Question

1. what is the image of a(3,-1) after a reflection, first across the line y = 3, and then across the line x = -1? (1,-5) (-5,-1) (3,-1) (-5,7)

Explanation:

Step1: Reflection across \(y = 3\)

The formula for reflection across the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k = 3\), and the point is \((3,-1)\).
So, \(x\) - coordinate remains \(x = 3\), and \(y\) - coordinate becomes \(2\times3-(-1)=6 + 1=7\). The point after reflection across \(y = 3\) is \((3,7)\).

Step2: Reflection across \(x=-1\)

The formula for reflection across the line \(x = h\) is \((x,y)\to(2h - x,y)\). Here \(h=-1\), and the point is \((3,7)\).
So, \(x\) - coordinate becomes \(2\times(-1)-3=-2 - 3=-5\), and \(y\) - coordinate remains \(y = 7\).

Answer:

\((-5,7)\)