Question

determine the coordinates of ( a^{prime prime} ) after a reflection of pre - image ( a(7,-6) ) across the line ( y = x ) followed by a reflection across the line ( x=-3 ). graph ( a^{prime prime} ) below.

Explanation:

Step1: Reflect across \( y = x \)

When reflecting a point \((x,y)\) across the line \( y = x \), the rule is \((x,y)\to(y,x)\).
For point \(A(7,-6)\), after reflection across \(y = x\), we get \(A'(-6,7)\).

Step2: Reflect across \( x=-3 \)

The formula for reflecting a point \((x,y)\) across the vertical line \(x = a\) is \((x,y)\to(2a - x,y)\).
Here \(a=-3\) and the point is \(A'(-6,7)\).
Substitute into the formula: \(2\times(-3)-(-6)=-6 + 6=0\).
So the new \(x\)-coordinate is \(0\) and the \(y\)-coordinate remains \(7\).

Answer:

\((0,7)\)