Question

question 4 of 10

which sequence of transformations will result in an image that maps onto itself?

a. rotate 180 degrees counterclockwise about the origin, and then reflect across the \\(y\\)-axis.
b. reflect over the \\(y\\)-axis, and then reflect over the \\(x\\)-axis.
c. reflect over the \\(y\\)-axis, and then reflect again over the \\(y\\)-axis.
d. rotate 180 degrees counterclockwise about the origin, and then reflect across the \\(x\\)-axis.

Explanation:

Analyze the identity transformation condition

An image maps onto itself if the sequence of transformations results in the identity transformation:

$$ (x, y) \to (x, y) $$

Evaluate each option using coordinate rules

• Option A: Rotate \(180^\circ\) counterclockwise about the origin, then reflect across the \(y\)-axis.

$$ (x, y) \xrightarrow{\text{Rotate } 180^\circ} (-x, -y) \xrightarrow{\text{Reflect } y\text{-axis}} (x, -y) eq (x, y) $$

• Option B: Reflect over the \(y\)-axis, then reflect over the \(x\)-axis.

$$ (x, y) \xrightarrow{\text{Reflect } y\text{-axis}} (-x, y) \xrightarrow{\text{Reflect } x\text{-axis}} (-x, -y) eq (x, y) $$

• Option C: Reflect over the \(y\)-axis, then reflect again over the \(y\)-axis.

$$ (x, y) \xrightarrow{\text{Reflect } y\text{-axis}} (-x, y) \xrightarrow{\text{Reflect } y\text{-axis}} (x, y) $$

• Option D: Rotate \(180^\circ\) counterclockwise about the origin, then reflect across the \(x\)-axis.

$$ (x, y) \xrightarrow{\text{Rotate } 180^\circ} (-x, -y) \xrightarrow{\text{Reflect } x\text{-axis}} (-x, y) eq (x, y) $$

Identify the correct sequence

Option C results in the original coordinates \((x, y)\), meaning the image maps onto itself.

Answer:

• (A) Rotate 180 degrees counterclockwise about the origin, and then reflect across the y-axis.
• (B) Reflect over the y-axis, and then reflect over the x-axis.
• (C) Reflect over the y-axis, and then reflect again over the y-axis. (Correct answer)
• (D) Rotate 180 degrees counterclockwise about the origin, and then reflect across the x-axis.