Question

which single transformation maps figure f onto its image? a reflection across the x-axis how can you map figure f onto its image using a reflection and a rotation? complete the statement. figure f is reflected across the x-axis then rotated ? around the or 90° clockwise 90° counterclockwise 180° counterclockwise

Explanation:

First Sub - Question (Which single transformation maps figure F onto its image?)

To determine the single transformation, we analyze the position of figure F and its image. A reflection across the x - axis changes the sign of the y - coordinate of each point \((x,y)\) to \((x, - y)\). By observing the coordinates of the vertices of figure F and its image, we can see that a reflection across the x - axis maps figure F onto its image.

1. First, we consider the reflection across the x - axis. After reflecting figure F across the x - axis, we then need to rotate the resulting figure.
2. When we reflect figure F across the x - axis, and then rotate the reflected figure \(180^{\circ}\) counterclockwise around the origin, we can map it onto the final image. Let's recall the rotation rules: A \(180^{\circ}\) counterclockwise (or clockwise) rotation around the origin transforms a point \((x,y)\) to \((-x,-y)\). After the reflection across the x - axis (which gives \((x, - y)\)), a \(180^{\circ}\) counterclockwise rotation will transform \((x, - y)\) to \((-x,y)\) (wait, no, let's correct: The reflection across x - axis: \((x,y)\to(x, - y)\). Then a \(180^{\circ}\) counterclockwise rotation around the origin: \((x, - y)\to(-x,y)\)? No, the correct rule for \(180^{\circ}\) rotation (clockwise or counterclockwise) around the origin is \((x,y)\to(-x,-y)\). Wait, maybe we made a mistake. Let's re - examine. If we first reflect across x - axis: \((x,y)\to(x, - y)\). Then, a \(180^{\circ}\) counterclockwise rotation: the formula for \(180^{\circ}\) counterclockwise rotation is \((x,y)\to(-x,-y)\). So applying it to \((x, - y)\), we get \((-x,y)\). But maybe by looking at the figure, after reflecting across x - axis, a \(180^{\circ}\) counterclockwise rotation around the origin will map the figure to its image. Alternatively, let's think about the orientation. After reflection across x - axis, a \(180^{\circ}\) counterclockwise rotation (which is the same as \(180^{\circ}\) clockwise in terms of the final position for some figures) will align the figure. From the given options, the correct rotation after reflection across x - axis is \(180^{\circ}\) counterclockwise.

Answer:

A reflection across the x - axis

Second Sub - Question (How to map figure F onto its image using reflection and rotation)