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 first apply the first transformation rule to the original coordinates 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, we apply the second transformation rule to the coordinates of ΔA'B'C':
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)
Therefore, ΔA'B'C' has the following coordinates: A' (1, -5), B' (4, -1), and C' (6, -6).
To find the rule for the composition transformation, we simply combine the two transformation rules:
Composition rule: (x,y) → (x+4+5, y-6-3) = (x+9, y-9)
The image of ΔABC after the composition transformation 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.
For the first transformation (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)
For the second transformation (x, y) → (x + 5, y - 3):
A" = (-4 + 5, -2 - 3) = (1, -5)
B" = (-1 + 5, 2 - 3) = (4, -1)
C" = (1 + 5, -3 - 3) = (6, -6)
Now, let's predict the quadrant in which ΔA"B"C" will be after the transformations. To do this, we need to look at the signs of the x and y coordinates.
Since the x-coordinate of A" is positive (1) and the y-coordinate is negative (-5), it will be in the fourth quadrant.
Similarly, the x-coordinate of B" is positive (4) and the y-coordinate is negative (-1), so it will be in the fourth quadrant.
Lastly, the x-coordinate of C" is positive (6) and the y-coordinate is negative (-6), indicating that it will also be in the fourth quadrant.
Therefore, we can predict that ΔA"B"C" will be in the fourth quadrant after the transformations.
To find the coordinates of ΔA"B"C" after the given transformations, we need to apply the function rules to each vertex of ΔABC.
Given function rule 1: (x,y) → (x+4, y-6)
Applying this rule to point A(-8, 4):
x = -8 + 4 = -4
y = 4 - 6 = -2
So, A (-8, 4) transforms to A' (-4, -2).
Applying the same rule to point B (-5, 8):
x = -5 + 4 = -1
y = 8 - 6 = 2
So, B (-5, 8) transforms to B' (-1, 2).
Applying the same rule to point C (-3, 3):
x = -3 + 4 = 1
y = 3 - 6 = -3
So, C (-3, 3) transforms to C' (1, -3).
Next, we need to apply the second given function rule to ΔA'B'C'.
Given function rule 2: (x,y) → (x+5, y-3)
Applying this rule to vertex A' (-4, -2):
x = -4 + 5 = 1
y = -2 - 3 = -5
So, A' (-4, -2) transforms to A" (1, -5).
Applying the same rule to vertex B' (-1, 2):
x = -1 + 5 = 4
y = 2 - 3 = -1
So, B' (-1, 2) transforms to B" (4, -1).
Applying the same rule to vertex C' (1, -3):
x = 1 + 5 = 6
y = -3 - 3 = -6
So, C' (1, -3) transforms to C" (6, -6).
The composition transformation rule is obtained by combining the two given rules:
(x,y) → (x+4, y-6) → (x+9, y-9)
We can predict the quadrant in which the image will be located by observing the signs of the x and y coordinates of a point in the image. Since the x-coordinate in the composition transformation rule increases by 9 and the y-coordinate decreases by 9, the resulting point C" (6, -6) will be located in the fourth quadrant.