Find a coordinate rule for a reflection across the line y = 2x. Then use the rule to find the image of triangle ABC

= [ 2 6 0 ]
[ 3 -4 4 ].
Be sure to clearly show your work indicating how you found your rule. In addition write the coordinate matrix for triangle A'B'C'. Graph and label both the pre-image and the image triangle on graph paper.

To find a coordinate rule for a reflection across the line y = 2x, we can follow these steps:

Step 1: Determine the equation of the reflection line.
In this case, the equation of the reflection line is y = 2x.

Step 2: Find the slope of the reflection line.
The slope of the reflection line can be found by comparing the equation y = mx + b with y = 2x. In this case, the slope (m) is 2.

Step 3: Find the negative reciprocal of the slope.
The negative reciprocal of 2 is -1/2.

Step 4: Write the reflection rule in point-slope form.
The reflection rule in point-slope form is y = mx + b, where m is the negative reciprocal of the slope and b is the y-intercept. Since the line y = 2x does not intersect the y-axis, the reflection rule can be simplified to y = -1/2x.

Now that we have the reflection rule, we can find the image of triangle ABC = [2 6 0] [3 -4 4] by applying the reflection rule to each coordinate point.

For point A (2, 6, 0):
x' = x - 2y = 2 - 2(6) = -10
y' = y - 2x = 6 - 2(2) = 2
z' = z remains the same = 0

Therefore, the image of point A is A'(-10, 2, 0).

For point B (3, -4, 4):
x' = x - 2y = 3 - 2(-4) = 11
y' = y - 2x = -4 - 2(3) = -10
z' = z remains the same = 4

Therefore, the image of point B is B'(11, -10, 4).

Now, for point C (0, 0, 0):
x' = x - 2y = 0 - 2(0) = 0
y' = y - 2x = 0 - 2(0) = 0
z' = z remains the same = 0

Therefore, the image of point C is C'(0, 0, 0).

The coordinate matrix for the image triangle A'B'C' is:
A' = [-10 2 0]
B' = [ 11 -10 4]
C' = [ 0 0 0]

To graph and label both the pre-image and the image triangle on graph paper, plot the points A(2, 6, 0), B(3, -4, 4), and C(0, 0, 0) for the pre-image triangle. Then plot the points A'(-10, 2, 0), B'(11, -10, 4), and C'(0, 0, 0) for the image triangle. Connect the points with lines to form the triangles.