EXAMPLE 1 Write a cubic function

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EXAMPLE 1 Write a cubic function

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Presentation transcript:

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.

EXAMPLE 1 Write a cubic function – 6 = a (0 + 4) (0 –1) (0 –3) – 6 = 12a – = a 2 1 2 1 The function is f (x) = (x + 4) (x – 1) (x – 3). ANSWER 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 f (n) = (n2 + n). Show that this function has constant second-order differences. 1 2

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 Each second-order difference is 1, so the second-order differences are constant. ANSWER

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 the 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 = 40a = a 4 1 4 1 The function is f (x) = (x + 4) (x – 2) (x – 5). ANSWER y = 0.25x3 – 0.75x2 – 4.5x +10

GUIDED PRACTICE for Examples 1 and 2 2. (– 1, 0), (0, – 12), (2, 0), (3, 0) SOLUTION STEP 1 Use the 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 = 6a – 2 = a The function is f (x) = – 2 (x + 1) (x – 2) (x – 3). ANSWER y = – 2 x3 – 8x2 – 2x – 12

GUIDED PRACTICE for Examples 1 and 2 3. GEOMETRY Show that f (n) = n(3n – 1), a 1 2 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 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. ANSWER