In the xy-plane, the graph of the function g(x) is obtained by taking the graph of f(x) = … — GRE General Test

In the $xy$ -plane, the graph of the function $g(x)$ is obtained by taking the graph of $f(x) = (x + 2)^3 - 1$ , reflecting it across the $y$ -axis, translating it $4$ units to the right, and then translating it $3$ units upward. What is the $y$ -intercept of the graph of $y = g(x)$ ?

A
$-214$
B
$-6$
C
$12$
D
$212$
$218$ Answer

Answer

The

y

-intercept of the graph of

y = g(x)

218

Applying the transformations step-by-step to

f(x) = (x + 2)^3 - 1

gives

g(x) = (-x + 6)^3 + 2

. Substituting

x = 0

yields

g(0) = 6^3 + 2 = 218

, making

218

the correct

y

-intercept.

Step-by-Step Solution

Reflect the function

f(x) = (x + 2)^3 - 1

across the

y

-axis.

Replacing

x

with

-x

yields

y_1 = f(-x) = (-x + 2)^3 - 1

Reflecting a graph across the

y

-axis corresponds to replacing

x

with

-x

in the function rule.

Translate the reflected graph

4

units to the right.

Replacing

x

with

x - 4

yields

y_2 = (-(x - 4) + 2)^3 - 1 = (-x + 4 + 2)^3 - 1 = (-x + 6)^3 - 1

Translating a graph

h

units to the right replaces

x

with

x - h

Translate the graph

3

units upward to form

g(x)

g(x) = (-x + 6)^3 - 1 + 3 = (-x + 6)^3 + 2

Translating a graph

k

units upward adds

k

to the expression.

Find the

y

-intercept by evaluating

g(0)

g(0) = (-0 + 6)^3 + 2 = 6^3 + 2 = 216 + 2 = 218

The

y

-intercept occurs where

x = 0

Key Concept

Function Transformations in Coordinate Geometry

Estimated Time:1m 30s

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