For the quadratic function f, the table shows some values of x and their corresponding val… — SAT

For the quadratic function $f$ , the table shows some values of $x$ and their corresponding values of $f(x)$ .

$x$	$f(x)$
$1$	$15$
$3$	$3$
$5$	$15$

What is the value of $f(0)$ ?

Answer: 30

Answer

Since the function values at

x = 1

and

x = 5

are both

15

, the axis of symmetry of the quadratic function must be located at the midpoint of these values, which is

x = 3

. The vertex of the function must therefore have an

x

-coordinate of

3

. From the table,

f(3) = 3

, which means the vertex is

(3, 3)

. The vertex form of the quadratic function is

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

. To find the value of

a

, substitute the point

(1, 15)

into the equation:

15 = a(1 - 3)^2 + 3

, which simplifies to

15 = 4a + 3

, giving

a = 3

. The equation of the function is

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

. Substituting

x = 0

yields

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 30

Step-by-Step Solution

Find the axis of symmetry.

The axis of symmetry is

x = 3

Since

f(1) = 15

and

f(5) = 15

, the function values are equal at these points. Because a quadratic function is symmetric, the axis of symmetry is the midpoint of these two

x

-values:

\frac{1 + 5}{2} = 3

Identify the vertex.

The vertex is

(3, 3)

The vertex of a quadratic function lies on its axis of symmetry. Since the axis of symmetry is

x = 3

, the

x

-coordinate of the vertex is

3

. From the table,

f(3) = 3

, so the vertex is

(3, 3)

Write the function in vertex form and solve for the leading coefficient.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Substituting

(3, 3)

, we get

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

. Using the point

(1, 15)

from the table, we substitute

x = 1

and

f(x) = 15

15 = a(1 - 3)^2 + 3 \Rightarrow 15 = 4a + 3 \Rightarrow 12 = 4a \Rightarrow a = 3

Evaluate the function at

x = 0

f(0) = 30

Substitute

x = 0

into the function:

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 27 + 3 = 30

Key Concept

Quadratic Functions and Graphs

Question

Answer

Step-by-Step Solution

Key Concept

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