Question

Use the function rules to find the coordinates of A"B" C"𝐴" 𝐵"𝐶" and predict the quadrant it will be in after the transformations. Draw a picture to help.△ABC△𝐴𝐵𝐶 coordinates: A(−8,4)𝐴(−8,4), B(−5,8)𝐵(−5,8), C(−3,3)𝐶(−3,3)△ABC△𝐴𝐵𝐶 to △A′B′C′△𝐴′𝐵′𝐶′ rule: (x,y)→(x+4, y−6)(𝑥,𝑦)→(𝑥+4, 𝑦−6)△A′B′C′△𝐴′𝐵′𝐶′ to △A"B" C"△𝐴" 𝐵"𝐶" rule: (x,y)→(x+5, y−3)(𝑥,𝑦)→(𝑥+5, 𝑦−3)Show your work for each transformation. Create the rule for the composition transformation. Predict what quadrant the image will be in.(4 points)

Answer 1

To find the coordinates of points \( A'', B'', C'' \) after applying the transformations as specified, we'll follow these steps:

1. **Initial Coordinates of Triangle \(\triangle ABC\)**:
- \( A(-8, 4) \)
- \( B(-5, 8) \)
- \( C(-3, 3) \)

2. **First Transformation Rule: \((x,y) \rightarrow (x+4, y-6)\)**:
- For point \( A \):
\[
A' = A(-8, 4) \rightarrow (-8 + 4, 4 - 6) = (-4, -2)
\]
- For point \( B \):
\[
B' = B(-5, 8) \rightarrow (-5 + 4, 8 - 6) = (-1, 2)
\]
- For point \( C \):
\[
C' = C(-3, 3) \rightarrow (-3 + 4, 3 - 6) = (1, -3)
\]

After the first transformation, the new coordinates are:
\[
A'(-4, -2), \quad B'(-1, 2), \quad C'(1, -3)
\]

3. **Second Transformation Rule: \((x,y) \rightarrow (x+5, y-3)\)**:
- For point \( A' \):
\[
A'' = A'(-4, -2) \rightarrow (-4 + 5, -2 - 3) = (1, -5)
\]
- For point \( B' \):
\[
B'' = B'(-1, 2) \rightarrow (-1 + 5, 2 - 3) = (4, -1)
\]
- For point \( C' \):
\[
C'' = C'(1, -3) \rightarrow (1 + 5, -3 - 3) = (6, -6)
\]

After the second transformation, the new coordinates are:
\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]

4. **Composition Rule**:
The entire transformation can be represented as a single combined rule.

The first transformation is represented by:
\[
(x, y) \rightarrow (x + 4, y - 6)
\]
The second transformation takes the output from the first:
\[
(x + 4, y - 6) \rightarrow ((x + 4) + 5, (y - 6) - 3)
\]
which simplifies to:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]

5. **Predicting the Quadrant**:
The coordinates of \( A'', B'', C'' \) are:
- \( A''(1, -5) \)
- \( B''(4, -1) \)
- \( C''(6, -6) \)

A point in the coordinate plane is in quadrant IV if \( x > 0 \) and \( y < 0 \).

Since all the transformed points have positive \( x \)-coordinates and negative \( y \)-coordinates, it follows that:
\[
A'', B'', C'' \text{ are all in Quadrant IV.}
\]

6. **Sketching the Transformed Triangle**:
Unfortunately, I can't draw directly here, but you would simply plot points \( A''(1, -5) \), \( B''(4, -1) \), and \( C''(6, -6) \) to visualize that they lie in quadrant IV.

**Final notation of coordinates after transformations**:
\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]

**Final Prediction**: The image \( \triangle A''B''C'' \) will be located in **Quadrant IV**.