Question

which series of transformations occurred to move the down arrow figure to the left arrow figure? a translation and then a 180° rotation; a translation and then a 90° clockwise rotation; a reflection over the x - axis and then a 180° rotation; a reflection over the x - axis and then a 90° clockwise rotation

Explanation:

Step1: Analyze the down arrow (red) and left arrow (blue)

The down arrow (red) has a vertical orientation (pointing down), the left arrow (blue) has a horizontal orientation (pointing left). Let's check transformations:
- A 90° clockwise rotation of a down - pointing arrow: A down - pointing arrow (let's consider its shape as a rectangle with a triangle at the bottom) when rotated 90° clockwise, the triangle (the "point") will face left, which matches the left - pointing arrow. But we also need to check if translation is involved? Wait, no, let's check the other options. Wait, no, first, let's check the reflection and rotation. Wait, no, let's think about the steps.
Wait, the down arrow (red) is at x = - 4 (approximate center) and the left arrow (blue) is at x = 2 (approximate center). Wait, no, maybe first, let's check the rotation. A 90° clockwise rotation of a down - pointing figure: if we have a figure pointing down, rotating 90° clockwise (around a center) will make it point left. But also, is there a translation? Wait, no, maybe the correct series is a translation and then a 90° clockwise rotation? Wait, no, let's re - examine.
Wait, the red arrow is down - pointing, the blue is left - pointing. Let's consider the transformation:
- A 90° clockwise rotation of the down - pointing arrow: When you rotate a down - pointing arrow 90° clockwise, its direction becomes left - pointing. But also, is there a translation? Wait, the red arrow is on the left side of the y - axis, the blue is on the right side? Wait, no, the red arrow is at x=-4 (center), blue at x = 2 (center). So maybe first, a translation (moving the red arrow towards the blue's position) and then a 90° clockwise rotation? Wait, no, let's check the options.
Wait, the options are:
1. a translation and then a 180° rotation: A 180° rotation of a down - pointing arrow would make it point up, not left. So this is wrong.
2. a translation and then a 90° clockwise rotation: A 90° clockwise rotation of a down - pointing arrow makes it point left. And translation can move the figure to the correct position.
3. a reflection over the x - axis and then a 180° rotation: Reflecting a down - pointing arrow over the x - axis makes it point up, then a 180° rotation would make it point down again. Wrong.
4. a reflection over the x - axis and then a 90° clockwise rotation: Reflecting over x - axis (down→up), then 90° clockwise rotation (up→right). Wrong.

Wait, maybe I made a mistake. Wait, the red arrow is down - pointing, blue is left - pointing. Let's consider the center of the figure. Let's take a point on the red arrow, say the tip (the bottom of the triangle) is at ( - 4, 2). The tip of the blue arrow (the left - most point of the triangle) is at (2, 2). Wait, no, the red arrow's tip is at ( - 4, 2) (down - pointing), the blue arrow's tip is at (2, 2) (left - pointing). Wait, if we translate the red arrow 6 units to the right (from x=-4 to x = 2) and then rotate 90° clockwise: rotating a down - pointing arrow 90° clockwise around its center (now at (2, 2)) will make it point left. That matches. So the correct series is a translation and then a 90° clockwise rotation.

Step2: Eliminate other options

- Option 1 (translation and 180° rotation): 180° rotation of down - pointing arrow gives up - pointing, not left - pointing. Eliminate.
- Option 3 (reflection over x - axis and 180° rotation): Reflection over x - axis (down→up), 180° rotation (up→down). Eliminate.
- Option 4 (reflection over x - axis and 90° clockwise rotation): Reflection over x - axis (down→up), 90° clockwise rotation (up→right). Eliminate.
- Option 2 (translation and 90° clockwise rotation): Translation (to move the figure to the correct position) and then 90° clockwise rotation (to change the direction from down to left) works.

Answer:

a translation and then a 90° clockwise rotation (the option: "a translation and then a 90° clockwise rotation")