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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.
EXAMPLE 1
Write a cubic function
– 6 = a (0 + 4) (0 –1) (0 –3)
– 6 = 12a
– 1 = a
2
ANSWER
The function is f (x) = 12 (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 (n2 + 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 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
1 = a
4
ANSWER
The function is f (x) = 1 (x + 4) (x – 2) (x – 5).
4
y = 0.25x3 – 0.75x2 – 4.5x +10
GUIDED PRACTICE
2.
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)
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
ANSWER
The function is f (x) = – 2 (x + 1) (x – 2) (x – 3).
y = – 2 x3 – 8x2 – 2x – 12
GUIDED PRACTICE
3.
for Examples 1 and 2
GEOMETRY Show that f (n) = 12 n(3n – 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.