The quadratic function f is defined by f(x) = (x - 3)(x - 7). The function g is defined by… — SAT

The quadratic function $f$ is defined by $f(x) = (x - 3)(x - 7)$ . The function $g$ is defined by $g(x) = f(x - 2) + k$ , where $k$ is a constant. In the $xy$ -plane, the vertex of the graph of $y = g(x)$ is $(7, 5)$ . What is the value of $k$ ?

Answer: 9

Answer

To find the vertex of the function

f(x) = (x - 3)(x - 7)

, we calculate the midpoint of the

x

-intercepts,

x = 5

, and evaluate the function at this value to get

f(5) = -4

, giving the vertex

(5, -4)

. The graph of

g(x) = f(x - 2) + k

shifts the graph of

f

right by

2

units and up by

k

units, resulting in a vertex of

(7, -4 + k)

. Since the vertex of

g(x)

(7, 5)

, we have

-4 + k = 5

, which solves to

k = 9

Step-by-Step Solution

Find the vertex of the quadratic function

f(x) = (x - 3)(x - 7)

The vertex of

f(x)

(5, -4)

The

x

-coordinate of the vertex of a quadratic function in factored form

f(x) = a(x - r_1)(x - r_2)

lies midway between the roots

r_1

and

r_2

. Here,

x = \frac{3 + 7}{2} = 5

. Evaluating the function at

x = 5

gives the

y

-coordinate:

f(5) = (5 - 3)(5 - 7) = -4

Apply the transformations to find the coordinates of the vertex of

g(x) = f(x - 2) + k

The vertex of

g(x)

(7, -4 + k)

The transformation

g(x) = f(x - h) + k

shifts the graph of

f(x)

horizontally by

h

units to the right and vertically by

k

units up. For

g(x) = f(x - 2) + k

, the vertex of

f

(5, -4)

, is shifted

2

units to the right and

k

units up, resulting in the new vertex

(5 + 2, -4 + k) = (7, -4 + k)

Equate the

y

-coordinate of the vertex of

g(x)

to the given value and solve for

k

k = 9

We are given that the vertex of the graph of

y = g(x)

(7, 5)

. Setting the

y

-coordinate of the transformed vertex equal to

5

gives

-4 + k = 5

. Adding

4

to both sides yields

k = 9

Key Concept

Function Transformations of Quadratic Graphs

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