Use the function rules to find the coordinates of A"B" C"𝐴" 𝐵"𝐶" and predict the quadrant it will be in after the transformations. Draw a picture to help.△ABC△𝐴𝐵𝐶 coordinates: A(−8,4)𝐴(−8,4), B(−5,8)𝐵(−5,8), C(−3,3)𝐶(−3,3)△ABC△𝐴𝐵𝐶 to △A′B′C′△𝐴′𝐵′𝐶′ rule: (x,y)→(x+4, y−6)(𝑥,𝑦)→(𝑥+4, 𝑦−6)△A′B′C′△𝐴′𝐵′𝐶′ to △A"B" C"△𝐴" 𝐵"𝐶" rule: (x,y)→(x+5, y−3)(𝑥,𝑦)→(𝑥+5, 𝑦−3)Show your work for each transformation. Create the rule for the composition transformation. Predict what quadrant the image will be in.(4 points)

To find the coordinates of points \( A'', B'', C'' \) after applying the transformations as specified, we'll follow these steps:

1. **Initial Coordinates of Triangle \(\triangle ABC\)**:
- \( A(-8, 4) \)
- \( B(-5, 8) \)
- \( C(-3, 3) \)

2. **First Transformation Rule: \((x,y) \rightarrow (x+4, y-6)\)**:
- For point \( A \):
\[
A' = A(-8, 4) \rightarrow (-8 + 4, 4 - 6) = (-4, -2)
\]
- For point \( B \):
\[
B' = B(-5, 8) \rightarrow (-5 + 4, 8 - 6) = (-1, 2)
\]
- For point \( C \):
\[
C' = C(-3, 3) \rightarrow (-3 + 4, 3 - 6) = (1, -3)
\]

After the first transformation, the new coordinates are:
\[
A'(-4, -2), \quad B'(-1, 2), \quad C'(1, -3)
\]

3. **Second Transformation Rule: \((x,y) \rightarrow (x+5, y-3)\)**:
- For point \( A' \):
\[
A'' = A'(-4, -2) \rightarrow (-4 + 5, -2 - 3) = (1, -5)
\]
- For point \( B' \):
\[
B'' = B'(-1, 2) \rightarrow (-1 + 5, 2 - 3) = (4, -1)
\]
- For point \( C' \):
\[
C'' = C'(1, -3) \rightarrow (1 + 5, -3 - 3) = (6, -6)
\]

After the second transformation, the new coordinates are:
\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]

4. **Composition Rule**:
The entire transformation can be represented as a single combined rule.

The first transformation is represented by:
\[
(x, y) \rightarrow (x + 4, y - 6)
\]
The second transformation takes the output from the first:
\[
(x + 4, y - 6) \rightarrow ((x + 4) + 5, (y - 6) - 3)
\]
which simplifies to:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]

5. **Predicting the Quadrant**:
The coordinates of \( A'', B'', C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)

A point in the coordinate plane is in quadrant IV if \( x > 0 \) and \( y < 0 \).

Since all the transformed points have positive \( x \)-coordinates and negative \( y \)-coordinates, it follows that:
\[
A'', B'', C'' \text{ are all in Quadrant IV.}
\]

6. **Sketching the Transformed Triangle**:
Unfortunately, I can't draw directly here, but you would simply plot points \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \) to visualize that they lie in quadrant IV.

**Final notation of coordinates after transformations**:
\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]

**Final Prediction**: The image \( \triangle A''B''C'' \) will be located in **Quadrant IV**.