1. EXAMPLE 1 Write a cubic function SOLUTION STEP 1 Use the three given x - intercepts to write the function in factored form. f (x) = a (x + 4)(x – 1)(x – 3) Write the cubic function whose graph is shown. STEP 2 Find the value of a by substituting the coordinates of the fourth point.

2. EXAMPLE 1 Write a cubic function –6 = a (0 + 4) (0 – 1) (0 – 3) –6 = 12a – = a 2 1 CHECK Check the end behavior of f. The degree of f is odd and a < 0. So f (x) +∞ as x → – ∞ and f (x) → – ∞ as x → +∞ which matches the graph. 2 1 The function is f (x) = – (x + 4)(x – 1)(x – 3). ANSWER

3. EXAMPLE 2 Find finite differences The first five triangular numbers are shown below. A formula for the n the triangular number is f (n) = (n 2 + n). Show that this function has constant second-order differences. 1 2

4. EXAMPLE 2 SOLUTION 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. Find finite differences

5. EXAMPLE 2 Each second-order difference is 1, so the second- order differences are constant. ANSWER Find finite differences

6. GUIDED PRACTICE for Examples 1 and 2 Write a cubic function whose graph passes through the given points. 1. ( – 4, 0), (0, 10), (2, 0), (5, 0) 4 1 The function is f (x) = (x + 4) (x – 2) (x – 5). ANSWER y = 0.25x 3 – 0.75x 2 – 4.5x +10

7. GUIDED PRACTICE for Examples 1 and 2 2. (– 1, 0), (0, – 12), (2, 0), (3, 0) The function is f (x) = – 2 (x + 1) (x – 2) (x – 3). ANSWER y = – 2 x 3 – 8x 2 – 2x – 12

8. GUIDED PRACTICE for Examples 1 and 2 3. GEOMETRY Show that f (n) = n(3n – 1), a 1 2 formula for the n th pentagonal number, has constant second-order differences. Write function values for equally-spaced n - values. First-order differences Second-order differences Each second-order difference is 3, so the second- order differences are constant. ANSWER

9. EXAMPLE 3 Model with finite differences The first seven triangular pyramidal numbers are shown below. Find a polynomial function that gives the n th triangular pyramidal number. SOLUTION Begin by finding the finite differences.

10. EXAMPLE 3 Model with finite differences Because the third-order differences are constant, you know that the numbers can be represented by a cubic function of the form f (n) = an 3 + bn 2 + cn + d. By substituting the first four triangular pyramidal numbers into the function, you obtain a system of four linear equations in four variables.

11. EXAMPLE 3 Model with finite differences Write the linear system as a matrix equation AX = B. Enter the matrices A and B into a graphing calculator, and then calculate the solution X = A – 1 B. a(1) 3 + b(1) 2 + c(1) + d = 1 a(2) 3 + b(2) 2 + c(2) + d = 4 a + b + c + d = 1 8a + 4b + 2c + d = 4 a(3) 3 + b(3) 2 + c(3) + d = 10 a(4) 3 + b(4) 2 + c(4) + d = 20 27a + 9b + 3c + d = 10 64a + 16b + 4c + d = 20

12. EXAMPLE 3 Model with finite differences The solution is a =,b =, c =, and d = 0. So, the nth triangular pyramidal number is given by f (n) = n 3 + n 2 + n. 1 62 1 1 3 1 1 1 6 23 1 8 27 64 1 4 9 16 12341234 11111111 abcdabcd Ax = 1 4 10 20 B

13. GUIDED PRACTICE for Example 3 4. Use finite differences to find a polynomial function that fits the data in the table. f (x) = –x 3 + 5x 2 + x + 1. ANSWER

14. EXAMPLE 4 Solve a multi-step problem The table shows the typical speed y (in feet per second) of a space shuttle x seconds after launch. Find a polynomial 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. Space Exploration

15. EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Enter the data into a graphing calculator and make a scatter plot. The points suggest a cubic model.

16. EXAMPLE 4 Solve a multi-step problem y = 0.00650x 3 – 0.739x 2 + 49.0x – 236 STEP 2 Use cubic regression to obtain this polynomial model:

17. EXAMPLE 4 Solve a multi-step problem STEP 3 Check the model by graphing it and the data in the same viewing window.

18. EXAMPLE 4 Solve a multi-step problem STEP 4 Graph the model and y = 4400 in the same viewing window. Use the intersect feature. The booster rockets detach about 106 seconds after launch.

19. GUIDED PRACTICE for Example 4 Use a graphing calculator to find a polynomial function that fits the data. 5. y = 2.71x 3 – 25.5x 2 + 71.8x – 45.7 ANSWER

20. GUIDED PRACTICE for Example 4 6. ANSWER y = –0.587x 3 + 8.99x 2 – 23.4x + 9.62