Question

reflect the figure over the line $y = -1$. plot all of the points of the reflected figure. you may click a plotted point to delete it.

Explanation:

To solve the reflection of a figure over the line \( y = -1 \), we follow these steps for each vertex of the original figure:

Step 1: Identify the original vertices

First, we need to determine the coordinates of the vertices of the original triangle. From the graph, let's assume the vertices are:

• \( A(-9, -2) \)
• \( B(-6, -7) \)
• \( C(-4, -9) \) (Wait, actually, looking at the graph, let's re - identify the correct coordinates. Let's look at the grid:

Looking at the plotted points:

• Let's say the top vertex (the one at \( x=-9 \)): its \( y \)-coordinate is - 2, so \( (x_1,y_1)=(-9,-2) \)
• The middle vertex (at \( x = - 6 \)): \( y=-7 \), so \( (x_2,y_2)=(-6,-7) \)
• The bottom - right vertex (at \( x=-4 \)): \( y = - 9 \), and the other vertex (let's say \( D(-3,-9) \))? Wait, maybe the original figure has vertices: Let's check the graph again. The original figure is a triangle (or a quadrilateral? Let's assume it's a triangle with vertices \( (-9,-2) \), \( (-6,-7) \), and \( (-3,-9) \))

Step 2: Recall the reflection formula over a horizontal line \( y = k \)

The formula for reflecting a point \( (x,y) \) over the line \( y = k \) is \( (x,2k - y) \). In our case, \( k=-1 \), so the formula becomes \( (x,2\times(-1)-y)=(x,-2 - y) \)

Step 3: Reflect each vertex

• For the point \( (-9,-2) \):

Using the formula \( (x,-2 - y) \), substitute \( x=-9 \) and \( y = - 2 \)
\( -2-(-2)=-2 + 2=0 \)
So the reflected point is \( (-9,0) \)

• For the point \( (-6,-7) \):

Substitute \( x = - 6 \) and \( y=-7 \) into the formula \( (x,-2 - y) \)
\( -2-(-7)=-2 + 7 = 5 \)
So the reflected point is \( (-6,5) \)

• For the point \( (-3,-9) \):

Substitute \( x=-3 \) and \( y = - 9 \) into the formula \( (x,-2 - y) \)
\( -2-(-9)=-2 + 9 = 7 \)
So the reflected point is \( (-3,7) \)

To plot the reflected figure, we plot the points \( (-9,0) \), \( (-6,5) \), and \( (-3,7) \) (assuming the original figure is a triangle with the above - identified vertices). If the original figure has more vertices, we follow the same reflection formula for each vertex.

Final Answer

The reflected points (assuming the original vertices are \( (-9,-2) \), \( (-6,-7) \), \( (-3,-9) \)) are \( (-9,0) \), \( (-6,5) \), and \( (-3,7) \). We plot these points on the coordinate plane to get the reflected figure.

Answer:

To solve the reflection of a figure over the line \( y = -1 \), we follow these steps for each vertex of the original figure:

Step 1: Identify the original vertices

First, we need to determine the coordinates of the vertices of the original triangle. From the graph, let's assume the vertices are:

• \( A(-9, -2) \)
• \( B(-6, -7) \)
• \( C(-4, -9) \) (Wait, actually, looking at the graph, let's re - identify the correct coordinates. Let's look at the grid:

Looking at the plotted points:

• Let's say the top vertex (the one at \( x=-9 \)): its \( y \)-coordinate is - 2, so \( (x_1,y_1)=(-9,-2) \)
• The middle vertex (at \( x = - 6 \)): \( y=-7 \), so \( (x_2,y_2)=(-6,-7) \)
• The bottom - right vertex (at \( x=-4 \)): \( y = - 9 \), and the other vertex (let's say \( D(-3,-9) \))? Wait, maybe the original figure has vertices: Let's check the graph again. The original figure is a triangle (or a quadrilateral? Let's assume it's a triangle with vertices \( (-9,-2) \), \( (-6,-7) \), and \( (-3,-9) \))

Step 2: Recall the reflection formula over a horizontal line \( y = k \)

The formula for reflecting a point \( (x,y) \) over the line \( y = k \) is \( (x,2k - y) \). In our case, \( k=-1 \), so the formula becomes \( (x,2\times(-1)-y)=(x,-2 - y) \)

Step 3: Reflect each vertex

• For the point \( (-9,-2) \):

Using the formula \( (x,-2 - y) \), substitute \( x=-9 \) and \( y = - 2 \)
\( -2-(-2)=-2 + 2=0 \)
So the reflected point is \( (-9,0) \)

• For the point \( (-6,-7) \):

Substitute \( x = - 6 \) and \( y=-7 \) into the formula \( (x,-2 - y) \)
\( -2-(-7)=-2 + 7 = 5 \)
So the reflected point is \( (-6,5) \)

• For the point \( (-3,-9) \):

Substitute \( x=-3 \) and \( y = - 9 \) into the formula \( (x,-2 - y) \)
\( -2-(-9)=-2 + 9 = 7 \)
So the reflected point is \( (-3,7) \)

To plot the reflected figure, we plot the points \( (-9,0) \), \( (-6,5) \), and \( (-3,7) \) (assuming the original figure is a triangle with the above - identified vertices). If the original figure has more vertices, we follow the same reflection formula for each vertex.

Final Answer

The reflected points (assuming the original vertices are \( (-9,-2) \), \( (-6,-7) \), \( (-3,-9) \)) are \( (-9,0) \), \( (-6,5) \), and \( (-3,7) \). We plot these points on the coordinate plane to get the reflected figure.