Write Polynomial Functions and Models

Transcription

1 TEKS 5.9 a.3, A.1.B, A.3.B; P.3.B Write Polnomial Functions and Models Before You wrote linear and quadratic functions. Now You will write higher-degree polnomial functions. Wh? So ou can model launch speed, as in Example 4. Ke Vocabular finite differences You know that two points determine a line and that three points determine a parabola. In Example 1, ou will see that four points determine the graph of a cubic function. E XAMPLE 1 Write a cubic function Write the cubic function whose graph is shown. STEP 1 STEP Use the three given x-intercepts to write the function in factored form. f(x) 5 a(x 1 4)(x 1)(x 3) Find the value of a b substituting the coordinates of the fourth point. 6 5 a(0 1 4)(0 1)(0 3) 6 5 1a (4, 0) (3, 0) (1, 0) 4 x (0, 6) 1 } 5 a c The function is f(x) 5 1 } (x 1 4)(x 1)(x 3). CHECK Check the end behavior of f. The degree of f is odd and a < 0. So f(x) 1` as x ` and f(x) ` as x 1`, which matches the graph. FINITE DIFFERENCES In Example 1, ou found a function given its graph. Functions can also be written from a set of data using finite differences. When the x-values in a data set are equall spaced, the differences of consecutive -values are called finite differences. For example, some finite differences for the function f(x) 5 x are shown below. f(1) f() f(3) f(4) Values of f(x) for equall-spaced x-values Finite differences The finite differences above are called first-order differences. You can also calculate higher-order differences, as shown in the next example. 5.9 Write Polnomial Functions and Models 393

2 E XAMPLE Find finite differences The first five triangular numbers are shown below. A formula for the nth triangular number is f(n) 5 } 1 (n 1 n). Show that this function has constant second-order differences. f(1) 5 1 f() 5 3 f(3) 5 6 f(4) 5 10 f(5) 5 15 Write the first several triangular numbers. Find the first-order differences b subtracting consecutive triangular numbers. Then find the second-order differences b subtracting consecutive first-order differences. f(1) f() f(3) f(4) f(5) f(6) f(7) Write function values for equall-spaced n-values First-order differences Second-order differences c Each second-order difference is 1, so the second-order differences are constant. GUIDED PRACTICE for Examples 1 and Write a cubic function whose graph passes through the given points. 1. (4, 0), (0, 10), (, 0), (5, 0). (1, 0), (0, 1), (, 0), (3, 0) 3. GEOMETRY Show that f(n) 5 } 1 n(3n 1), a formula for the nth pentagonal number, has constant second-order differences. PROPERTIES OF FINITE DIFFERENCES In Example, notice that the function has degree two and that the second-order differences are constant. This illustrates the first of the following two properties of finite differences. KEY CONCEPT For Your Notebook Properties of Finite Differences 1. If a polnomial function f(x) has degree n, then the nth-order differences of function values for equall-spaced x-values are nonzero and constant.. Conversel, if the nth-order differences of equall-spaced data are nonzero and constant, then the data can be represented b a polnomial function of degree n. The second propert of finite differences allows ou to write a polnomial function that models a set of equall-spaced data. 394 Chapter 5 Polnomials and Polnomial Functions

3 E XAMPLE 3 Model with finite differences The first seven triangular pramidal numbers are shown below. Find a polnomial function that gives the nth triangular pramidal number. f(1) 5 1 f() 5 4 f(3) 5 10 f(4) 5 0 f(5)5 35 f(6) 5 56 f(7) 5 84 Begin b finding the finite differences. f(1) f() f(3) f(4) f(5) f(6) f(7) Write function values for equall-spaced n-values First-order differences Second-order differences Third-order differences Because the third-order differences are constant, ou know that the numbers can be represented b a cubic function of the form f(n) 5 an 3 1 bn 1 cn 1 d. B substituting the first four triangular pramidal numbers into the function, ou obtain a sstem of four linear equations in four variables. a(1) 3 1 b(1) 1 c(1) 1 d 5 1 a 1 b 1 c 1 d 5 1 a() 3 1 b() 1 c() 1 d 5 4 8a 1 4b 1 c 1 d 5 4 a(3) 3 1 b(3) 1 c(3) 1 d a 1 9b 1 3c 1 d 5 10 REVIEW SYSTEMS For help with using matrices to solve linear sstems, see p. 10. a(4) 3 1 b(4) 1 c(4) 1 d a 1 16b 1 4c 1 d 5 0 Write the linear sstem as a matrix equation AX 5 B. Enter the matrices A and B into a graphing calculator, and then calculate the solution X 5 A 1 B a 1 3 d b c [A]-1[B] [[ ] [.5 ] [ ] [0 ]] A X B Calculate X 5 A 1 B. c The solution is a 5 1 } 6, b 5 1 }, c 5 1 } 3, and d 5 0. So, the nth triangular pramidal number is given b f(n) 5 1 } 6 n } n 1 1 } 3 n. GUIDED PRACTICE for Example 3 4. Use finite differences to find a polnomial function that fits the data in the table. f(x) Write Polnomial Functions and Models 395

4 CUBIC REGRESSION In Examples 1 and 3, ou found a cubic model that exactl fits a set of data points. In man real-life situations, ou cannot find a simple model to fit data points exactl. Instead, ou can use the regression feature of a graphing calculator to find an nth-degree polnomial model that best fits the data. E XAMPLE 4 Solve TAKS a REASONING: multi-step problem Multi-Step Problem SPACE EXPLORATION The table shows the tpical speed (in feet per second) of a space shuttle x seconds after launch. Find a polnomial model for the data. Use the model to predict the time when the shuttle s speed reaches 4400 feet per second, at which point its booster rockets detach. x STEP 1 Enter the data into a graphing calculator and make a scatter plot. The points suggest a cubic model. STEP Use cubic regression to obtain this polnomial model: x x x 36 CubicReg =ax3+bx+cx+d a= b= c= d= ANOTHER WAY You can also find the value of x for which b subtracting 4400 from the right side of the cubic model, graphing the resulting function, and using the zero feature to find the graph s x-intercept. STEP 3 Check the model b graphing it and the data in the same viewing window. STEP 4 Graph the model and in the same viewing window. Use the intersect feature. Intersection X= Y=4400 c The booster rockets detach about 106 seconds after launch. at classzone.com GUIDED PRACTICE for Example 4 Use a graphing calculator to find a polnomial function that fits the data f(x) x f(x) Chapter 5 Polnomials and Polnomial Functions

5 5.9 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 9, 15, and 7 5 TAKS PRACTICE AND REASONING Exs. 10,, 3, 8, 31, and 3 1. VOCABULARY Cop and complete: When the x-values in a data set are equall spaced, the differences of consecutive -values are called?.. WRITING Describe first-order differences and second-order differences. EXAMPLE 1 on p. 393 for Exs WRITING CUBIC FUNCTIONS Write the cubic function whose graph is shown. 3. (0, 3) 4. 4 s0, 3 d 5. (0, ) 1 x x 1 x CUBIC MODELS Write a cubic function whose graph passes through the points. 6. (3, 0), (1, 10), (0, 0), (4, 0) 7. (, 0), (1, 0), (0, 8), (, 0) 8. (3, 0), (1, 0), (3, ), (4, 0) 9. (5, 0), (0, 0), (1, 1), (6, 0) 10. MULTIPLE TAKS REASONING CHOICE Which cubic function s graph passes through the points (3, 0), (1, 0), (3, 0), and (0, 3)? A f(x) 5 (x 3)(x 1 3)(x 1) B f(x) 5} 1 (x 3)(x 1 3)(x 1 1) 3 C f(x) 5(x 3)(x 1 3)(x 1) D f(x) 5 (x 3)(x 1 3)(x 1 1) 11. ERROR ANALYSIS A student tried to write a cubic function whose graph has x-intercepts 1,, and 5, and passes through (1, 3). Describe and correct the error in the student s calculation of the leading coefficient a. 1 5 a(3 1 1)(3 )(3 5) 1 58a 1 } 8 5 a EXAMPLE on p. 394 for Exs EXAMPLE 3 on p. 395 for Exs FINDING FINITE DIFFERENCES Show that the nth-order differences for the given function of degree n are nonzero and constant. 1. f(x) 5 5x f(x) 5x 1 5x 14. f(x) 5 x 4 3x f(x) 5 4x 9x f(x) 5 x 3 4x x f(x) 5 x 5 3x 1 x FINDING A MODEL Use finite differences and a sstem of equations to find a polnomial function that fits the data in the table f(x) f(x) f(x) f(x) Write Polnomial Functions and Models 397

6 . OPEN-ENDED TAKS REASONING MATH Write two different cubic functions whose graphs pass through the points (3, 0), (1, 0), and (, 6). 3. SHORT TAKS REASONING RESPONSE How man points do ou need to determine a quartic function? a quintic (fifth-degree) function? Justif our answers. 4. CHALLENGE Substitute the expressions k, k 1 1, k 1,..., k 1 5 for x in the function f(x) 5 ax 3 1 bx 1 cx 1 d to generate six equall-spaced ordered pairs. Then show that third-order differences are constant. PROBLEM SOLVING EXAMPLE 3 on p. 395 for Ex GEOMETRY Find a polnomial function that gives the number of diagonals d of a polgon with n sides. Number of sides, n Number of diagonals, d EXAMPLE 4 on p. 396 for Exs AVIATION The table shows the number of active pilots (in thousands) with airline transport licenses in the United States for the ears 1997 to 004. Use a graphing calculator to find a polnomial model for the data. Years since 1997, t Transport pilots, p MULTI-STEP PROBLEM The table shows the average U.S. movie ticket price (in dollars) for various ears from 1983 to 003. Years since 1983, t Movie ticket price, m a. Use a graphing calculator to find a polnomial model for the data. b. Estimate the average U.S. movie ticket price in 010. c. In which ear was the average U.S. movie ticket price about $4.50? 8. SHORT TAKS REASONING RESPONSE Based on data collected from friends, ou estimate the cumulative profits (in dollars) after each of six months for two potential businesses. Find a polnomial function that models the profit for each business. Which business will ield the greatest long-term profit? Wh? Yard work Pet care Month, t Profit, p Month, t Profit, p WORKED-OUT SOLUTIONS 398 Chapter 5 Polnomials on p. WS1 and Polnomial Functions 5 TAKS PRACTICE AND REASONING

7 9. GEOMETRY The maximum number of regions R into which space can be divided b n intersecting spheres is given b R(n) 5 } 1 n 3 n 1 } 8 n. Show 3 3 that this function has constant third-order differences. 30. CHALLENGE A clindrical cake is divided into the maximum number of pieces p b c planes. When c 5 1,, 3, 4, 5, and 6 the values of p(c) are, 4, 8, 15, 6, and 4 respectivel. What is the maximum number of pieces into which the cake can be divided when it is cut b 8 planes? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 3.1; TAKS Workbook REVIEW Lesson 4.5; TAKS Workbook 31. TAKS PRACTICE Graph the linear sstem. What is the solution of the sstem? TAKS Obj. 4 3x 58 x 5 10 A (4, 18) B (4,) C (1, 14) D No solution 3. TAKS PRACTICE The height h above the ground (in feet) of a stuntman falling from a window is given b h 516t 1 90 where t is the time (in seconds). An air cushion that is 9 feet high is positioned on the ground below the window. About how man seconds will the stuntman fall before he hits the air cushion? TAKS Obj. 5 F.5 sec G.37 sec H 8.66 sec J 9.48 sec QUIZ for Lessons Find all zeros of the polnomial function. (p. 379) 1. f(x) 5 x 3 4x 11x f(x) 5 x 4 x 3 49x 1 9x Write a polnomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. (p. 379) 3. 4, 1, 4. 4, 1 1 i 5. 3, 5, 7 1 Ï } 6. 1, i, 3 Ï } 6 Graph the function. (p. 387) 7. f(x) 5(x 3)(x )(x 1 ) 8. f(x) 5 3(x 1)(x 1 1)(x 4) 9. f(x) 5 x(x 4)(x 1)(x 1 ) 10. f(x) 5 (x 3)(x 1 ) (x 1 3) Write a cubic function whose graph passes through the given points. (p. 393) 11. (5, 0), (, 0), (1, 9), (, 0) 1. (1, 0), (0, 16), (, 0), (4, 0) 13. DRIVE-INS The table shows the number of U.S. drive-in movie theaters for the ears 1995 to 00. Find a polnomial model that fits the data. (p. 393) Years since 1995, t Drive-in movie theaters, D EXTRA PRACTICE for Lesson 5.9, p Write ONLINE Polnomial QUIZFunctions at classzone.com and Models 399