Question

what is the image point of $(-1,-3)$ after the transformation $r_{y=-x}circ t_{3,4}$?

Explanation:

Step1: Translation

First, perform the translation \(T_{3,4}\). The rule for translation \(T_{a,b}\) is \((x,y)\to(x + a,y + b)\).
For the point \((-1,-3)\) and \(a = 3,b = 4\), we have \((-1+3,-3 + 4)=(2,1)\).

Step2: Reflection

Next, perform the reflection \(r_{y=-x}\). The rule for reflection over the line \(y=-x\) is \((x,y)\to(-y,-x)\).
For the point \((2,1)\), we substitute \(x = 2\) and \(y = 1\) into the rule: \((-1,-2)\).

Answer:

\((-1,-2)\)