The graph of the quadratic function f in the xy-plane has x-intercepts at (-2, 0) and (8, … — SAT

The graph of the quadratic function $f$ in the $xy$ -plane has $x$ -intercepts at $(-2, 0)$ and $(8, 0)$ . If the maximum value of $f(x)$ is $25$ , what is the value of $f(0)$ ?

Answer: 16

Answer

The axis of symmetry of the quadratic function

f

lies halfway between the

x

-intercepts

x = -2

and

x = 8

, which is at

x = \frac{-2 + 8}{2} = 3

. Since the function has a maximum value of

25

, this maximum must occur at the vertex, giving the vertex coordinates

(3, 25)

. In vertex form, the function is

f(x) = a(x - 3)^2 + 25

. Substituting the

x

-intercept

(8, 0)

into the function yields

0 = a(8 - 3)^2 + 25

, which simplifies to

25a = -25

, or

a = -1

. Therefore, the equation of the function is

f(x) = -(x - 3)^2 + 25

. Evaluating this at

x = 0

gives

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

Step-by-Step Solution

Find the

x

-coordinate of the vertex (axis of symmetry)

x = 3

The axis of symmetry of a parabola is located exactly halfway between its

x

-intercepts:

x = \frac{-2 + 8}{2} = 3

Determine the vertex coordinates

(3, 25)

The maximum value of the quadratic function occurs at its vertex, so the

y

-coordinate of the vertex is the maximum value

25

Write the vertex form of the quadratic function

f(x) = a(x - 3)^2 + 25

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex.

Solve for the leading coefficient

a

a = -1

Substitute the

x

-intercept

(8, 0)

into the vertex form:

0 = a(8 - 3)^2 + 25 \implies 25a = -25 \implies a = -1

Find the value of

f(0)

f(0) = 16

Substitute

x = 0

into the function:

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

Key Concept

Using

x

-intercepts and the maximum value to determine the vertex and equation of a quadratic function.

Question

Answer

Step-by-Step Solution

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