Question

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.

Answer 1

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.