Question

1. what is the image point of $(-6, 7)$ after the transformation $r_{y=-x} \circ t_{-3, -5}$?
2. what is the image point of $(-2, 7)$ after the transformation $r_{x\text{-axis}} \circ d_3$?

Explanation:

Step1: Apply the translation \(T_{-3,-5}\)

For a point \((x,y)\), the translation \(T_{a,b}\) gives \((x + a,y + b)\).
For the point \((-6,7)\) and \(a=-3,b = -5\), we have \((-6-3,7 - 5)=(-9,2)\)

Step2: Apply the reflection \(r_{y=-x}\)

The rule for reflection over the line \(y=-x\) is \((x,y)\to(-y,-x)\)
For the point \((-9,2)\), we get \((-2,9)\)

Step3: Apply the dilation \(D_3\) for problem 13

The rule for dilation \(D_k\) is \((x,y)\to(kx,ky)\). For the point \((-2,7)\) and \(k = 3\), we have \((-2\times3,7\times3)=(-6,21)\)

Step4: Apply the reflection \(r_{x - axis}\)

The rule for reflection over the \(x\)-axis is \((x,y)\to(x,-y)\)
For the point \((-6,21)\), we get \((-6,-21)\)

Answer:

For problem 12: \((-2,9)\)
For problem 13: \((-6,-21)\)