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.
A

z

no pictures

Answer 1

To find the coordinates of \( \Delta A"B"C" \) after the transformations, we will apply each transformation step by step.

### Initial Coordinates
The original coordinates of the triangle \( \Delta ABC \) are:
- \( A(-8, 4) \)
- \( B(-5, 8) \)
- \( C(-3, 3) \)

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

We apply this to each vertex of \( \Delta ABC \):

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

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

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

So, the coordinates of \( \Delta A'B'C' \) are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)

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

We will now apply this transformation to each vertex of \( \Delta A'B'C' \):

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

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

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

Thus, the coordinates of \( \Delta A"B"C" \) after both transformations are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)

### Composition Rule
To create the composition transformation from the original coordinates \( (x, y) \) to \( (x + 9, y - 9) \) (adding both transformations together):
- The total transformation can be viewed as:
\[
(x, y) \to (x + 4 + 5, y - 6 - 3) \to (x + 9, y - 9)
\]

### Quadrant Prediction
Now, let's analyze the final coordinates \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \):
- All these points have a positive x-coordinate and a negative y-coordinate. This means:
- \( x > 0 \) and \( y < 0 \)

Therefore, the points lie in **Quadrant IV**.

### Conclusion
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 in **Quadrant IV** after the transformations.