Question

the vertices of trapezoid abcd are a(3,-3), b(7,-3), c(8,-8), d(2,-8). which is the graph of abcd and abcd, its image after a reflection across the y - axis.

Explanation:

Step1: Recall reflection over y - axis rule

The rule for reflecting a point \((x,y)\) across the \(y\) - axis is \((x,y)\to(-x,y)\).

Step2: Find coordinates of \(A',B',C',D'\)

• For point \(A(3,-3)\): Using the reflection rule, \(A'\) has coordinates \((- 3,-3)\).
• For point \(B(7,-3)\): Using the reflection rule, \(B'\) has coordinates \((-7,-3)\).
• For point \(C(8,-8)\): Using the reflection rule, \(C'\) has coordinates \((-8,-8)\).
• For point \(D(2,-8)\): Using the reflection rule, \(D'\) has coordinates \((-2,-8)\).

Now, let's analyze the original trapezoid \(ABCD\). The \(y\) - coordinates of \(A\) and \(B\) are \(- 3\), and the \(y\) - coordinates of \(C\) and \(D\) are \(-8\). So, \(AB\) and \(CD\) are horizontal lines (since the \(y\) - coordinates are constant for each pair). The length of \(AB\) is \(|7 - 3|=4\), and the length of \(CD\) is \(|8 - 2| = 6\).

After reflection, the image \(A'B'C'D'\) should have vertices with \(x\) - coordinates negated. So, the original trapezoid \(ABCD\) is in the fourth quadrant (since \(x>0\) and \(y<0\)) and its reflection \(A'B'C'D'\) should be in the third quadrant (since \(x<0\) and \(y<0\)).

Looking at the options:

• The green - colored option: The blue trapezoid (original) is in the fourth quadrant and the red trapezoid (reflection) is in the second quadrant (incorrect, since reflection over \(y\) - axis of fourth quadrant points should be in third quadrant).
• The purple - colored option: The blue trapezoid (original) is in the fourth quadrant and the red trapezoid (reflection) is in the first quadrant (incorrect).
• The orange - colored option: The blue trapezoid (original) is in the fourth quadrant and the red trapezoid (reflection) is in the third quadrant (correct, as the \(x\) - coordinates of the red trapezoid are negative and \(y\) - coordinates are negative, matching the reflection rule).

Answer:

The orange - colored graph (where the original trapezoid \(ABCD\) is blue in the fourth quadrant and its reflection \(A'B'C'D'\) is red in the third quadrant)