The function f is defined by f(x) = -x^2 + 6x - 1. The function g is defined by g(x) = f(x… — SAT

The function $f$ is defined by $f(x) = -x^2 + 6x - 1$ . The function $g$ is defined by $g(x) = f(x + 2) - 5$ . If the maximum value of $g(x)$ in the $xy$ -plane occurs at the point $(h, k)$ , what is the value of $h + k$ ?

Answer: 4

Answer

The vertex of the original quadratic function

f(x) = -x^2 + 6x - 1

is located at

(3, 8)

. The transformation

g(x) = f(x + 2) - 5

translates the graph horizontally to the left by

2

units and vertically down by

5

units. This moves the vertex from

(3, 8)

(3 - 2, 8 - 5) = (1, 3)

. Therefore,

h = 1

and

k = 3

, and their sum

h + k

equals

4

Step-by-Step Solution

Find the vertex of the function

f(x) = -x^2 + 6x - 1

The vertex of

f(x)

is at

(3, 8)

By writing

f(x)

in vertex form,

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

, we find that the maximum value of

f(x)

occurs at

(3, 8)

Determine the vertex

(h, k)

of the transformed function

g(x) = f(x + 2) - 5

(h, k) = (1, 3)

The horizontal shift of

f(x + 2)

translates the graph to the left by

2

units, changing the

x

-coordinate from

3

3 - 2 = 1

. The vertical shift of

- 5

translates the graph down by

5

units, changing the

y

-coordinate from

8

8 - 5 = 3

Calculate the value of

h + k

Adding the coordinates of the transformed vertex yields

h + k = 1 + 3 = 4

Key Concept

Vertex form of a quadratic function and translation of functions.

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