Question

which single transformation maps figure f onto its image? a reflection across the y-axis; a 90° clockwise rotation around the origin; a reflection across the x-axis; a translation 4 units down

Explanation:

To determine the transformation, we analyze each option:

• Reflection across y - axis: A reflection over the y - axis would change the x - coordinates' sign. But looking at the figure, the x - coordinates of the vertices of F and its image do not show this pattern. For example, a vertex of F at (1,5) (assuming grid - based coordinates) would go to (- 1,5) after y - axis reflection, which does not match the image.
• 90° clockwise rotation around origin: The rule for a 90° clockwise rotation about the origin is \((x,y)\to(y, - x)\). Let's take a vertex of F, say (1,5). After rotation, it would be (5, - 1), which does not match the coordinates of the image vertices.
• Reflection across x - axis: The rule for reflection across the x - axis is \((x,y)\to(x, - y)\). Let's consider the vertices of figure F. If we take a vertex of F with coordinates \((x,y)\), after reflecting across the x - axis, the y - coordinate becomes the negative of its original value. Looking at the figure, the image \(F''\) is a mirror image of F with respect to the x - axis. For example, if a vertex of F is at (1,5), after reflection across the x - axis, it would be at (1, - 5), which matches the position of the corresponding vertex in \(F''\).
• Translation 4 units down: A translation 4 units down would subtract 4 from the y - coordinates of all vertices of F. But the shape of the image \(F''\) is a mirror image, not just a vertical shift. For example, a vertex of F at (1,5) would go to (1,1) after a 4 - unit down translation, which does not match the image.

Answer:

A reflection across the \(x\) - axis