Use the function rules to find the coordinates of ΔA"B"C"
and predict the quadrant it will be in after the transformations. ΔABC
, with coordinates A(−8,4)
, B(−5,8)
, and C(−3,3)
, is transformed by (x,y)→(x+4,y−6)
to produce ΔA′B′C′
. ΔA′B′C′
is transformed by the rule (x,y)→(x+5,y−3)
. Create the rule for the composition transformation and predict what quadrant the image will be in.(4 points)
To find the coordinates of ΔA'B'C', we can apply the transformation rules step by step.
Applying the first transformation rule (x,y)→(x+4,y−6) to ΔABC:
A' = (-8+4, 4-6) = (-4, -2)
B' = (-5+4, 8-6) = (-1, 2)
C' = (-3+4, 3-6) = (1, -3)
Now, applying the second transformation rule (x,y)→(x+5,y−3) to ΔA'B'C':
A" = (-4+5, -2-3) = (1, -5)
B" = (-1+5, 2-3) = (4, -1)
C" = (1+5, -3-3) = (6, -6)
The coordinates of ΔA"B"C" are A" = (1, -5), B" = (4, -1), and C" = (6, -6).
To predict the quadrant that ΔA"B"C" will be in, we can look at the signs of the x and y coordinates.
Since the x coordinates are positive and the y coordinates are negative, ΔA"B"C" will be in the fourth quadrant.
To find the coordinates of ΔA"B"C", we need to apply the given transformations to each vertex of ΔABC.
The first transformation is (x,y)→(x+4,y−6). Applying this transformation to the coordinates of A(−8,4), we get A'(-8+4,4-6) = A'(-4,-2).
Similarly, applying the transformation to the coordinates of B(-5,8), we get B'(-5+4,8-6) = B'(-1,2).
Finally, applying the transformation to the coordinates of C(-3,3), we get C'(-3+4,3-6) = C'(1,-3).
So, the coordinates of ΔA'B'C' are A'(-4,-2), B'(-1,2), and C'(1,-3).
To find the composition transformation, we need to find the rule that combines the two given transformations.
The first transformation is (x,y)→(x+4,y−6) and the second transformation is (x,y)→(x+5,y−3).
To combine these two transformations, we can simply add the corresponding values of x and y in each transformation. This gives us the composition transformation rule: (x,y)→(x+4+5,y−6-3) = (x+9,y-9).
Now, let's predict the quadrant where the image of ΔABC (ΔA'B'C') will be after the transformations.
The original triangle ΔABC is in the third quadrant (the negative x-axis and negative y-axis region).
Applying the composition transformation (x,y)→(x+9,y-9) to each vertex of ΔABC, we can determine the positions of the image vertices relative to the original triangle.
A'(-4,-2) + (9, -9) = (5, -11)
B'(-1,2) + (9, -9) = (8, -7)
C'(1,-3) + (9, -9) = (10, -12)
The image ΔA'B'C' will be in the fourth quadrant (the positive x-axis and negative y-axis region) after the transformations.
To find the coordinates of ΔA"B"C", we need to apply the given transformation rules to the original coordinates of ΔABC.
1. Applying the first transformation rule (x,y)→(x+4,y−6), we get:
A' = (-8 + 4, 4 - 6) = (-4, -2)
B' = (-5 + 4, 8 - 6) = (-1, 2)
C' = (-3 + 4, 3 - 6) = (1, -3)
These are the coordinates of ΔA'B'C' after the first transformation.
2. Now, we need to apply the second transformation rule (x,y)→(x+5,y−3) to ΔA'B'C':
A" = (-4 + 5, -2 - 3) = (1, -5)
B" = (-1 + 5, 2 - 3) = (4, -1)
C" = (1 + 5, -3 - 3) = (6, -6)
These are the coordinates of ΔA"B"C" after the composition of the two transformations.
To predict the quadrant it will be in, we need to analyze the signs of the x and y coordinates of the final points:
A": x = 1 (positive), y = -5 (negative) -> Quadrant IV
B": x = 4 (positive), y = -1 (negative) -> Quadrant IV
C": x = 6 (positive), y = -6 (negative) -> Quadrant III
Based on these signs, it can be predicted that ΔA"B"C" will be in Quadrant III.