A drone starts at position P on a grid. The drone's path is programmed with a sequence of … — ACT

A drone starts at position $P$ on a grid. The drone's path is programmed with a sequence of two movements: first, it is translated $6$ units to the right and $5$ units up, and then its position is reflected across the $y$ -axis. If the drone's final position is $(-4, 8)$ , what were the coordinates of its starting position $P$ ?

A
$(10, 3)$
B
$(-2, 13)$
$(-2, 3)$ Answer
D
$(10, 13)$
E
$(-10, -13)$

Answer

The correct answer is

(-2, 3)

because working backward from the final position

(-4, 8)

by undoing the reflection across the

y

-axis gives

(4, 8)

, and then undoing the translation by subtracting

6

from the

x

-coordinate and

5

from the

y

-coordinate yields

(-2, 3)

The correct answer is

(-2, 3)

because we can find the starting position by working backward from the final position. The last transformation applied was a reflection across the

y

-axis. Undoing this reflection maps the final coordinates

(-4, 8)

(4, 8)

because reflecting across the

y

-axis changes the sign of the

x

-coordinate. The first transformation was a translation of

6

units to the right and

5

units up. To undo this translation, we subtract

6

from the

x

-coordinate and subtract

5

from the

y

-coordinate of

(4, 8)

, giving

(4 - 6, 8 - 5) = (-2, 3)

Step-by-Step Solution

Identify the final transformation and undo it. The final transformation is a reflection across the

y

-axis, which maps

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

The coordinates before reflection are

(4, 8)

Undoing a reflection across the

y

-axis on

(-4, 8)

means changing the sign of the

x

-coordinate:

-(-4) = 4

, while keeping the

y

-coordinate the same.

Identify the first transformation and undo it. The first transformation is a translation of

6

units right and

5

units up, which maps

(x, y) \rightarrow (x+6, y+5)

The starting coordinates

P

are

(-2, 3)

To undo the translation, subtract

6

from the

x

-coordinate and

5

from the

y

-coordinate of the intermediate point

(4, 8)

4 - 6 = -2

and

8 - 5 = 3

Key Concept

Transformations in the Coordinate Plane

Alternative Method

Instead of working backward, you can test the options by applying the transformations forward. For

(-2, 3)

, translating 6 units right and 5 units up gives

(-2+6, 3+5) = (4, 8)

. Reflecting across the

y

-axis then gives

(-4, 8)

, which matches the final position.

Estimated Time:1m 15s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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