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’ 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 need to apply the transformation rules to each of the vertices of ΔABC.
For point A:
(x, y) → (x + 4, y - 6)
(-8, 4) → (-8 + 4, 4 - 6) → (-4, -2)
For point B:
(x, y) → (x + 4, y - 6)
(-5, 8) → (-5 + 4, 8 - 6) → (-1, 2)
For point C:
(x, y) → (x + 4, y - 6)
(-3, 3) → (-3 + 4, 3 - 6) → (1, -3)
The coordinates of ΔA"B"C" are A"(-4, -2), B"(-1, 2), and C"(1, -3).
Next, we need to apply the transformation rule (x, y) → (x + 5, y - 3) to ΔA"B"C".
For point A":
(x, y) → (x + 5, y - 3)
(-4, -2) → (-4 + 5, -2 - 3) → (1, -5)
For point B":
(x, y) → (x + 5, y - 3)
(-1, 2) → (-1 + 5, 2 - 3) → (4, -1)
For point C":
(x, y) → (x + 5, y - 3)
(1, -3) → (1 + 5, -3 - 3) → (6, -6)
The coordinates of ΔA"B"C" are A"(1, -5), B"(4, -1), and C"(6, -6).
From the coordinates of the transformed triangle, we can predict the quadrant it will be in. Since all three vertices have positive x-values and negative y-values, ΔA"B"C" will likely be in Quadrant IV. However, please note that the exact quadrant also depends on the scale of the graph and the position of the original triangle.
To find the coordinates of ΔA"B"C", we can apply the given function rules to the coordinates of ΔABC.
First, let's apply the transformation rule (x,y)→(x+4,y-6) to each vertex of ΔABC:
Vertex A(−8,4): (−8+4,4-6) = (-4,-2)
Vertex B(−5,8): (−5+4,8-6) = (-1,2)
Vertex C(−3,3): (−3+4,3-6) = (1,-3)
So, the coordinates of ΔA"B"C" are A"(-4,-2), B"(-1,2), and C"(1,-3).
Now, let's apply the second transformation rule (x,y)→(x+5,y-3) to each vertex of ΔA"B"C":
Vertex A"(-4,-2): (-4+5,-2-3) = (1, -5)
Vertex B"(-1,2): (-1+5,2-3) = (4, -1)
Vertex C"(1,-3): (1+5,-3-3) = (6, -6)
The coordinates of ΔA"B"C" after the composition transformation are A"(1, -5), B"(4, -1), and C"(6, -6).
To predict the quadrant it will be in after the transformations, we need to look at the signs of the x and y coordinates. Based on the coordinates we found, the image ΔA"B"C" will be in the fourth quadrant.
To find the coordinates of ΔA"B"C" after the given transformations, we need to apply the transformation rules to each vertex of ΔABC.
First, let's apply the first transformation rule (x, y) → (x + 4, y - 6) to each coordinate of ΔABC:
A' = (x + 4, y - 6) = (-8 + 4, 4 - 6) = (-4, -2)
B' = (x + 4, y - 6) = (-5 + 4, 8 - 6) = (-1, 2)
C' = (x + 4, y - 6) = (-3 + 4, 3 - 6) = (1, -3)
Now, let's apply the second transformation rule (x, y) → (x + 5, y - 3) to these newly obtained coordinates:
A" = (x + 5, y - 3) = (-4 + 5, -2 - 3) = (1, -5)
B" = (x + 5, y - 3) = (-1 + 5, 2 - 3) = (4, -1)
C" = (x + 5, y - 3) = (1 + 5, -3 - 3) = (6, -6)
So, the coordinates of ΔA"B"C" are A"(1, -5), B"(4, -1), and C"(6, -6).
To predict the quadrant the image will be in after the transformations, we need to identify the quadrant of one of the vertices. Let's take A" as an example. The x-coordinate of A" is 1, which is positive, and the y-coordinate is -5, which is negative.
Since the x-coordinate is positive and the y-coordinate is negative, the vertex A" is located in the fourth quadrant.
Therefore, we predict that ΔA"B"C" will be in the fourth quadrant after the transformations.