The graph of the polynomial function h in the xy-plane has x-intercepts only at (-4, 0), (… — SAT

The graph of the polynomial function $h$ in the $xy$ -plane has $x$ -intercepts only at $(-4, 0)$ , $(1, 0)$ , and $(3, 0)$ . If the $y$ -intercept of the graph of $h$ is $(0, -12)$ and the degree of $h$ is $3$ , what is the value of $h(2)$ ?

6Answer
B
-6
C
-30
D
30

Answer

Since the degree of the polynomial function

h

is 3 and its only

x

-intercepts are

(-4, 0)

(1, 0)

, and

(3, 0)

, the function can be expressed in factored form as

h(x) = a(x + 4)(x - 1)(x - 3)

, where

a

is a constant. We can find the value of

a

by using the

y

-intercept of

(0, -12)

. Substituting

x = 0

into the equation gives

-12 = a(0 + 4)(0 - 1)(0 - 3)

, which simplifies to

-12 = 12a

, so

a = -1

. The complete function is therefore

h(x) = -(x + 4)(x - 1)(x - 3)

. To find

h(2)

, substitute

2

for

x

h(2) = -(2 + 4)(2 - 1)(2 - 3) = -(6)(1)(-1) = 6

. This matches the correct answer.

Step-by-Step Solution

Write the general form of the cubic polynomial function

h(x)

using its

x

-intercepts.

h(x) = a(x + 4)(x - 1)(x - 3)

Since the degree of

h

is 3 and its only

x

-intercepts are

(-4, 0)

(1, 0)

, and

(3, 0)

, the corresponding factors must be

(x - (-4)) = (x + 4)

(x - 1)

, and

(x - 3)

, multiplied by a constant leading coefficient

a

Use the

y

-intercept

(0, -12)

to solve for the constant coefficient

a

a = -1

Substituting

x = 0

and

h(0) = -12

into the equation gives

-12 = a(0 + 4)(0 - 1)(0 - 3)

, which simplifies to

-12 = 12a

. Solving for

a

gives

a = -1

Evaluate

h(2)

using the fully defined function

h(x) = -(x + 4)(x - 1)(x - 3)

h(2) = 6

Substitute

x = 2

into the polynomial:

h(2) = -(2 + 4)(2 - 1)(2 - 3) = -(6)(1)(-1) = 6

Key Concept

Polynomial Factors and Graphs

Estimated Time:1m 30s

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