A point P in a coordinate plane undergoes a translation of 4 units to the left and 5 units… — ACT

A point $P$ in a coordinate plane undergoes a translation of 4 units to the left and 5 units up, followed by a reflection across the $y$ -axis. If the coordinates of the image point after both transformations are $(2, -3)$ , what are the coordinates of the original point $P$ ?

A
$(-6, 2)$
B
$(6, -2)$
$(2, -8)$ Answer
D
$(-6, -8)$
E
$(-2, 8)$

Answer

The coordinates of the original point

P

are

(2, -8)

To find the coordinates of the original point, we must work backward from the final image point

(2, -3)

by reversing each transformation in reverse order. First, we reverse the reflection across the

y

-axis. Reflecting across the

y

-axis negates the x-coordinate, so negating the x-coordinate of

(2, -3)

gives the intermediate point

(-2, -3)

. Second, we reverse the translation of 4 units left and 5 units up by translating the intermediate point 4 units right and 5 units down. This gives

x = -2 + 4 = 2

and

y = -3 - 5 = -8

. Thus, the original coordinates of point

P

are

(2, -8)

Step-by-Step Solution

Reverse the reflection across the

y

-axis by applying it to the final image point

(2, -3)

The intermediate point is

(-2, -3)

Reflecting a point across the

y

-axis negates its x-coordinate while keeping its y-coordinate the same. Reversing this reflection also negates the x-coordinate.

Reverse the translation of 4 units left and 5 units up by translating the intermediate point

(-2, -3)

4 units right and 5 units down.

The original point

P

(2, -8)

To undo a translation, apply the opposite operations: add 4 to the x-coordinate and subtract 5 from the y-coordinate.

Key Concept

Reversing composite transformations in the coordinate plane

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