Question

graph the image of rectangle abcd after a reflection over the x-axis.

Explanation:

Step1: Identify coordinates of ABCD

First, find the coordinates of the vertices of rectangle \(ABCD\). From the graph:

• \(A\): Let's assume the grid. Looking at the points, \(A\) is at \((-8, -9)\) (wait, no, looking at the y-axis, the bottom is -10, so let's check again. Wait, the rectangle: \(A\) is at \((-8, -9)\)? Wait, no, the y-coordinate for \(A\) and \(B\) is -9? Wait, no, the grid lines: each square is 1 unit. Let's see: \(D\) and \(C\) are at y = -3? Wait, no, the original rectangle: \(D\) is at \((-8, -3)\)? Wait, no, the blue rectangle: \(A\) is at \((-8, -9)\), \(B\) at \((-4, -9)\), \(C\) at \((-4, -3)\), \(D\) at \((-8, -3)\)? Wait, no, the y-axis: the bottom is -10, so the points:

Wait, let's look at the coordinates:

• \(A\): x = -8, y = -9 (since it's on the line y = -9, x = -8)
• \(B\): x = -4, y = -9 (same y as A, x = -4)
• \(C\): x = -4, y = -3 (same x as B, y = -3)
• \(D\): x = -8, y = -3 (same x as D, y = -3)

Wait, no, maybe I misread. Wait, the original rectangle: \(A\) is at \((-8, -9)\)? Wait, the y-axis has -10 at the bottom. Let's check the vertical distance. From \(A\) to \(B\) is horizontal, \(B\) to \(C\) is vertical, \(C\) to \(D\) is horizontal, \(D\) to \(A\) is vertical.

Alternatively, maybe the coordinates are:

• \(A(-8, -9)\)
• \(B(-4, -9)\)
• \(C(-4, -3)\)
• \(D(-8, -3)\)

Step2: Reflect over x-axis

The rule for reflection over the x-axis is \((x, y) \to (x, -y)\). So we apply this to each vertex:

• For \(A(-8, -9)\): Reflect to \(A'(-8, -(-9)) = (-8, 9)\)
• For \(B(-4, -9)\): Reflect to \(B'(-4, -(-9)) = (-4, 9)\)
• For \(C(-4, -3)\): Reflect to \(C'(-4, -(-3)) = (-4, 3)\)
• For \(D(-8, -3)\): Reflect to \(D'(-8, -(-3)) = (-8, 3)\)

Step3: Plot the reflected points

Now, plot the points \(A'(-8, 9)\), \(B'(-4, 9)\), \(C'(-4, 3)\), \(D'(-8, 3)\) and connect them to form the reflected rectangle.

Answer:

The reflected rectangle has vertices at \(A'(-8, 9)\), \(B'(-4, 9)\), \(C'(-4, 3)\), \(D'(-8, 3)\). (To graph, plot these points and connect them in order.)