In a Cartesian coordinate system, the point A(2, -4) is translated 4 units to the left and… — ACT

In a Cartesian coordinate system, the point $A(2, -4)$ is translated $4$ units to the left and $3$ units up to form point $B$ . If point $B$ is then reflected across the line $y = -x$ to form point $C$ , what are the coordinates of point $C$ ?

$(1, 2)$ Answer
B
$(-1, -2)$
C
$(7, -6)$
D
$(7, 2)$
E
$(-2, 1)$

Answer

(1, 2)

First, the point

(2, -4)

is translated

4

units left and

3

units up by subtracting

4

from the

x

-coordinate and adding

3

to the

y

-coordinate, which yields

(-2, -1)

. Next, this point is reflected across the line

y = -x

by swapping the coordinates and negating both, which results in the point

(1, 2)

Step-by-Step Solution

Apply the translation to point

A(2, -4)

Point

B

has coordinates

(-2, -1)

Translating a point

4

units to the left subtracts

4

from the

x

-coordinate, and translating

3

units up adds

3

to the

y

-coordinate:

(2 - 4, -4 + 3) = (-2, -1)

Apply the reflection across the line

y = -x

to point

B(-2, -1)

Point

C

has coordinates

(1, 2)

Reflecting a point

(x, y)

across the line

y = -x

swaps the coordinates and negates both, mapping

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

. Thus,

(-2, -1) \rightarrow (-(-1), -(-2)) = (1, 2)

Key Concept

Composite transformations involving translations and reflections in the coordinate plane.

Question

Answer

Step-by-Step Solution

Key Concept

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