Question

question 10 of 10
which of the following transformations will result in an image that maps onto itself?
a. reflect across the y - axis, and then reflect across the x - axis.
b. rotate 360 degrees about the origin.
c. rotate 270 degrees counterclockwise about the origin, and the reflect it across the x - axis.
d. rotate 90 degrees counterclockwise about the origin, and then rotate 180 degrees.

Explanation:

Step1: Analyze Option A

Reflecting across \(y -\)axis and then across \(x -\)axis changes the orientation (unless the figure has both \(x\) and \(y\) - axis symmetry). For a general figure, this is a non - identity transformation.

Step2: Analyze Option B

A rotation of \(360^{\circ}\) about the origin is equivalent to the identity transformation. By the definition of a full - circle rotation, any point \((x,y)\) rotated \(360^{\circ}\) about the origin \((0,0)\) using the rotation formula \(x'=x\cos\theta - y\sin\theta\), \(y'=x\sin\theta + y\cos\theta\) with \(\theta = 360^{\circ}\) (\(\cos360^{\circ}=1\), \(\sin360^{\circ}=0\)) gives \((x',y')=(x,y)\).

Step3: Analyze Option C

A \(270^{\circ}\) counter - clockwise rotation about the origin followed by a reflection across the \(x -\)axis. A \(270^{\circ}\) counter - clockwise rotation \((x,y)\to(y, - x)\) and then reflecting across \(x -\)axis \((y, - x)\to(y,x)\) (for a non - symmetric figure about \(x\) - axis), is a non - identity transformation.

Step4: Analyze Option D

A \(90^{\circ}\) counter - clockwise rotation \((x,y)\to(-y,x)\) and then a \(180^{\circ}\) rotation \((-y,x)\to(y, - x)\) is a non - identity transformation.

Answer:

B. Rotate 360 degrees about the origin.