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.
And please make your answer as simple and easy to understand as possible

Answer 1

what is the rule for the composition transformation?

Answer 2

To find the coordinates of triangle A" B" C" after the transformations, we can follow the steps below:

### Step 1: Transform Triangle ABC to A'B'C'
The transformation rule is given as (x,y) → (x + 4, y - 6).

**Apply to A (-8, 4)**:
- A' = (-8 + 4, 4 - 6) = (-4, -2)

**Apply to B (5, 8)**:
- B' = (5 + 4, 8 - 6) = (9, 2)

**Apply to C (-3, 3)**:
- C' = (-3 + 4, 3 - 6) = (1, -3)

The new coordinates after the first transformation are:
- A' (-4, -2)
- B' (9, 2)
- C' (1, -3)

### Step 2: Transform Triangle A'B'C' to A''B''C''
The next transformation rule is (x, y) → (x + 5, y - 3).

**Apply to A' (-4, -2)**:
- A'' = (-4 + 5, -2 - 3) = (1, -5)

**Apply to B' (9, 2)**:
- B'' = (9 + 5, 2 - 3) = (14, -1)

**Apply to C' (1, -3)**:
- C'' = (1 + 5, -3 - 3) = (6, -6)

The new coordinates after the second transformation are:
- A'' (1, -5)
- B'' (14, -1)
- C'' (6, -6)

### Step 3: Determine the Quadrant
The coordinates A''(1, -5), B''(14, -1), and C''(6, -6) reveal that all points have a positive x-coordinate and a negative y-coordinate.

- A'' (1, -5) is in Quadrant IV.
- B'' (14, -1) is also in Quadrant IV.
- C'' (6, -6) is likewise in Quadrant IV.

### Final Answer:
- The coordinates of triangle A''B''C'' are:
- A'' (1, -5)
- B'' (14, -1)
- C'' (6, -6)

- **Prediction**: Triangle A''B''C'' will be located in **Quadrant IV**, where the x-coordinates are positive and the y-coordinates are negative.

Answer 3

To find the rule for the composition transformation, we need to combine the two given transformations step by step.

### Given Transformations:

1. First Transformation:
\[(x, y) \rightarrow (x + 4, y - 6)\]

2. Second Transformation:
\[(x, y) \rightarrow (x + 5, y - 3)\]

### Step-by-Step Composition:

1. Start with an initial point \((x, y)\).
2. Apply the first transformation:
- New coordinates after the first transformation:
\[(x', y') = (x + 4, y - 6)\]

3. Apply the second transformation to \((x', y')\):
- Substituting \(x'\) and \(y'\) into the second transformation gives:
\[
(x'', y'') = ((x + 4) + 5, (y - 6) - 3)
\]
Simplifying:
\[
x'' = x + 4 + 5 = x + 9
\]
\[
y'' = y - 6 - 3 = y - 9
\]

### Final Composition Rule:

The composition transformation can be represented as:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]

### Summary:

The rule for the composition of the two transformations is:
\[
(x, y) \rightarrow (x + 9, y - 9)
\]