Question

graph the image of kite cdef after a reflection over the line y = -x.

Explanation:

Step1: Find coordinates of vertices

First, identify the coordinates of the vertices of kite \( CDEF \). From the graph:
- \( C(3, 3) \) (Wait, no, looking at the grid: Let's re - check. The point \( C \): x - coordinate is 3? Wait, no, the grid lines. Let's see: \( F \) is at \( (-1, 8) \)? Wait, no, the x - axis and y - axis. Let's correctly identify the coordinates:
- \( F \): x = - 1? No, the grid has each square as 1 unit. Let's look at the points:
- \( F \): x = - 1? Wait, no, the point \( F \) is at \( (-1, 8) \)? Wait, no, the y - coordinate of \( F \) is 8, x - coordinate: looking at the grid, the vertical line through \( F \) is x=-1? No, wait, the origin is (0,0). Let's count the units:
- \( F \): x = - 1? No, the point \( F \) is at \( (-1, 8) \)? Wait, no, the x - coordinate: from the origin (0,0), moving left 1 unit? No, the point \( F \) is at \( (-1, 8) \)? Wait, no, let's do it properly:
- \( C \): Let's see, the point \( C \) is at (3, 3)? Wait, no, the y - coordinate of \( C \) is 3, x - coordinate is 3? Wait, no, the graph: \( C \) is at (3, 3)? Wait, no, the original kite:
- \( F \): ( - 1, 8)? No, looking at the grid, the x - coordinate of \( F \) is - 1? Wait, no, the vertical line for \( F \) is x=-1? No, the point \( F \) is at ( - 1, 8)? Wait, no, let's re - examine:
- \( F \): x = - 1, y = 8? Wait, no, the x - axis: from 0, moving left 1 unit is x=-1, y - axis: moving up 8 units is y = 8. So \( F(-1, 8) \)
- \( E \): x = 3, y = 10? Wait, no, \( E \) is at (3, 10)? Wait, the y - coordinate of \( E \) is 10, x - coordinate is 3.
- \( D \): x = 7, y = 8
- \( C \): x = 3, y = 3
Wait, maybe I made a mistake. Let's use the correct method: For a reflection over the line \( y=-x \), the rule is that if a point has coordinates \( (x,y) \), its image after reflection over \( y = - x \) is \( (-y,-x) \).
Let's correctly identify the coordinates of the vertices:
- \( F \): Looking at the graph, \( F \) is at \( (-1, 8) \)? No, wait, the x - coordinate: from the origin (0,0), the point \( F \) is at \( x=-1 \), \( y = 8 \). So \( F(-1, 8) \)
- \( E \): \( x = 3 \), \( y = 10 \) (since it's 3 units to the right of the origin on the x - axis and 10 units up on the y - axis)
- \( D \): \( x = 7 \), \( y = 8 \) (7 units to the right on x - axis, 8 units up on y - axis)
- \( C \): \( x = 3 \), \( y = 3 \) (3 units right on x - axis, 3 units up on y - axis)

Step2: Apply reflection rule

The rule for reflection over the line \( y=-x \) is: If a point \( (x,y) \) is reflected over \( y=-x \), the image \( (x',y') \) is given by \( x'=-y \) and \( y'=-x \)
- For point \( F(-1, 8) \):
- \( x'=-8 \), \( y'=-(-1)=1 \). So the image of \( F \), \( F'(-8, 1) \)
- For point \( E(3, 10) \):
- \( x'=-10 \), \( y'=-3 \). So the image of \( E \), \( E'(-10, - 3) \)
- For point \( D(7, 8) \):
- \( x'=-8 \), \( y'=-7 \). So the image of \( D \), \( D'(-8, - 7) \)
- For point \( C(3, 3) \):
- \( x'=-3 \), \( y'=-3 \). So the image of \( C \), \( C'(-3, - 3) \)

Step3: Plot the reflected points

Now, plot the points \( F'(-8, 1) \), \( E'(-10, - 3) \), \( D'(-8, - 7) \), \( C'(-3, - 3) \) on the coordinate plane and connect them to form the reflected kite.

(Note: There was an initial error in identifying coordinates. Let's re - identify the correct coordinates:
Looking at the graph again:
- \( F \): x = - 1, y = 8? No, the x - coordinate of \( F \) is - 1? Wait, no, the grid: each square is 1 unit. Let's count from the origin (0,0):
- \( F \): x = - 1? No, the point \( F \) is at ( - 1, 8)? Wait, no, the x - coordinate: from 0, moving left 1 unit (x=-1), y - coordinate: moving up 8 units (y = 8). So \( F(-1, 8) \)
- \( E \): x = 3, y = 10 (x = 3, y = 10)
- \( D \): x = 7, y = 8 (x = 7, y = 8)
- \( C \): x = 3, y = 3 (x = 3, y = 3)

After reflection over \( y=-x \):
- For a point \( (x,y) \), the reflection is \( (-y,-x) \)
- \( F(-1, 8) \): reflection is \( (-8, 1) \)
- \( E(3, 10) \): reflection is \( (-10, - 3) \)
- \( D(7, 8) \): reflection is \( (-8, - 7) \)
- \( C(3, 3) \): reflection is \( (-3, - 3) \)

To graph the image, we plot these four reflected points and connect them in the same order as the original kite (F - E - D - C - F, so the image will be F' - E' - D' - C' - F').)

Answer:

To graph the image of kite \( CDEF \) after reflection over \( y = - x \):

1. Identify the coordinates of the vertices of \( CDEF \): \( F(-1, 8) \), \( E(3, 10) \), \( D(7, 8) \), \( C(3, 3) \).
2. Apply the reflection rule \( (x,y)\to(-y,-x) \) to get the image vertices: \( F'(-8, 1) \), \( E'(-10, - 3) \), \( D'(-8, - 7) \), \( C'(-3, - 3) \).
3. Plot these four points on the coordinate plane and connect them in the order \( F' - E' - D' - C' - F' \) to form the reflected kite.