In the standard (x, y) coordinate plane, a vertex V of a square is translated 3 units to t… — ACT

In the standard $(x, y)$ coordinate plane, a vertex $V$ of a square is translated $3$ units to the left and $4$ units up, and then reflected across the $x$ -axis. If the coordinates of the final image of $V$ are $(2, -5)$ , what are the coordinates of the original vertex $V$ ?

A
$(-1, 9)$
B
$(5, 9)$
C
$(1, -9)$
$(5, 1)$ Answer
E
$(5, -9)$

Answer

The coordinates of the original vertex V are (5, 1)

The correct answer is the coordinates

(5, 1)

. Starting with the original vertex

V(x, y)

, a horizontal translation of

3

units left results in

x - 3

, and a vertical translation of

4

units up results in

y + 4

, yielding the intermediate point

(x - 3, y + 4)

. Reflecting this point across the

x

-axis negates the entire y-coordinate expression, giving

(x - 3, -y - 4)

. Equating this to the final coordinates

(2, -5)

results in

x - 3 = 2

, which solves to

x = 5

, and

-y - 4 = -5

, which simplifies to

-y = -1

, or

y = 1

Step-by-Step Solution

Represent the transformations algebraically starting from the original vertex

V(x, y)

Translating

3

units left and

4

units up maps

V(x, y)

to an intermediate point

V'(x - 3, y + 4)

Horizontal translation left subtracts from the x-coordinate, and vertical translation up adds to the y-coordinate.

Apply the second transformation, reflecting the intermediate point

V'

across the

x

-axis.

Reflecting

V'(x - 3, y + 4)

across the

x

-axis negates the y-coordinate, resulting in the final image

V''(x - 3, -(y + 4)) = (x - 3, -y - 4)

A reflection across the x-axis transforms any point

(a, b)

(a, -b)

Set the algebraic coordinates of the final image equal to the given coordinates

(2, -5)

and solve for

x

and

y

x - 3 = 2 \implies x = 5

and

-y - 4 = -5 \implies -y = -1 \implies y = 1

This determines the coordinates of the original pre-image vertex

V(5, 1)

Key Concept

Performing composite transformations in reverse order to find the pre-image coordinates.

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