Question

write the coordinates of the vertices after a reflection over the line y = -2.

Explanation:

1. Recall the rule for reflection over the horizontal line \(y = k\):

• The rule for reflecting a point \((x,y)\) over the horizontal line \(y = k\) is \((x,2k - y)\). Here, \(k=-2\), so the transformation for a point \((x,y)\) is \((x,- 4 - y)\).

1. Assume the coordinates of the vertices:

• Let's assume the coordinates of the vertices of the quadrilateral are \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), and \(D(x_4,y_4)\). From the graph, if we assume \(A(-8,-9)\), \(B(-4,-9)\), \(C(0,-5)\), \(D(-4,-5)\).
• For point \(A(-8,-9)\):
• Using the formula \((x,-4 - y)\), we substitute \(x=-8\) and \(y = - 9\). Then \(x\) - coordinate remains the same, and the \(y\) - coordinate is \(-4-(-9)=-4 + 9 = 5\). So the new - coordinate of \(A\) is \(A'(-8,5)\).
• For point \(B(-4,-9)\):
• Substitute \(x=-4\) and \(y=-9\) into the formula \((x,-4 - y)\). The \(x\) - coordinate remains \(-4\), and the \(y\) - coordinate is \(-4-(-9)=5\). So the new - coordinate of \(B\) is \(B'(-4,5)\).
• For point \(C(0,-5)\):
• Substitute \(x = 0\) and \(y=-5\) into the formula \((x,-4 - y)\). The \(x\) - coordinate is \(0\), and the \(y\) - coordinate is \(-4-(-5)=-4 + 5 = 1\). So the new - coordinate of \(C\) is \(C'(0,1)\).
• For point \(D(-4,-5)\):
• Substitute \(x=-4\) and \(y=-5\) into the formula \((x,-4 - y)\). The \(x\) - coordinate is \(-4\), and the \(y\) - coordinate is \(-4-(-5)=1\). So the new - coordinate of \(D\) is \(D'(-4,1)\).

Answer:

The new coordinates of the vertices (assuming the original vertices as above) are \(A'(-8,5)\), \(B'(-4,5)\), \(C'(0,1)\), \(D'(-4,1)\)