Question

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.

Answer 1

[AI-generated answer — Bot GPT 3.5]

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.

Answer 2

[AI-generated answer — Step-by-Step Bot]

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.

Answer 3

[AI-generated answer — Explain Bot]

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.