A function f has exactly two local extrema: a local maximum at the point (-2, 5) and a loc… — SAT

A function $f$ has exactly two local extrema: a local maximum at the point $(-2, 5)$ and a local minimum at the point $(2, -3)$ . A second function $g$ is defined by $g(x) = 1 - 3f(2x - 4)$ . If the local minimum of the graph of $y = g(x)$ occurs at the point $(h, k)$ , what is the value of $h + k$ ?

A
-17
B
-15
-13Answer
D
-5

Answer

-13

To locate the local minimum of the transformed function

g(x) = 1 - 3f(2x - 4)

, we analyze the vertical reflection and scaling. The negative coefficient in

-3f(2x - 4)

reflects the graph vertically, which means the local maximum of

f(x)

at the point

(-2, 5)

becomes the local minimum of

g(x)

. To find the corresponding

x

-coordinate

h

, we solve

2h - 4 = -2

, which gives

h = 1

. To find the corresponding

y

-coordinate

k

, we evaluate

g(1) = 1 - 3f(-2) = 1 - 3(5) = -14

. Thus, the local minimum occurs at

(1, -14)

, and the sum of these coordinates is

1 + (-14) = -13

Step-by-Step Solution

Analyze how the vertical reflection in

g(x) = 1 - 3f(2x - 4)

affects the extrema.

Due to the negative coefficient in

-3f(2x - 4)

, the graph is reflected vertically. Therefore, the local maximum of

f(x)

(-2, 5)

transforms into the local minimum of

g(x)

, while the local minimum of

f(x)

transforms into the local maximum of

g(x)

A vertical reflection inverts the relative heights of the outputs, converting peaks to valleys and vice versa.

Determine the horizontal transformation to find the

x

-coordinate

h

of the new local minimum.

Set the input of

f

in the definition of

g(x)

equal to the

x

-coordinate of the maximum of

f(x)

, which is

-2

2h - 4 = -2

. Solving this equation gives

2h = 2

, which yields

h = 1

The horizontal shift and compression require solving for the new input variable that produces the same argument for the inner function.

Determine the vertical transformation to find the

y

-coordinate

k

of the new local minimum.

Substitute

h = 1

into

g(x)

to find the output value:

k = g(1) = 1 - 3f(2(1) - 4) = 1 - 3f(-2)

. Since the maximum value of

f(x)

f(-2) = 5

, we compute

k = 1 - 3(5) = 1 - 15 = -14

The vertical transformations (stretch, reflection, and shift) are applied directly to the function output.

Calculate the sum of the coordinates

h + k

Compute

h + k = 1 + (-14) = -13

The question asks for the sum of the coordinates of the local minimum of

g(x)

Key Concept

Analyzing function transformations including horizontal compression, horizontal translation, vertical stretch, reflection, and vertical translation to determine the coordinates of key features (local extrema) of a transformed function.

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