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EXAMPLE 1 Write a cubic function Write the cubic function whose graph is shown. 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) STEP 2 Find the value of a by substituting the coordinates of the fourth point.

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

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EXAMPLE 2 Find finite differences The first five triangular numbers are shown below. A formula for the n the triangular number is f (n) = (n2 + n). Show that this function has constant second-order differences. 1 2

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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.

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

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GUIDED PRACTICE for Examples 1 and 2 Write a cubic function whose graph passes through the given points. (– 4, 0), (0, 10), (2, 0), (5, 0) 4 1 The function is f (x) = (x + 4) (x – 2) (x – 5). ANSWER y = 0.25x3 – 0.75x2 – 4.5x +10

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GUIDED PRACTICE for Examples 1 and 2 (– 1, 0), (0, – 12), (2, 0), (3, 0) The function is f (x) = – 2 (x + 1) (x – 2) (x – 3). ANSWER y = – 2 x3 – 8x2 – 2x – 12

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**3. GEOMETRY Show that f (n) = n(3n – 1), a 1 2 **

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

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EXAMPLE 1 Find a positive slope Let (x 1, y 1 ) = (–4, 2) = (x 2, y 2 ) = (2, 6). m = y 2 – y 1 x 2 – x 1 6 – 2 2 – (–4) = = 4 6 2 3 = Simplify. Substitute.

EXAMPLE 1 Find a positive slope Let (x 1, y 1 ) = (–4, 2) = (x 2, y 2 ) = (2, 6). m = y 2 – y 1 x 2 – x 1 6 – 2 2 – (–4) = = 4 6 2 3 = Simplify. Substitute.

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