1. 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.

2. 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.

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) = (n2 + n). Show that this function has constant second-order differences. 1 2

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

6. 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)
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)

7. 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

8. GUIDED PRACTICE
for Examples 1 and 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)

9. 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

10. 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.
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. 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