EXAMPLE 1 Write a cubic function Write the

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

EXAMPLE 1 Write a cubic function – 6 = a (0 + 4) (0 – 1) (0 – 3) – 6 = 12 a – 1 = a 2 ANSWER 1 The function is f (x) = 2 (x + 4) (x – 1) (x – 3). 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.

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

EXAMPLE 2 Find finite differences 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.

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

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) SOLUTION STEP 1 Use three given x-intercepts to write the function in factored form. f (x) = a (x + 4) (x – 2) (x – 5)

GUIDED PRACTICE for Examples 1 and 2 STEP 2 Find the value of a by substituting the coordinates of the fourth point. 10 = a (0 + 4) (0 – 2) (0 – 5) 10 = 40 a 1 = a 4 ANSWER 1 The function is f (x) = 4 (x + 4) (x – 2) (x – 5). y = 0. 25 x 3 – 0. 75 x 2 – 4. 5 x +10

GUIDED PRACTICE 2. for Examples 1 and 2 (– 1, 0), (0, – 12), (2, 0), (3, 0) SOLUTION STEP 1 Use three given x - intercepts to write the function in factored form. f (x) = a (x + 1) (x – 2) (x – 3)

GUIDED PRACTICE for Examples 1 and 2 STEP 2 Find the value of a by substituting the coordinates of the fourth point. – 12 = a (0 + 1) (0 – 2) (0 – 3) – 12 = 6 a – 2 = a ANSWER The function is f (x) = – 2 (x + 1) (x – 2) (x – 3). y = – 2 x 3 – 8 x 2 – 2 x – 12

GUIDED PRACTICE 3. for Examples 1 and 2 GEOMETRY Show that f (n) = 21 n(3 n – 1), a formula for the nth pentagonal number, has constant second-order differences. 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.

GUIDED PRACTICE for Examples 1 and 2 Write function values for equally-spaced n - values. First-order differences Second-order differences ANSWER Each second-order difference is 3, so the secondorder differences are constant.