Question

Use the function rules to find the coordinates of A " В" C" and predict the quadrant it will be in after the transformations. ДАВС, with coordinates A (-8, 4), B (-5, 8), and C (-3, 3), is transformed
b (х, у) → (x+4,4-6) to
produce AA'B'C". AA'B'C" is transformed by the
rule (x, y) → (x+ 5,1-3). Create the rule for the composition transformation and predict what quadrant the image will be in. (4 points)

Answer 1

Make this shorter

Answer 2

Take away the “#” and the “*”s

Answer 3

To solve this problem, we first need to apply the given transformations to the original coordinates of points A, B, and C.

### Step 1: Apply the first transformation \( (x, y) \rightarrow (x + 4, 4 - y) \).

**For point A (-8, 4)**:
- \( A' = (-8 + 4, 4 - 4) = (-4, 0) \)

**For point B (-5, 8)**:
- \( B' = (-5 + 4, 4 - 8) = (-1, -4) \)

**For point C (-3, 3)**:
- \( C' = (-3 + 4, 4 - 3) = (1, 1) \)

Now the new coordinates after the first transformation are:
- \( A'(-4, 0) \)
- \( B'(-1, -4) \)
- \( C'(1, 1) \)

### Step 2: Apply the second transformation \( (x, y) \rightarrow (x + 5, 1 - y) \).

**For point A' (-4, 0)**:
- \( A'' = (-4 + 5, 1 - 0) = (1, 1) \)

**For point B' (-1, -4)**:
- \( B'' = (-1 + 5, 1 - (-4)) = (4, 5) \)

**For point C' (1, 1)**:
- \( C'' = (1 + 5, 1 - 1) = (6, 0) \)

**New coordinates after the second transformation**:
- \( A''(1, 1) \)
- \( B''(4, 5) \)
- \( C''(6, 0) \)

### Step 3: Determine the quadrant of the transformed points

- **Point A''(1, 1)** is in Quadrant I (both x and y are positive).
- **Point B''(4, 5)** is also in Quadrant I.
- **Point C''(6, 0)** lies on the positive x-axis, which is technically not in any quadrant.

### Conclusion
The coordinates of the transformed points \( A''(1, 1) \), \( B''(4, 5) \), and \( C''(6, 0) \) suggest that the transformed figure AA'B'C'' will primarily be in Quadrant I, with point C'' on the positive x-axis.

### Composition Transformation Rule
The composition transformation combining both transformations can be derived as follows:
1. Initial transformation: \( (x, y) \to (x + 4, 4 - y) \)
2. Second transformation: \( (x, y) \to (x + 5, 1 - y) \)

Combining these two results gives:
- \( x' = (x + 4) + 5 = x + 9 \)
- \( y' = 1 - (4 - y) = y - 3 \)

Thus, the composition transformation rule is:
\[
(x, y) \rightarrow (x + 9, y - 3)
\]

The prediction for the quadrant of the image after the transformations is that the image will be primarily in Quadrant I, with point C'' on the positive x-axis.

Answer 4

Sure! Here’s a concise version:

### Transformations:

1. **First Transformation:** \( (x, y) \rightarrow (x + 4, 4 - y) \)

- \( A(-8, 4) \rightarrow A'(-4, 0) \)
- \( B(-5, 8) \rightarrow B'(-1, -4) \)
- \( C(-3, 3) \rightarrow C'(1, 1) \)

2. **Second Transformation:** \( (x, y) \rightarrow (x + 5, 1 - y) \)

- \( A'(-4, 0) \rightarrow A''(1, 1) \)
- \( B'(-1, -4) \rightarrow B''(4, 5) \)
- \( C'(1, 1) \rightarrow C''(6, 0) \)

### Final Coordinates:
- \( A''(1, 1) \) in Quadrant I
- \( B''(4, 5) \) in Quadrant I
- \( C''(6, 0) \) on the positive x-axis

### Composition Transformation Rule:
Combining the transformations gives:
\[
(x, y) \rightarrow (x + 9, y - 3)
\]

### Quadrant Prediction:
The final figure will primarily be in **Quadrant I**, with point C'' on the positive x-axis.

Answer 5

Sure! Here’s the revised version without any special characters:

Transformations:

1. First Transformation: (x, y) → (x + 4, 4 - y)

A(-8, 4) → A'(-4, 0)
B(-5, 8) → B'(-1, -4)
C(-3, 3) → C'(1, 1)

2. Second Transformation: (x, y) → (x + 5, 1 - y)

A'(-4, 0) → A''(1, 1)
B'(-1, -4) → B''(4, 5)
C'(1, 1) → C''(6, 0)

Final Coordinates:
- A''(1, 1) in Quadrant I
- B''(4, 5) in Quadrant I
- C''(6, 0) on the positive x-axis

Composition Transformation Rule:
Combining the transformations gives:
(x, y) → (x + 9, y - 3)

Quadrant Prediction:
The final figure will primarily be in Quadrant I, with point C'' on the positive x-axis.