In the standard (x, y) coordinate plane, point P undergoes two transformations. First, it … — ACT

In the standard $(x, y)$ coordinate plane, point $P$ undergoes two transformations. First, it is reflected across the $y$ -axis. Second, it is translated $3$ units down and $4$ units to the right, resulting in the image point $(1, -2)$ . What are the coordinates of the original point $P$ ?

A
$(-3, -1)$
B
$(3, -5)$
C
$(-5, -5)$
$(3, 1)$ Answer
E
$(1, -3)$

Answer

The coordinates of the original point

P

are

(3, 1)

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

(1, -2)

. First, we undo the translation (which was

3

units down and

4

units to the right) by performing the opposite actions: moving

3

units up and

4

units to the left. This shifts

(1, -2)

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

. Next, we undo the reflection across the

y

-axis. Reflecting a point across the

y

-axis negates its

x

-coordinate. The reflection of

(-3, 1)

across the

y

-axis yields the original point

(3, 1)

Step-by-Step Solution

Identify the transformations in reverse order to work backward from the final image

(1, -2)

to the original point

P

The final image is

(1, -2)

. We must first undo the translation (

3

units down,

4

units right) and then undo the reflection across the

y

-axis.

To find the pre-image, we apply the inverse transformations in the reverse order of the original operations.

Undo the translation by applying the opposite operations: move

3

units up and

4

units to the left.

The intermediate point is

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

Undoing a translation of

+4

in the

x

-direction and

-3

in the

y

-direction requires subtracting

4

from the

x

-coordinate and adding

3

to the

y

-coordinate.

Undo the reflection across the

y

-axis by reflecting the intermediate point

(-3, 1)

across the

y

-axis.

The original point

P

(3, 1)

Reflecting a point across the

y

-axis negates its

x

-coordinate. Since reflection is its own inverse, applying it again returns the original coordinates:

-(-3) = 3

Key Concept

Working backward through composite transformations in the coordinate plane.

Alternative Method

Instead of working backward step-by-step, write the transformation equations. If the original point is

P(x, y)

, the reflection across the

y

-axis gives

P'(-x, y)

. The subsequent translation of

3

units down and

4

units right gives

P''(-x + 4, y - 3)

. Set this expression equal to the final coordinates:

-x + 4 = 1 \implies x = 3

and

y - 3 = -2 \implies y = 1

, which gives the original point

(3, 1)

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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