Question

1. which of the following transformations maps figure 1 to figure 2? translate (x,y)→(x + 7,y - 8), then reflect over the y - axis rotate 180°

Explanation:

Step1: Analyze translation

If we consider the translation $(x,y)\to(x + 7,y - 8)$. For a point $(x_1,y_1)$ on figure 1, after translation, its new - coordinates will be $(x_1 + 7,y_1-8)$. Then reflecting over the y - axis changes the x - coordinate sign, so $(x_1 + 7,y_1-8)$ becomes $(-(x_1 + 7),y_1-8)$.

Step2: Analyze rotation

A $180^{\circ}$ rotation about the origin of a point $(x,y)$ has the transformation rule $(x,y)\to(-x,-y)$.
Let's take a key - point on figure 1, say the top - left corner of the green shape at approximately $(-6,6)$.
For the translation and reflection:
First, after translation $(x,y)\to(x + 7,y - 8)$, the point $(-6,6)$ becomes $(-6 + 7,6 - 8)=(1,-2)$. Then reflecting over the y - axis, it becomes $(-1,-2)$.
For the $180^{\circ}$ rotation:
The point $(-6,6)$ after a $180^{\circ}$ rotation about the origin becomes $(6,-6)$.
By observing the orientation and position of figure 2 relative to figure 1, we can see that a $180^{\circ}$ rotation about the origin maps figure 1 to figure 2.

Answer:

Rotate $180^{\circ}$