Question

the endpoints of (overline{pq}) are (p(-4,6)) and (q(3,-7)). find the endpoints of the image of (overline{pq}) after the composition.
translation: ((x,y)\to(x - 8,y))
translation: ((x,y)\to(x + 1,y + 6))
after the composition, the final image of (overline{pq}), (overline{pq}), has endpoints (p) at ((-11, 12)) and (q) at (\begin{array}{l}(12, -1)\\(-5, -7)\\(-4, -1)\\(4, -9)\\(4, -1)end{array})

Explanation:

Step1: Analyze the composition of translations

First, we have two translations. The first translation is \((x,y)\to(x - 8,y)\) and the second is \((x,y)\to(x + 1,y + 6)\). We can combine these two translations into a single composition. For a point \((x,y)\), after the first translation, the coordinates become \((x-8,y)\), and then after the second translation, we substitute \(x'=x - 8\) and \(y'=y\) into the second translation formula. So the composition of the two translations is \((x,y)\to((x - 8)+1,y + 6)=(x-7,y + 6)\).

Step2: Apply the composition to point \(Q(3,-7)\)

We apply the combined translation \((x,y)\to(x-7,y + 6)\) to the point \(Q(3,-7)\).
For the \(x\)-coordinate: \(x=3\), so \(x-7=3 - 7=-4\)
For the \(y\)-coordinate: \(y=-7\), so \(y + 6=-7+6=-1\)

Answer:

\((-4,-1)\)