use the function rules to find the coordinates of triangle A’’ B’’ C’’ and predict the quadrant. It will be in after the transformations. triangle ABC, with coordinates A (-8, 4), B( -5, 8), and C(-3, 3), is transformed by (x, y) —> (x + 4,y -6) to produce triangle A’B’C. triangle A’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.
To find the coordinates of triangle A''B''C'', we need to apply the transformation rule to the vertices of triangle A'B'C'.
Let's start by finding the coordinates of A', B', and C'.
For A':
Applying the transformation rule (x, y) —> (x + 4, y - 6) to point A(-8, 4):
A' = (-8 + 4, 4 - 6) = (-4, -2)
For B':
Applying the transformation rule (x, y) —> (x + 4, y - 6) to point B(-5, 8):
B' = (-5 + 4, 8 - 6) = (-1, 2)
For C':
Applying the transformation rule (x, y) —> (x + 4, y - 6) to point C(-3, 3):
C' = (-3 + 4, 3 - 6) = (1, -3)
Now, let's apply the second transformation rule to find the coordinates of A''B''C''.
For A'':
Applying the transformation rule (x, y) —> (x + 5, y - 3) to point A'(-4, -2):
A'' = (-4 + 5, -2 - 3) = (1, -5)
For B'':
Applying the transformation rule (x, y) —> (x + 5, y - 3) to point B'(-1, 2):
B'' = (-1 + 5, 2 - 3) = (4, -1)
For C'':
Applying the transformation rule (x, y) —> (x + 5, y - 3) to point C'(1, -3):
C'' = (1 + 5, -3 - 3) = (6, -6)
To predict the quadrant of the image A''B''C'', we need to consider the signs of the x-coordinate and y-coordinate.
The x-coordinate of A'' is positive, and the y-coordinate is negative. Therefore, A'' is in the fourth quadrant.
The x-coordinate of B'' is positive, and the y-coordinate is negative. Therefore, B'' is in the fourth quadrant.
The x-coordinate of C'' is positive, and the y-coordinate is negative. Therefore, C'' is in the fourth quadrant.
Hence, the image triangle A''B''C'' will be in the fourth quadrant.