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Modeling with Polynomial Functions 381 Page 3 of 7 Page 1 of 7 EXPLORING DATA A N D STATISTICS Modeling with 6.9 Polynomial Functions GOAL 1 USING FINITE DIFFERENCES You know that two points determine a line and that three points determine a parabola. What you should learn In Example 1 you will see that four points determine the graph of a cubic function. GOAL 1 Use finite differences to determine the degree of a polynomial EXAMPLE 1 Writing a Cubic Function function that will fit a set of data. Write the cubic function whose graph is shown at the right. y GOAL 2 Use technology to SOLUTION 2 find polynomial models for (–3, 0) (5, 0) real-life data, as applied in Use the three given x-intercepts to write the following: 4 x Example 4 . ƒ( x) = a(x + 3)( x º 2)( x º 5) (2, 0) Why you should learn it To find a, substitute the coordinates of the fourth point. ᭢ To model real-life º15 = a(0 + 3)( 0 º 2)( 0 º 5), so a = º ᎏ1ᎏ quantities, such as the speed 2 (0, Ϫ15) of the space shuttle ᭤ ƒ( x) = º ᎏ1ᎏ(x + 3)( x º 2)( x º 5) 2 in Ex. 49 . L LIF A E E R ✓CHECK Check the graph’s end behavior. The degree of ƒ is odd and a < 0, so ƒ( x) ˘ +‡ as x ˘ º‡ and ƒ( x) ˘ º‡ as x ˘ +‡. To decide whether y-values for equally-spaced x-values can be modeled by a polynomial function, you can use finite differences . EXAMPLE 2 Finding Finite Differences The first three triangular numbers are shown at the right. Triangular Numbers A formula for the nth triangular number is ƒ( n) = ᎏ1ᎏ(n2 + n). 2 ƒ(1) = 1 Show that this function has constant second-order differences. SOLUTION ƒ(2) = 3 Write the first several triangular numbers. Find the first-order differences by subtracting consecutive triangular numbers. Then find the second-order differences by subtracting consecutive first-order differences. ƒ(3) = 6 ƒ(1) ƒ(2) ƒ(3) ƒ(4) ƒ(5) ƒ(6) ƒ(7) Function values for 1 3 6 10 15 21 28 equally-spaced n-values 2 3 4 5 6 7 First-order differences 1 1 1 1 1 Second-order differences 380 Chapter 6 Polynomials and Polynomial Functions Page 2 of 7 In Example 2 notice that the function has degree two and that the second -order differences are constant. This illustrates the first property of finite differences. PROPERTIES OF FINITE DIFFERENCES 1. If a polynomial function ƒ( x) has degree n, then the nth-order differences of function values for equally spaced x-values are nonzero and constant. 2. Conversely, if the nth-order differences of equally-spaced data are nonzero and constant, then the data can be represented by a polynomial function of degree n. The following example illustrates the second property of finite differences. EXAMPLE 3 Modeling with Finite Differences STUDENT HELP The first six triangular pyramidal numbers are shown below. Find a polynomial RNET E function that gives the nth triangular pyramidal number. T HOMEWORK HELP N I Visit our Web site www.mcdougallittell.com for extra examples. ƒ(1) = 1 ƒ(2) = 4 ƒ(3) = 10 ƒ(4) = 20 ƒ(5) = 35 ƒ(6) = 56 ƒ(7) = 84 SOLUTION Begin by finding the finite differences. f(1) f(2) f(3) f(4) f(5) f(6) f(7) Function values for 1 4 10 20 35 56 84 equally-spaced n-values 3 6 10 15 21 28 First-order differences 3 4 5 6 7 Second-order differences 1 1 1 1 Third-order differences Because the third-order differences are constant, you know that the numbers can be represented by a cubic function which has the form ƒ( n) = an 3 + bn 2 + cn + d. By substituting the first four triangular pyramidal numbers into the function, you can obtain a system of four linear equations in four variables. a(1)3 + b(1)2 + c(1) + d = 1 a + b + c + d = 1 a(2)3 + b(2)2 + c(2) + d = 4 8a + 4 b + 2 c + d = 4 a(3)3 + b(3)2 + c(3) + d = 10 27 a + 9 b + 3 c + d = 10 a(4)3 + b(4)2 + c(4) + d = 20 64 a + 16 b + 4 c + d = 20 Using a calculator to solve the system gives a = ᎏ1ᎏ, b = ᎏ1ᎏ, c = ᎏ1ᎏ, and d = 0. 6 2 3 ᭤ The nth triangular pyramidal number is given by ƒ( n) = ᎏ1ᎏn3 + ᎏ1ᎏn2 + ᎏ1ᎏn. 6 2 3 6.9 Modeling with Polynomial Functions 381 Page 3 of 7 FOCUS ON APPLICATIONS GOAL 2 POLYNOMIAL MODELING WITH TECHNOLOGY In Examples 1 and 3 you found a cubic model that exactly fits a set of data points. In many real-life situations, you cannot find a simple model to fit data points exactly. Instead you can use the regression feature on a graphing calculator to find an nth- degree polynomial model that best fits the data. EXAMPLE 4 Modeling with Cubic Regression BOATING The data in the table give the average speed y (in knots) of the Trident motor yacht for several different engine speeds x (in hundreds of revolutions per L LIF A E minute, or RPMs). E MOTORBOATS R R often have a. Find a polynomial model for the data. tachometers instead of speedometers. The b. Estimate the average speed of the Trident for an engine speed of 2400 RPMs. tachometer measures the engine speed in revolutions c. What engine speed produces a boat speed of 14 knots? per minute, which can then be used to determine the Engine speed, x 9 11 13 15 17 19 21.5 speed of the boat. Boat speed, y 6.43 7.61 8.82 9.86 10.88 12.36 15.24 SOLUTION a. Enter the data in a graphing calculator and make a scatter plot. From the scatter plot, it appears that a cubic function will fit the data better than a linear or quadratic function. Use cubic regression to obtain a model. ᭤ y = 0.00475 x3 º 0.194 x2 + 3.13 x º 9.53 ✓CHECK By graphing the model in the same viewing window as the scatter plot, you can see that it is a good fit. b. Substitute x = 24 into the model from part (a). y = 0.00475( 24 )3 º 0.194( 24 )2 + 3.13( 24 ) º 9.53 = 19.51 ᭤ The Trident ’s speed for an engine speed of 2400 RPMs is about 19.5 knots. c. Graph the model and the equation y = 14 on the same screen. Use the Intersect feature to find the point where the graphs intersect. Intersection ᭤ An engine speed of about 2050 RPMs X=20.475611 Y=14 produces a boat speed of 14 knots. 382 Chapter 6 Polynomials and Polynomial Functions Page 4 of 7 GUIDED PRACTICE Vocabulary Check ✓ 1. Describe what first-order differences and second-order differences are. Concept Check ✓ 2. How many points do you need to determine a quartic function? 3. Why can’t you use finite differences to find a model for the data in Example 4? Skill Check ✓ 4. Write the cubic function whose graph passes through (3, 0), (º1, 0), (º2, 0), and (1, 2). Show that the nth-order finite differences for the given function of degree n are nonzero and constant. 5. ƒ( x) = 5 x2 º 2 x + 1 6. ƒ( x) = x3 + x2 º 1 7. ƒ( x) = x4 + 2 x 8. ƒ( x) = 2 x3 º 12 x2 º 5 x + 3 Use finite differences to determine the degree of the polynomial function that will fit the data. 9. x 1 2 3 4 5 6 10. x 1 2 3 4 5 6 ƒ(x) º1 3 3 5 15 39 ƒ(x) 0 8 12 12 8 0 Find a polynomial function that fits the data. 11. x 1 2 3 4 5 6 12. x 1 2 3 4 5 6 ƒ(x) 6 15 22 21 6 –29 ƒ(x) º1 º4 º3 8 35 84 13. GEOMETRY CONNECTION Find a polynomial function that gives the number of diagonals of a polygon with n sides. Number of sides, n 3 4 5 6 7 8 Number of diagonals, d 0 2 5 9 14 20 PRACTICE AND APPLICATIONS STUDENT HELP WRITING CUBIC FUNCTIONS Write the cubic function whose graph is shown. Extra Practice 14.y 15.y 16. y to help you master skills is on p. 949. 1 1 1 2 x 1 x 1 x FINDING A CUBIC MODEL Write a cubic function whose graph passes through the given points. 17. (º1, 0), (º2, 0), (0, 0), (1, º3) 18. (3, 0), (2, 0), (º3, 0), (1, º1) 19. (1, 0), (3, 0), (º2, 0), (2, 1) 20. (º1, 0), (º4, 0), (4, 0), (0, 3) 21. (3, 0), (2, 0), (º1, 0), (1, 4) 22. (0, 0), (º3, 0), (5, 0), (º2, 3) 6.9 Modeling with Polynomial Functions 383 Page 5 of 7 STUDENT HELP FINDING FINITE DIFFERENCES Show that the nth-order differences for the given function of degree n are nonzero and constant. HOMEWORK HELP 2 3 2 3 2 Example 1: Exs. 14–22 23. ƒ( x) = x º 3 x + 7 24. ƒ( x) = 2 x º 5 x º x 25. ƒ( x) = º x +3 x º2 xº3 Example 2: Exs. 23–31, 4 3 4 2 44, 45 26.
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