Use the function rules to find the coordinates of ΔA"B"C"

Question

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.(4 points)

Answer 1

To find the coordinates of triangle \( \Delta A"B"C" \) after the transformations, we will apply the transformations step by step to the original triangle \( \Delta ABC \) with vertices \( A(-8, 4) \), \( B(-5, 8) \), and \( C(-3, 3) \).

Step 1: First Transformation

The first transformation is given by the rule \((x,y) \to (x+4,y-6)\).

Applying this transformation to each vertex:

• Point A: \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]

• Point B: \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]

• Point C: \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]

After the first transformation, the coordinates of \( \Delta A'B'C' \) are \( A'(-4, -2) \), \( B'(-1, 2) \), and \( C'(1, -3) \).

Step 2: Second Transformation

The second transformation is given by the rule \((x,y) \to (x+5,y-3)\).

Applying this transformation to each point of \( \Delta A'B'C' \):

• Point A': \[ A" = (-4 + 5, -2 - 3) = (1, -5) \]

• Point B': \[ B" = (-1 + 5, 2 - 3) = (4, -1) \]

• Point C': \[ C" = (1 + 5, -3 - 3) = (6, -6) \]

After the second transformation, the coordinates of \( \Delta A"B"C" \) are:

• \( A"(1, -5) \)
• \( B"(4, -1) \)
• \( C"(6, -6) \)

Rule for the Composition Transformation

To create the overall transformation rule, we can combine both transformations:

1. The first transformation adds 4 to the x-coordinate and subtracts 6 from the y-coordinate.
2. The second transformation adds 5 to the x-coordinate and subtracts 3 from the y-coordinate.

Combining both transformations: \[ (x, y) \to (x + 4 + 5, y - 6 - 3) \Rightarrow (x + 9, y - 9) \]

Thus, the composition transformation rule is: \[ (x,y) \to (x + 9, y - 9) \]

Predicting the Quadrant

Now we will predict the quadrant for the triangle \( \Delta A"B"C" \) with the coordinates \( (1, -5) \), \( (4, -1) \), and \( (6, -6) \):

• The x-coordinates are all positive: \( 1, 4, 6 \)
• The y-coordinates are all negative: \( -5, -1, -6 \)

Given that the x-coordinates are positive and the y-coordinates are negative, all three points are located in the fourth quadrant.

Summary

The coordinates of \( \Delta A"B"C" \) are:

• \( A"(1, -5) \)
• \( B"(4, -1) \)
• \( C"(6, -6) \)

The triangle \( \Delta A"B"C" \) will be located entirely in the fourth quadrant.