In the xy-plane, the graph of the quadratic function f is a parabola with vertex (3, -8) a… — SAT

In the $xy$ -plane, the graph of the quadratic function $f$ is a parabola with vertex $(3, -8)$ and passes through the point $(1, 0)$ . If the graph intersects the $y$ -axis at $(0, c)$ , what is the value of $c$ ?

Answer: 10

Answer

The correct answer is 10.

By writing the function in vertex form

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

and substituting the vertex

(3, -8)

, we obtain

f(x) = a(x - 3)^2 - 8

. Substituting the point

(1, 0)

into the equation gives

0 = a(1 - 3)^2 - 8

, which simplifies to

4a = 8

and yields

a = 2

. With the function fully defined as

f(x) = 2(x - 3)^2 - 8

, the

y

-intercept is found by calculating

f(0) = 2(0 - 3)^2 - 8 = 2(9) - 8 = 10

. Thus,

c = 10

Step-by-Step Solution

Express the quadratic function in vertex form using the given vertex

(3, -8)

f(x) = a(x - 3)^2 - 8

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

Substitute the coordinates of the point

(1, 0)

into the function to solve for the leading coefficient

a

a = 2

Since the graph passes through

(1, 0)

, substituting

x = 1

and

f(x) = 0

must satisfy the equation.

Substitute

x = 0

into the completed function

f(x) = 2(x - 3)^2 - 8

to find the value of

c

c = 10

The intersection with the

y

-axis occurs where the input

x

is 0.

Key Concept

Writing and evaluating quadratic functions using the vertex form

Question

Answer

Step-by-Step Solution

Key Concept

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