use the function rules to find the coordinates of triangle A", B", C" and predict the quadrant it will be in after the transformations. triangleABC, with coordinates A(-8,4), B(-5,8), and C(-3,3), is transformed be (x,y) arrow (x+4, y-6) to produce triangleA'B'C'. triangleA', B', C' is transformed by the rule (x,y) arrow (x+5, y-3). Create the rule for the composition transformation and predict what quadrant the image will be in

To find the coordinates of the transformed triangle \( A''B''C'' \) and determine in which quadrant it will lie, we need to apply the transformations step-by-step.

### Step 1: First Transformation

The first transformation is given by the rule:

\[
(x, y) \rightarrow (x + 4, y - 6)
\]

Let's apply this transformation to each vertex of triangle \( ABC \):

1. **Point :
\[
A(-8, 4) \rightarrow (-8 + 4, 4 - 6) = (-4, -2)
\]

2. **Point B**:
\[
B(-5, 8) \rightarrow (-5 + 4, 8 - 6) = (-1, 2)
\]

3. **Point C**:
\[
C(-3, 3) \rightarrow (-3 + 4, 3 - 6) = (1, -3)
\]

So, after the first transformation, the coordinates of triangle \( A'B'C' \) are:
- \( A'(-4, -2) \)
- \( B'(-1, 2) \)
- \( C'(1, -3) \)

### Step 2: Second Transformation

The second transformation is defined by the rule:

\[
(x, y) \rightarrow (x + 5, y - 3)
\]

Now, we apply this transformation to each point of triangle \( A'B'C' \):

1. **Point A'**:
\[
A'(-4, -2) \rightarrow (-4 + 5, -2 - 3) = (1, -5)
\]

2. **Point B'**:
\[
B'(-1, 2) \rightarrow (-1 + 5, 2 - 3) = (4, -1)
\]

3. **Point C'**:
\[
C'(1, -3) \rightarrow (1 + 5, -3 - 3) = (6, -6)
\]

So, after the second transformation, the coordinates of triangle \( A''B''C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)

### Step 3: Predicting the Quadrant

Now, we need to determine the quadrant where triangle \( A''B''C'' \) is located.

1. **Point A''(1, -5)**: \( x > 0 \) and \( y < 0 \) (Quadrant IV)
2. **Point B''(4, -1)**: \( x > 0 \) and \( y < 0 \) (Quadrant IV)
3. **Point C''(6, -6)**: \( x > 0 \) and \( y < 0 \) (Quadrant IV)

All three points are in Quadrant IV.

### Conclusion

The coordinates of triangle \( A''B''C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)

After the transformations, triangle \( A''B''C'' \) will be located entirely in **Quadrant IV**.