In the standard (x, y) coordinate plane, rectangle ABCD has vertices A(-3, 1), B(-1, 1), C… — ACT

In the standard $(x, y)$ coordinate plane, rectangle $ABCD$ has vertices $A(-3, 1)$ , $B(-1, 1)$ , $C(-1, 4)$ , and $D(-3, 4)$ . The rectangle is first reflected across the $y$ -axis and then translated $3$ units to the left and $2$ units down. What are the coordinates of the final image of vertex $C$ ?

$(-2, 2)$ Answer
B
$(4, 2)$
C
$(-2, 6)$
D
$(-4, -6)$
E
$(-4, 2)$

Answer

(-2, 2)

To find the final image of vertex

C(-1, 4)

, we first apply the reflection across the

y

-axis. The rule for reflecting a point across the

y

-axis is

(x, y) \rightarrow (-x, y)

. Applying this to

C(-1, 4)

gives

C'(1, 4)

. Next, we apply the translation of

3

units to the left and

2

units down. The rule for this translation is

(x, y) \rightarrow (x - 3, y - 2)

. Applying this to

C'(1, 4)

gives

C''(1 - 3, 4 - 2) = (-2, 2)

Step-by-Step Solution

Identify the initial coordinates of vertex

C

C(-1, 4)

We need to find the final image of vertex

C

, so we start with its given coordinates.

Apply the reflection across the

y

-axis to vertex

C

C'(1, 4)

Reflecting a point

(x, y)

across the

y

-axis negates the x-coordinate, changing

(x, y)

(-x, y)

Apply the translation of

3

units to the left and

2

units down to the reflected point

C'

C''(-2, 2)

Translating a point

(x, y)

3

units to the left subtracts

3

from the x-coordinate (

x - 3

), and translating

2

units down subtracts

2

from the y-coordinate (

y - 2

Key Concept

Composite transformations in the coordinate plane involve applying multiple geometric transformations in a specific sequence.

Question

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Key Concept

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