Question

use the function rules to find the coordinates of A"B"C" and predict the quadrant it will be in after the transformations ABC with the coordinates A(-8,4), B(-5.8), and C-3,3) is transformed by (x,y)-(x+4,y-6) to produce A'B'C'. A'B'C' is transformed by the rule (x,y)-(x+3,y-2) to produce A"B"C" create the rule for the composition transformation (x,y) (____, ______)

Answer 1

I assume the original points are A(-8, 4), B(-5, 8), C(-3, 3).

First translation: (x,y) -> (x+4, y-6)
- A' = (-8+4, 4-6) = (-4, -2)
- B' = (-5+4, 8-6) = (-1, 2)
- C' = (-3+4, 3-6) = (1, -3)

Second translation: (x,y) -> (x+3, y-2)
You can apply it to A',B',C' or compose the two translations at once.

Composition: (x,y) -> (x+4+3, y-6-2) = (x+7, y-8)

Apply composition to the originals:
- A" = (-8+7, 4-8) = (-1, -4)
- B" = (-5+7, 8-8) = (2, 0)
- C" = (-3+7, 3-8) = (4, -5)

So the composition rule is (x,y) -> (x+7, y-8).

Quadrant locations:
- A" (-1, -4): Quadrant III
- B" (2, 0): on the positive x-axis (not in any quadrant)
- C" (4, -5): Quadrant IV

Thus the image triangle has vertices in Quadrant III and IV and one vertex on the x-axis.