In the standard (x, y) coordinate plane, a point P is reflected across the y-axis and then… — ACT

In the standard $(x, y)$ coordinate plane, a point $P$ is reflected across the $y$ -axis and then translated 4 units down and 3 units right. The coordinates of the resulting image point, $P'$ , are $(1, -2)$ . What are the coordinates of the original point $P$ ?

A
$(-2, -2)$
B
$(-2, 2)$
C
$(2, -6)$
D
$(2, -2)$
$(2, 2)$ Answer

Answer

The coordinates of the original point

P

are

(2, 2)

To find the coordinates of the original point

P(x, y)

, we apply the inverse transformations in reverse order to the final image point

P'(1, -2)

. First, we undo the translation of 4 units down and 3 units right by translating

P'

4 units up and 3 units left. This results in the intermediate coordinates

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

. Next, we undo the reflection across the

y

-axis by reflecting

(-2, 2)

across the

y

-axis again (since a reflection is its own inverse). Changing the sign of the

x

-coordinate gives

(2, 2)

. Alternatively, setting up the equations

-x + 3 = 1

and

y - 4 = -2

and solving them yields

x = 2

and

y = 2

, which corresponds to the point

(2, 2)

Step-by-Step Solution

Set up equations to express the composite transformations of the point

P(x, y)

P'(1, -2)

Reflecting

P(x, y)

across the

y

-axis changes the sign of the

x

-coordinate, yielding

(-x, y)

. Translating this point 4 units down and 3 units right yields

(-x + 3, y - 4)

Establishing the mathematical relationship for each transformation is necessary to work backward to find the coordinates of the pre-image.

Equate the coordinates of the transformed point to the coordinates of the final image point

P'(1, -2)

-x + 3 = 1

and

y - 4 = -2

This sets up two independent linear equations that can be solved for the original coordinates

x

and

y

Solve the equations for

x

and

y

From

-x + 3 = 1

, we get

-x = -2

, which means

x = 2

. From

y - 4 = -2

, we get

y = 2

. Therefore, the coordinates of

P

are

(2, 2)

Solving these equations gives the exact coordinates of the original pre-image point.

Key Concept

Working backward with composite transformations in the coordinate plane

Estimated Time:1m 15s

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