Question

Use the function rules to find the coordinates of ΔA"B"C" and predict the quadrant it will be in after the transformations. ΔABC , with coordinates A(−8,4) , B(−5,8) , and C(−3,3) , is transformed by (x,y)→(x+4,y−6) to produce ΔA′B′C′ . ΔA′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

Let's first find the coordinates of \(\Delta A'B'C'\) after applying the first transformation to \(\Delta ABC\).

1. **Original Coordinates:**
- \(A(-8, 4)\)
- \(B(-5, 8)\)
- \(C(-3, 3)\)

2. **Transformation 1: \((x, y) \to (x + 4, y - 6)\)**

Applying this transformation to each point:

- For point \(A\):
\[
A' = (-8 + 4, 4 - 6) = (-4, -2)
\]

- For point \(B\):
\[
B' = (-5 + 4, 8 - 6) = (-1, 2)
\]

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

Now the coordinates of \(\Delta A' B' C'\) are:
- \(A'(-4, -2)\)
- \(B'(-1, 2)\)
- \(C'(1, -3)\)

3. **Transformation 2: \((x, y) \to (x + 5, y - 3)\)**

Next, we apply this transformation to each point of \(\Delta A'B'C'\):

- For point \(A'\):
\[
A'' = (-4 + 5, -2 - 3) = (1, -5)
\]

- For point \(B'\):
\[
B'' = (-1 + 5, 2 - 3) = (4, -1)
\]

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

The coordinates of \(\Delta A''B''C''\) are:
- \(A''(1, -5)\)
- \(B''(4, -1)\)
- \(C''(6, -6)\)

4. **Create the Rule for the Composition Transformation:**

The overall transformation is a combination of both transformations:

1. First Transformation: \((x, y) \to (x + 4, y - 6)\)
2. Second Transformation: \((x, y) \to (x + 5, y - 3)\)

To carry out the composition:
- The combined transformation can be written as:

\[
(x, y) \to (x + 4 + 5, y - 6 - 3) = (x + 9, y - 9)
\]

5. **Predicting the Quadrant:**

Now let's analyze the final coordinates:
- \(A''(1, -5)\), \(B''(4, -1)\), and \(C''(6, -6)\).

All points have a positive x-coordinate and a negative y-coordinate. Thus, they are located in **Quadrant IV**, where \(x > 0\) and \(y < 0\).

**Final Result:**
- The coordinates of \(\Delta A''B''C''\) are \(A''(1, -5)\), \(B''(4, -1)\), and \(C''(6, -6)\).
- The transformation rule is \((x, y) \to (x + 9, y - 9)\).
- The resulting image will be in **Quadrant IV**.