The function f(x) is defined by f(x) = |x - 3| - 2. The graph of y = f(x) in the xy-plane … — GRE General Test

The function $f(x)$ is defined by $f(x) = |x - 3| - 2$ . The graph of $y = f(x)$ in the $xy$ -plane is reflected across the $y$ -axis to produce the graph of $y = g(x)$ . The graph of $y = g(x)$ is then translated $1$ unit to the left and $4$ units downward to produce the graph of $y = h(x)$ . Which of the following statements about the graph of $y = h(x)$ must be true? Select all that apply.

The $y$ -intercept of the graph of $y = h(x)$ is $(0, -2)$ .Answer
The graph of $y = h(x)$ is symmetric with respect to the line $x = -4$ .Answer
The area of the triangular region bounded by the graph of $y = h(x)$ and the $x$ -axis is $36$ .Answer
D
The minimum value of $h(x)$ is $-2$ .
E
The distance between the two $x$ -intercepts of the graph of $y = h(x)$ is $6$ .

Answer

The statements establishing that the

y

-intercept is

(0, -2)

, that the line of symmetry is

x = -4

, and that the bounded region with the

x

-axis has an area of

36

are all correct.

Reflecting

f(x) = |x - 3| - 2

across the

y

-axis yields

g(x) = |x + 3| - 2

. Translating

g(x)

left by

1

unit and down by

4

units yields

h(x) = |x + 4| - 6

. From

h(x) = |x + 4| - 6

, evaluating

h(0) = -2

verifies the

y

-intercept of

(0, -2)

. The vertex at

(-4, -6)

defines the axis of symmetry at

x = -4

. Setting

h(x) = 0

gives

x

-intercepts at

-10

and

2

, producing a triangle bounded by the

x

-axis with base

12

and height

6

, which has an area of

\frac{1}{2} \times 12 \times 6 = 36

Step-by-Step Solution

Apply reflection across the

y

-axis to determine

g(x)

g(x) = f(-x) = |-x - 3| - 2 = |x + 3| - 2

Reflecting

y = f(x)

across the

y

-axis replaces every occurrence of

x

with

-x

Apply horizontal and vertical translations to determine

h(x)

h(x) = g(x + 1) - 4 = |(x + 1) + 3| - 2 - 4 = |x + 4| - 6

Translating a graph

1

unit left adds

1

to the input variable, and translating

4

units down subtracts

4

from the expression.

Find the

y

-intercept of

y = h(x)

h(0) = |0 + 4| - 6 = 4 - 6 = -2

, so the intercept point is

(0, -2)

The

y

-intercept is found by setting

x = 0

Determine the vertex, axis of symmetry, and minimum value.

Vertex is at

(-4, -6)

, line of symmetry is

x = -4

, and minimum value is

-6

For

y = a|x - h| + k

, the vertex is at

(h, k)

and the axis of symmetry is

x = h

Find the

x

-intercepts and compute the bounded triangular area.

|x + 4| - 6 = 0 \implies |x + 4| = 6 \implies x = 2

x = -10

. Base

= 2 - (-10) = 12

, height

= |-6| = 6

. Area

= \frac{1}{2} \times 12 \times 6 = 36

The region bounded by the V-shaped graph below the

x

-axis forms a triangle with base on the

x

-axis and height equal to the magnitude of the vertex's

y

-coordinate.

Key Concept

Coordinate transformations of algebraic functions including

y

-axis reflection, horizontal and vertical translations, and analyzing key geometric features of the resulting graph.

Question

Answer

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