In the xy-plane, the graph of the quadratic function f is defined by f(x) = -x^2 + 6x - 5.… — SAT

In the $xy$ -plane, the graph of the quadratic function $f$ is defined by $f(x) = -x^2 + 6x - 5$ . If the graph of $f$ is translated $2$ units to the right and $4$ units up to form the graph of a new function $g$ , what is the maximum value of $g$ ?

A
0
B
5
8Answer
D
10

Answer

The maximum value of the quadratic function is the y-coordinate of its vertex. For

f(x) = -x^2 + 6x - 5

, the vertex occurs at

x = -\frac{6}{2(-1)} = 3

. Evaluating

f(3)

yields

-9 + 18 - 5 = 4

, which is the maximum value of

f

. Translating the graph

2

units to the right shifts the vertex horizontally to

x = 5

but does not change the maximum height. Translating the graph

4

units up increases the y-coordinate of the vertex by

4

, resulting in a new maximum value of

4 + 4 = 8

Step-by-Step Solution

Find the x-coordinate of the vertex of the function

f(x) = -x^2 + 6x - 5

x = 3

The x-coordinate of the vertex of a quadratic function in the standard form

y = ax^2 + bx + c

is given by

x = -\frac{b}{2a}

. For

f(x)

, we have

a = -1

and

b = 6

, so

x = -\frac{6}{2(-1)} = 3

Calculate the maximum value of

f(x)

by finding the y-coordinate of its vertex.

f(3) = 4

Substitute

x = 3

into the function:

f(3) = -(3)^2 + 6(3) - 5 = -9 + 18 - 5 = 4

. Since the coefficient of

x^2

is negative, the parabola opens downward, meaning the vertex represents the maximum value.

Apply the vertical translation to find the maximum value of the new function

g(x)

Maximum value of

g

8

Translating the graph

2

units to the right shifts the vertex horizontally but does not affect the vertical maximum value. Translating the graph

4

units up increases the maximum value by

4

. Thus, the new maximum value is

4 + 4 = 8

Key Concept

Determining the maximum value of a quadratic function and applying translations to its graph.

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