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.(4 points).

Answer 1

To find the coordinates of \( \Delta A"B"C" \) after two transformations, we proceed as follows.

Step 1: Apply the first transformation \( (x, y) \rightarrow (x+4, y-6) \)

Starting with the coordinates of \( \Delta ABC \):

• \( A(-8, 4) \)
• \( B(-5, 8) \)
• \( C(-3, 3) \)

We apply the transformation to each vertex:

1. For point \( A \): \[ A' = (-8 + 4, 4 - 6) = (-4, -2) \]

2. For point \( B \): \[ B' = (-5 + 4, 8 - 6) = (-1, 2) \]

3. For point \( C \): \[ C' = (-3 + 4, 3 - 6) = (1, -3) \]

After the first transformation, we have:

• \( A'(-4, -2) \)
• \( B'(-1, 2) \)
• \( C'(1, -3) \)

Step 2: Apply the second transformation \( (x, y) \rightarrow (x+5, y-3) \)

Now we apply the second transformation to \( A' \), \( B' \), and \( C' \):

1. For point \( A' \): \[ A" = (-4 + 5, -2 - 3) = (1, -5) \]

2. For point \( B' \): \[ B" = (-1 + 5, 2 - 3) = (4, -1) \]

3. For point \( C' \): \[ C" = (1 + 5, -3 - 3) = (6, -6) \]

Final coordinates for \( \Delta A"B"C" \):

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

Step 3: Determine the quadrant

The coordinates of the points \( A" \), \( B" \), and \( C" \) are:

• \( A"(1, -5) \) is in Quadrant IV (where \( x > 0 \) and \( y < 0 \)).
• \( B"(4, -1) \) is also in Quadrant IV.
• \( C"(6, -6) \) is in Quadrant IV.

Conclusion

All points \( A" \), \( B" \), and \( C" \) are in Quadrant IV. Therefore, the predicted quadrant for \( \Delta A"B"C" \) after the transformations is:

Quadrant IV.