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 one by one.
1. Transformation (x,y)→(x+4,y−6):
For point A(-8,4):
New x-coordinate = -8 + 4 = -4
New y-coordinate = 4 - 6 = -2
So, A'(-4, -2)
For point B(-5,8):
New x-coordinate = -5 + 4 = -1
New y-coordinate = 8 - 6 = 2
So, B'(-1, 2)
For point C(-3,3):
New x-coordinate = -3 + 4 = 1
New y-coordinate = 3 - 6 = -3
So, C'(1, -3)
2. Transformation (x,y)→(x+5,y−3):
For point A'(-4, -2):
New x-coordinate = -4 + 5 = 1
New y-coordinate = -2 - 3 = -5
So, A''(1, -5)
For point B'(-1, 2):
New x-coordinate = -1 + 5 = 4
New y-coordinate = 2 - 3 = -1
So, B''(4, -1)
For point C'(1, -3):
New x-coordinate = 1 + 5 = 6
New y-coordinate = -3 - 3 = -6
So, C''(6, -6)
The coordinates of ΔA''B''C'' are A''(1, -5), B''(4, -1), and C''(6, -6).
Since all the x-coordinates are positive and all the y-coordinates are negative, ΔA''B''C'' will be in the fourth quadrant after the transformations.
To find the coordinates of ΔA"B"C", we need to apply the given transformation rules to each vertex of ΔABC.
Applying the first transformation rule (x,y)→(x+4,y−6) to point A(-8,4):
A' = (-8 + 4, 4 - 6) = (-4, -2)
Applying the first transformation rule (x,y)→(x+4,y−6) to point B(-5,8):
B' = (-5 + 4, 8 - 6) = (-1, 2)
Applying the first transformation rule (x,y)→(x+4,y−6) to point C(-3,3):
C' = (-3 + 4, 3 - 6) = (1, -3)
Now, applying the second transformation rule (x,y)→(x+5,y−3) to point A'(-4, -2):
A" = (-4 + 5, -2 - 3) = (1, -5)
Applying the second transformation rule (x,y)→(x+5,y−3) to point B'(-1, 2):
B" = (-1 + 5, 2 - 3) = (4, -1)
Applying the second transformation rule (x,y)→(x+5,y−3) to point C'(1, -3):
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 in which ΔA"B"C" will be, we can look at the signs of the x and y coordinates. Based on the coordinates (1, -5), (4, -1), and (6, -6), we see that the x-coordinates are positive and the y-coordinates are mostly negative. Therefore, it is likely that ΔA"B"C" will be in the fourth quadrant.
To find the coordinates of ΔA"B"C", we need to apply the given transformation rules to each vertex of ΔABC step by step.
1. Applying the first transformation rule (x,y)→(x+4,y−6):
A' = A + (4, -6) = (-8 + 4, 4 - 6) = (-4, -2)
B' = B + (4, -6) = (-5 + 4, 8 - 6) = (-1, 2)
C' = C + (4, -6) = (-3 + 4, 3 - 6) = (1, -3)
2. Applying the second transformation rule (x,y)→(x+5,y−3):
A" = A' + (5, -3) = (-4 + 5, -2 - 3) = (1, -5)
B" = B' + (5, -3) = (-1 + 5, 2 - 3) = (4, -1)
C" = C' + (5, -3) = (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 ΔA"B"C" will be in, we can look at the signs of the x-coordinate and y-coordinate of any vertex. In this case, A"(1, -5) is in the fourth quadrant since both x and y are positive.
To find the coordinates of ΔA'B'C', we need to apply the function rules to each vertex of ΔABC one by one.
Let's start with vertex A(-8,4):
1. Apply the first transformation rule: (x, y) → (x+4, y-6)
For A(-8,4), we will have: (x+4, y-6) → (-8+4, 4-6) = (-4, -2)
Next, let's consider vertex B(-5,8):
2. Apply the first transformation rule: (x, y) → (x+4, y-6)
For B(-5,8), we will have: (x+4, y-6) → (-5+4, 8-6) = (-1, 2)
Lastly, let's examine vertex C(-3,3):
3. Apply the first transformation rule: (x, y) → (x+4, y-6)
For C(-3,3), we will have: (x+4, y-6) → (-3+4, 3-6) = (1, -3)
The coordinates of ΔA'B'C' are A'(-4,-2), B'(-1,2), and C'(1,-3).
To find the composition transformation rule, we need to combine the two given transformation rules.
The first transformation rule is (x, y) → (x+4, y-6).
The second transformation rule is (x, y) → (x+5, y-3).
To compose these transformations, we sequentially apply them.
So, for the composition transformation rule, we will have:
(x, y) → (x+4+5, y-6-3)
(x, y) → (x+9, y-9)
The composed transformation rule is (x, y) → (x+9, y-9).
Now, to predict the quadrant the image ΔA'B'C' will be in after the composition transformation, we need to consider the signs of the x and y values.
Since the x values will be increased by 9 and the y values will be decreased by 9, the image ΔA'B'C' will be shifted 9 units to the right (positive x-direction) and 9 units downwards (negative y-direction) from its original position.
Based on the first quadrant being the top-right quadrant and following the shifting, the image ΔA'B'C' will be in the fourth quadrant.