Use the function rules to find the coordinates of triangle A" B" C" and predict the quadrant it will be in after the transformations. Triangle ABC, with coordinates A (-8, 4), B (-5, 8), and C (-3, 3), is transformed by (x, y) -> (x + 4, y - 6) to produce triangle A'B'C'. Triangle A'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.

To find the coordinates of triangle \( A''B''C'' \) after the given transformations, we will first apply the transformation rules step by step and record the results.

### Step 1: Apply the first transformation

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

Starting with the original coordinates of triangle ABC:
- A(-8, 4)
- B(-5, 8)
- C(-3, 3)

Applying the transformation to each vertex:

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

### Step 2: Apply the second transformation

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

Now we will apply this transformation to each of the new vertices:

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

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

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

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

### Step 3: Determine the quadrant

Next, we need to predict which quadrant triangle \( A''B''C'' \) will be in. The key point is that all coordinates of the vertices \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \) have a positive x-coordinate and a negative y-coordinate.

### Conclusion

Since all new vertices have \( x > 0 \) and \( y < 0 \), this places triangle \( A''B''C'' \) in the **fourth quadrant** of the Cartesian plane.