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.
To find the coordinates of ΔA"B"C", we need to apply the given transformations to the coordinates of ΔABC.
First, we apply the transformation (x,y)→(x+4,y−6) to each vertex of ΔABC:
A' = (-8+4, 4-6) = (-4, -2)
B' = (-5+4, 8-6) = (-1, 2)
C' = (-3+4, 3-6) = (1, -3)
Then, we apply the second transformation (x,y)→(x+5,y−3) to each vertex of 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 the image ΔA"B"C" will be in, we can look at the signs of the x and y coordinates. Since the x-coordinate of all the vertices is positive and the y-coordinate is negative, the image will be in the fourth quadrant.
To find the coordinates of ΔA"B"C", we need to apply the given function rules to each vertex of ΔABC.
First, let's apply the rule (x,y) → (x+4, y-6) to each coordinate of ΔABC:
A' = (−8+4,4-6) = (−4,-2)
B' = (−5+4,8-6) = (−1,2)
C' = (−3+4,3-6) = (1,-3)
Now, let's apply the second rule (x,y) → (x+5, y-3) to each coordinate of ΔA'B'C':
A" = (−4+5,-2-3) = (1,-5)
B" = (−1+5,2-3) = (4,-1)
C" = (1+5,-3-3) = (6,-6)
Therefore, the coordinates of ΔA"B"C" are A"(1,-5), B"(4,-1), and C"(6,-6).
To predict the quadrant in which the image ΔA"B"C" will be, we need to analyze the signs of the x and y coordinates. In the coordinate plane, the quadrants are defined as follows:
1st Quadrant: x > 0, y > 0
2nd Quadrant: x < 0, y > 0
3rd Quadrant: x < 0, y < 0
4th Quadrant: x > 0, y < 0
Analyzing the coordinates of ΔA"B"C", we see that the x and y values are both positive for each vertex, (1,-5), (4,-1), and (6,-6). Therefore, the image ΔA"B"C" will be in the 1st Quadrant.
To find the coordinates of ΔA"B"C", we need to apply the given function rules to the original coordinates of ΔABC. Let's go step by step:
1. Transformation 1: (x, y) → (x+4, y-6)
Apply this rule to each vertex of ΔABC:
A(-8, 4) → A'(-8+4, 4-6) → A'(-4, -2)
B(-5, 8) → B'(-5+4, 8-6) → B'(-1, 2)
C(-3, 3) → C'(-3+4, 3-6) → C'(1, -3)
So, the transformed coordinates of ΔABC are:
A'(-4, -2), B'(-1, 2), C'(1, -3)
2. Transformation 2: (x, y) → (x+5, y-3)
Now apply this rule to the transformed coordinates of ΔABC:
A'(-4, -2) → A"(-4+5, -2-3) → A"(1, -5)
B'(-1, 2) → B'(-1+5, 2-3) → B'(4, -1)
C'(1, -3) → C"(1+5, -3-3) → C"(6, -6)
So, the transformed coordinates of ΔA'B'C' are:
A"(1, -5), B'(4, -1), C"(6, -6)
Now, let's predict the quadrant the image ΔA"B"C" will be in after the transformations.
Looking at the x-coordinate of point A", which is 1, we can see that it is positive. In quadrant I, the x-coordinate is positive, and in quadrant IV, the x-coordinate is negative. Since the x-coordinate of A" is positive, the image will be in quadrant I or II.
Looking at the y-coordinate of point A", which is -5, we can see that it is negative. In quadrants III and IV, the y-coordinate is negative. So, the image ΔA"B"C" will be in quadrant III.
Therefore, the image ΔA"B"C" will be in quadrant III after the composition transformation.