Question

graph the image of $\triangle def$ after a reflection over the $x$-axis.

Explanation:

Step1: Identify coordinates of vertices

First, find the coordinates of \( D \), \( E \), and \( F \) from the graph.

• \( D \): \( (2, -7) \) (since it's 2 units right on x - axis and 7 units down on y - axis)
• \( E \): \( (7, -7) \) (7 units right on x - axis and 7 units down on y - axis)
• \( F \): \( (-3, -3) \) (3 units left on x - axis and 3 units down on y - axis)

Step2: Apply reflection over x - axis rule

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

• For \( D(2, -7) \): New \( y \) - coordinate is \( -(-7)=7 \), so \( D'=(2, 7) \)
• For \( E(7, -7) \): New \( y \) - coordinate is \( -(-7)=7 \), so \( E'=(7, 7) \)
• For \( F(-3, -3) \): New \( y \) - coordinate is \( -(-3)=3 \), so \( F'=(-3, 3) \)

Step3: Plot the reflected points

Plot \( D'(2, 7) \), \( E'(7, 7) \), and \( F'(-3, 3) \) on the coordinate plane and connect them to form the reflected triangle \( \triangle D'E'F' \).

Answer:

The image of \( \triangle DEF \) after reflection over the x - axis has vertices \( D'(2, 7) \), \( E'(7, 7) \), and \( F'(-3, 3) \). The graph is formed by plotting these points and connecting them. (To graph, mark the points \( (2,7) \), \( (7,7) \), and \( (-3,3) \) and draw the triangle.)