Question

1. what is the image of $(-7,-2)$ after a reflection over the $y$-axis?
2. what is the image of $(-4,-2)$ after a reflection over the line $y=-x$?
3. what is the image of $(-6,9)$ after a reflection over the $y$-axis?
4. what is the image of $(7,8)$ after a reflection over the line $y=-x$?
5. what is the image of $(-8,2)$ after a reflection the $y$-axis?

Explanation:

Step1: Reflection over y - axis rule

When reflecting a point \((x,y)\) over the \(y\) - axis, the rule is \((x,y)\to(-x,y)\). For the point \((-7,-2)\), applying the rule: \(x=-7\), so \(-x = 7\), and \(y=-2\) remains the same.

Step2: Reflection over \(y=-x\) rule

When reflecting a point \((x,y)\) over the line \(y =-x\), the rule is \((x,y)\to(-y,-x)\). For the point \((-4,-2)\), \(x=-4\) and \(y =-2\). Then \(-y=2\) and \(-x = 4\).

Step3: Reflection over y - axis rule (again)

For the point \((-6,9)\) over the \(y\) - axis, using \((x,y)\to(-x,y)\), \(x=-6\) gives \(-x = 6\) and \(y = 9\) remains.

Step4: Reflection over \(y=-x\) rule (again)

For the point \((7,8)\) over \(y=-x\), using \((x,y)\to(-y,-x)\), \(x = 7\) and \(y=8\), so \(-y=-8\) and \(-x=-7\).

Step5: Reflection over y - axis rule (third time)

For the point \((-8,2)\) over the \(y\) - axis, using \((x,y)\to(-x,y)\), \(x=-8\) gives \(-x = 8\) and \(y = 2\) remains.

Answer:

1. \((7,-2)\)
2. \((2,4)\)
3. \((6,9)\)
4. \((-8,-7)\)
5. \((8,2)\)