Question

reflect the figure over the line y = -2. plot all of the points of the reflected figure. you may click a plotted point to delete it.

Explanation:

Step1: Identify Original Points

First, find the coordinates of the original triangle's vertices. From the graph, let's assume the original points (by estimating their positions) are: Let's say the three vertices are \( A(-8, -6) \), \( B(-6, -9) \), and \( C(0, -7) \) (we need to confirm the exact coordinates by looking at the grid. Wait, actually, looking at the grid, the y-axis has -2, -3, -4, -5, -6, -7, -8, -9. Let's re-express:

Wait, the horizontal line is \( y = -2 \)? No, the blue horizontal line is \( y = -2 \)? Wait, the problem says "reflect the figure over the line \( y = -2 \)". Let's find the original points. Let's look at the pink triangle:

First vertex: Let's see, x=-8, y=-6 (since it's at x=-8, y=-6? Wait, no, the y-coordinate: the grid lines. Let's check the distance from \( y = -2 \).

Wait, the formula for reflecting a point \( (x, y) \) over the line \( y = k \) is \( (x, 2k - y) \). So \( k = -2 \), so the reflection of \( (x, y) \) is \( (x, 2*(-2) - y) = (x, -4 - y) \).

Now, let's find the original points. Let's assume the original triangle has vertices:

1. Point 1: Let's say \( (-8, -6) \) (x=-8, y=-6)
2. Point 2: \( (-6, -9) \) (x=-6, y=-9)
3. Point 3: \( (0, -7) \) (x=0, y=-7)

Step2: Apply Reflection Formula

For each point, apply \( (x, -4 - y) \):

• For \( (-8, -6) \): \( y' = -4 - (-6) = -4 + 6 = 2 \). So reflected point: \( (-8, 2) \)
• For \( (-6, -9) \): \( y' = -4 - (-9) = -4 + 9 = 5 \). So reflected point: \( (-6, 5) \)
• For \( (0, -7) \): \( y' = -4 - (-7) = -4 + 7 = 3 \). So reflected point: \( (0, 3) \)

Wait, but let's confirm the original points. Wait, maybe I made a mistake in original coordinates. Let's re-examine the graph. Let's look at the pink triangle:

First vertex: x=-8, y=-6? Wait, no, the y-coordinate: the line \( y = -2 \) is the horizontal line. Let's check the vertical distance from \( y = -2 \) to the original point.

Wait, maybe the original points are:

1. \( (-8, -6) \): distance from \( y = -2 \) is \( |-6 - (-2)| = 4 \) units below. So reflection is 4 units above \( y = -2 \), so \( y = -2 + 4 = 2 \). So \( (-8, 2) \)
2. \( (-6, -9) \): distance from \( y = -2 \) is \( |-9 - (-2)| = 7 \) units below. So reflection is 7 units above \( y = -2 \), so \( y = -2 + 7 = 5 \). So \( (-6, 5) \)
3. \( (0, -7) \): distance from \( y = -2 \) is \( |-7 - (-2)| = 5 \) units below. So reflection is 5 units above \( y = -2 \), so \( y = -2 + 5 = 3 \). So \( (0, 3) \)

So now, we need to plot these reflected points: \( (-8, 2) \), \( (-6, 5) \), and \( (0, 3) \).

Step3: Plot the Reflected Points

Now, plot each of these points:

• \( (-8, 2) \): x=-8, y=2 (so on the grid, x=-8, y=2)
• \( (-6, 5) \): x=-6, y=5 (x=-6, y=5)
• \( (0, 3) \): x=0, y=3 (x=0, y=3)

Answer:

The reflected points are \( (-8, 2) \), \( (-6, 5) \), and \( (0, 3) \). To plot them, mark these coordinates on the graph.