Question

point ( f ) is the image when point ( f ) is reflected over the line ( x = -2 ) and then over the line ( y = 3 ). the location of ( f ) is ( (3,7) ). which of the following is the location of point ( f )?
o a. ( (-7,7) )
o b. ( (1,7) )
o c. ( (-7,-1) )
o d. ( (1,5) )

Explanation:

Step1: Find the point before reflection over \(y = 3\)

The formula for reflecting a point \((x,y)\) over the line \(y = k\) is \((x,2k - y)\).
Let the point before reflection over \(y=3\) be \((x_1,y_1)\). Given the reflected point \((3,7)\) over \(y = 3\), we have \(y_1=2\times3 - 7=-1\), and \(x_1 = 3\). So the point is \((3,-1)\).

Step2: Find the original point \(F\)

The formula for reflecting a point \((x,y)\) over the line \(x = h\) is \((2h - x,y)\).
Let the original point \(F\) be \((x_0,y_0)\). Given the point \((3,-1)\) is the result of reflection over \(x=-2\), we have \(x_0=2\times(-2)-3=-7\), and \(y_0=-1\).

Answer:

C. \((-7,-1)\)