3 point: (6, 9) point: (7, 4) point: (0, 9) ERROR ANALYSIS Describe and correct the error in writing a quadratic function whose graph has the given -intercepts or verte and passes through the given point. 6. -intercepts: 4, 3; point: (5, 5) 7. verte: (, 3); point: (, 5) 5 a( 5)( 5) 3 5 a(4 5)(4 5) 3 59a } 3 5 a, so 5 } 3 ( 5)( 5) 5 a( )( 3) 5 5 a( )( 3) 5 5 a 5 } 5 a, so 5 5 } ( )( 3) EXAMPLE 3 on p. 30 for Es. 8 39 WRITING IN STANDARD FORM Write a quadratic function in standard form for the parabola shown. 8. 3 (, ) (, 6) 4 (4, 3) 9. (6, ) (4, ) 6 (3, 4) 30. (4, 6) (, 6) (0, ) WRITING IN STANDARD FORM Write a quadratic function in standard form for the parabola that passes through the given points. 3. (4, 3), (0, ), (, 7) 3. (, 4), (0, 0), (3, 7) 33. (, 4), (0, 5), (, ) 34. (, ), (, ), (3, 7) 35. (, 9), (, ), (3, 7) 36. (6, ), (3, 4), (3, 8) 37. (, 3), (, 3), (4, 5) 38. (6, 9), (4, ), (, 3) 39. (3, ), (3, 0), (6, ) WRITING QUADRATIC FUNCTIONS Write a quadratic function whose graph has the given characteristics. 40. passes through: 4. -intercepts:, 3 4. verte: (4.5, 7.5) (0.5, ), (, 8), (, 5) passes through: (, 9) passes through: (7, 3) 43. TAKS REASONING Draw a parabola that passes through (, 3). Write a function for the parabola in standard form, intercept form, and verte form. 44. TAKS REASONING Suppose ou are given a set of data pairs (, ). Describe how ou can use ratios to determine whether the data can be modeled b a quadratic function of the form 5 a. 45. CHALLENGE Find a function of the form 5 a b c whose graph passes through (, 4), (3, 6), and (7, 4). Eplain what the model tells ou about the points. 4.0 Write Quadratic Functions and Models 33

PROBLEM SOLVING EXAMPLES and 3 on pp. 309 30 for Es. 46 47 46. ANTENNA DISH Three points on the parabola formed b the cross section of an antenna dish are (0, 4), (, 3.5), and (5, 3.065). Write a quadratic function that models the cross section. 47. FOOTBALL Two points on the parabolic path of a kicked football are (0, 0) and the verte (0, 5). Write a quadratic function that models the path. EXAMPLE 4 on p. 3 for Es. 48 50 48. MULTI-STEP PROBLEM The bar graph shows the average number of hours per person per ear spent on the Internet in the United States for the ears 997 00. a. Use a graphing calculator to create a scatter plot. b. Use the quadratic regression feature of the calculator to find the best-fitting quadratic model for the data. c. Use our model from part (b) to predict the average number of hours a person will spend on the Internet in 00. Hours per person Yearl Time on the Internet 50 34 06 00 8 54 50 34 0 997 998 999 000 00 49. RUNNING The table shows how wind affects a runner s performance in the 00 meter dash. Positive wind speeds correspond to tailwinds, and negative wind speeds correspond to headwinds. The change t in finishing time is the difference beween the runner s time when the wind speed is s and the runner s time when there is no wind. Wind speed (m/sec), s 6 4 0 4 6 Change in finishing time (sec), t.8.4 0.67 0 0.57.05.4 50. a. Use a graphing calculator to find the best-fitting quadratic model. b. Predict the change in finishing time when the wind speed is 0 m/sec. MULTIPLE REPRESENTATIONS The table shows the number of U.S. households (in millions) with color televisions from 970 through 000. Years since 970 0 5 0 5 0 5 30 Households with color TVs (millions) 47 63 78 90 94 0 a. Drawing a Graph Make a scatter plot of the data. Draw the parabola that ou think best fits the data. b. Writing a Function Estimate the coordinates of three points on the parabola. Use the points to write a quadratic function for the data. c. Making a Table Use our function from part (b) to make a table of data for the ears listed in the original table above. Compare the numbers of households given b our function with the numbers in the original table. 34 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

5. MULTIPLE TAKS REASONING CHOICE The Garabit Viaduct in France has a parabolic arch as part of its support. Three points on the parabola that models the arch are (0, 0), (40, 38.), and (65, 0) where and are measured in meters. Which point is also on the parabola? A (0,.84) B (6.74, 5) C (80, 5.95) D (5, 45) 5. CHALLENGE Let R be the maimum number of regions into which a circle can be divided using n chords. For eample, the diagram shows that R 5 4 when n 5. Cop and complete the table. Then write a quadratic model giving R as a function of n. n 0 3 4 5 6 4 3 R?? 4???? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 998; TAKS Workbook REVIEW Lesson.3; TAKS Workbook 53. TAKS PRACTICE Charlie receives some mone for his birthda. He deposits one third of the mone in the bank. He purchases a concert ticket for $45. Then he spends half of the remaining mone on dinner. Charlie has $8.50 left. How much mone did he receive for his birthda? TAKS Obj. 0 A $80 B $93 C $8 D $4 54. TAKS PRACTICE Which equation represents a line that is parallel to the line that passes through (4, 9) and (5, 3)? TAKS Obj. 7 F 4 3 5 9 G 3 5 9 H 4 3 5 J 3 5 QUIZ for Lessons 4.8 4.0 Use the quadratic formula to solve the equation. (p. 9). 4 5 5 0. 8 5 0 3. 3 5 4 5 0 Graph the inequalit. (p. 300) 4. < 3 5. > 6. 3 Solve the inequalit. (p. 300) 7. 0 5 8. 7 9. > 5 Write a quadratic function whose graph has the given characteristics. (p. 309) 0. verte: (5, 7). -intercepts: 3, 5. passes through: passes through: (3, ) passes through: (7, 40) (, ), (4, 3), (, 7) 3. 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? (p. 9) EXTRA PRACTICE for Lesson 4.0, p. 03 ONLINE QUIZ at classzone.com 35

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