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Cubic Function and Triangular Numbers Analysis

Learn how to write cubic functions using x-intercepts and analyze the constant second-order differences in triangular numbers.

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Cubic Function and Triangular Numbers Analysis

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

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

  3. 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

  4. 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.

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

  6. STEP 1 Use the three given x-intercepts to write the function in factored form. for Examples 1 and 2 GUIDED PRACTICE Write a cubic function whose graph passes through the given points. 1. (– 4, 0), (0, 10), (2, 0), (5, 0) SOLUTION f (x) = a (x + 4) (x – 2) (x – 5)

  7. STEP 2 Find the value of aby substituting the coordinates of the fourth point. 1 4 ANSWER = a 1 The function isf (x) = (x + 4) (x – 2) (x – 5). 4 y = 0.25x3 – 0.75x2 – 4.5x +10 for Examples 1 and 2 GUIDED PRACTICE 10 = a (0 + 4) (0 –2) (0 –5) 10 = 40a

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

  9. STEP 2 Find the value of aby substituting the coordinates of the fourth point. ANSWER The function isf (x) = – 2 (x + 1) (x – 2) (x – 3). y = – 2 x3 – 8x2 – 2x – 12 for Examples 1 and 2 GUIDED PRACTICE – 12 = a (0 + 1) (0 –2) (0 –3) – 12 = 6a – 2 = a

  10. 1 3. GEOMETRY Show that f (n) = n(3n – 1), a 2 formula for the nth pentagonal number, has constant second-order differences. for Examples 1 and 2 GUIDED PRACTICE 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.

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

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