Question

Use the function rules to find the coordinates of A" B" C" and predict the quadrant it will be in after the transformations. AABC, with coordinates A (-8, 4), B(-5,8), and C (-3, 3), is transformed by (x, y)(x+4, y6) to produce ΔΑ' Β'C'. ΔΑ' Β'C' is transformed by the rule (x, y)(x+5, y3). Create the rule for the composition transformation and predict what quadrant the image will be in. (4 points)

Answer 1

Let's start by applying the transformation rules step by step to find the coordinates of points A", B", and C" after the transformations.

1. **Initial Points**:
- A(-8, 4)
- B(-5, 8)
- C(-3, 3)

2. **First Transformation (to get A', B', C')**:
The transformation rule is given by \( (x, y) \to (x + 4, y + 6) \).

- For A:
\[
A' = (-8 + 4, 4 + 6) = (-4, 10)
\]

- For B:
\[
B' = (-5 + 4, 8 + 6) = (-1, 14)
\]

- For C:
\[
C' = (-3 + 4, 3 + 6) = (1, 9)
\]

So after the first transformation, we have:
- A'(-4, 10)
- B'(-1, 14)
- C'(1, 9)

3. **Second Transformation (to get A'', B'', C'')**:
The transformation rule is given by \( (x, y) \to (x + 5, y + 3) \).

- For A':
\[
A'' = (-4 + 5, 10 + 3) = (1, 13)
\]

- For B':
\[
B'' = (-1 + 5, 14 + 3) = (4, 17)
\]

- For C':
\[
C'' = (1 + 5, 9 + 3) = (6, 12)
\]

So after the second transformation, we have:
- A''(1, 13)
- B''(4, 17)
- C''(6, 12)

4. **Chain Rule (Composition of Transformations)**:
The composed transformation rule from the original points is calculated by combining the two transformations:

Starting from \( (x, y) \), we first apply \( (x + 4, y + 6) \) followed by \( (x + 5, y + 3) \):

- First transformation:
\[(x, y) \to (x + 4, y + 6)\]
- Second transformation applied to the result of the first:
\[(x + 4, y + 6) \to ((x + 4) + 5, (y + 6) + 3)\]
- This simplifies to:
\[
(x + 9, y + 9)
\]

Therefore, the overall transformation rule is:
\[
(x, y) \to (x + 9, y + 9)
\]

5. **Predicted Quadrant**:
Now let's consider the final coordinates of A'', B'', and C'':
- A''(1, 13) is in Quadrant I (both x and y are positive).
- B''(4, 17) is also in Quadrant I.
- C''(6, 12) is in Quadrant I as well.

Since all the transformed points A'', B'', and C'' have positive x and y coordinates, we can predict that the image will be in **Quadrant I**.