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)

To analyze the given transformations on triangle ΔABC with vertices A(−8,4), B(−5,8), and C(−3,3), let's first break down the transformations step by step.

1. **First transformation: (x,y) → (x + 4, y - 6)**

We apply this transformation to each vertex of ΔABC.

- A(−8, 4):
\[ A' = (−8 + 4, 4 - 6) = (−4, -2) \]
- B(−5, 8):
\[ B' = (−5 + 4, 8 - 6) = (−1, 2) \]
- C(−3, 3):
\[ C' = (−3 + 4, 3 - 6) = (1, -3) \]

After the first transformation, the coordinates of ΔA'B'C' are:
- A'(-4, -2)
- B'(-1, 2)
- C'(1, -3)

2. **Second transformation: (x,y) → (x + 5, y - 3)**

Now we apply the second transformation to the new vertices A', B', and C'.

- A'(-4, -2):
\[ A'' = (−4 + 5, −2 - 3) = (1, -5) \]
- B'(-1, 2):
\[ B'' = (−1 + 5, 2 - 3) = (4, -1) \]
- C'(1, -3):
\[ C'' = (1 + 5, −3 - 3) = (6, -6) \]

Now, we have the coordinates of ΔA"B"C":
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)

3. **Composition of transformations:**

The overall transformation can be represented as follows:
- Original point (x, y) → (x + 4, y - 6) → (x + 4 + 5, y - 6 - 3)
- Combining these we get:
\[ (x, y) → (x + 4 + 5, y - 6 - 3) \]
\[ (x, y) → (x + 9, y - 9) \]

Thus, the rule for the composition transformation is:
\[ (x, y) → (x + 9, y - 9) \]

4. **Prediction of the quadrant:**

Now let's check the coordinates of ΔA"B"C":
- A''(1, -5)
- B''(4, -1)
- C''(6, -6)

All points have a positive x-coordinate and a negative y-coordinate. This means the points are located in the fourth quadrant.

Thus, the final coordinates of ΔA"B"C will be A(1, -5), B(4, -1), C(6, -6), and it will be located in the **fourth quadrant**.

are you sure thats right

Let’s double-check the transformations step by step carefully to ensure accuracy.

### Step 1: First Transformation

The first transformation is given by the rule:

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

We will apply this to each of the points in \( \Delta ABC \):
- For \( A(-8, 4) \):
\[
A' = (-8 + 4, 4 - 6) = (-4, -2)
\]
- For \( B(-5, 8) \):
\[
B' = (-5 + 4, 8 - 6) = (-1, 2)
\]
- For \( C(-3, 3) \):
\[
C' = (-3 + 4, 3 - 6) = (1, -3)
\]

After this first transformation, we have:

\[
\Delta A'B'C' \text{ with coordinates } A'(-4, -2), B'(-1, 2), C'(1, -3)
\]

### Step 2: Second Transformation

The second transformation is given by the rule:

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

Now we apply this to the new vertices \( A', B', \) and \( C' \):

- For \( A'(-4, -2) \):
\[
A'' = (-4 + 5, -2 - 3) = (1, -5)
\]
- For \( B'(-1, 2) \):
\[
B'' = (-1 + 5, 2 - 3) = (4, -1)
\]
- For \( C'(1, -3) \):
\[
C'' = (1 + 5, -3 - 3) = (6, -6)
\]

Thus, the coordinates of \( \Delta A''B''C'' \) are:

\[
A''(1, -5), B''(4, -1), C''(6, -6)
\]

### Step 3: Composition of Transformations

The overall transformation can be summarized as:

\[
(x, y) \rightarrow (x + 9, y - 9)
\]

So each point \((x, y)\) after both transformations is shifted by \((+9, -9)\).

### Step 4: Prediction of Quadrant

Now, let’s analyze the final coordinates:

- \( A''(1, -5) \) (1 positive, -5 negative) → 4th quadrant
- \( B''(4, -1) \) (4 positive, -1 negative) → 4th quadrant
- \( C''(6, -6) \) (6 positive, -6 negative) → 4th quadrant

Since all the points \( A'', B'', \) and \( C'' \) have a positive x-coordinate and a negative y-coordinate, they indeed lie in the **fourth quadrant**.

### Final Confirmation

Upon reviewing the calculations and the transformations step by step, the original conclusion is confirmed to be correct. The coordinates of \( \Delta A''B''C'' \) are indeed \( (1, -5), (4, -1), (6, -6) \), and they are located in the **fourth quadrant**. Thank you for your patience, and let me know if you have any further questions!