Question

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

Answer 1

To find the coordinates of \(\Delta A"B"C"\) after the transformations, we will follow these steps:

1. **Apply the first transformation**: \((x,y) \to (x+4, y-6)\) to the vertices of \(\Delta ABC\).

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

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

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

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

2. **Apply the second transformation**: \((x,y) \to (x+5, y-3)\) to the coordinates of \(\Delta A'B'C'\).

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

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

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

3. **Coordinates of \(\Delta A"B"C"\)**:
- \(A"(1, -5)\)
- \(B"(4, -1)\)
- \(C"(6, -6)\)

4. **Determine the quadrant**:
- All final coordinates \((1, -5)\), \((4, -1)\), and \((6, -6)\) have positive \(x\)-values and negative \(y\)-values. Therefore, they are all in the **fourth quadrant**.

5. **Create the rule for the composition transformation**:
- The combined transformation can be represented as:
\[
(x, y) \to (x + 4, y - 6) \quad \text{then} \quad (x + 5, y - 3)
\]
- If we combine these transformations, we get:
\[
(x, y) \to ((x + 4) + 5, (y - 6) - 3) = (x + 9, y - 9)
\]

So, the composite transformation rule is:
\[
(x, y) \to (x + 9, y - 9)
\]

In conclusion, the final coordinates of \(\Delta A"B"C"\) are \(A"(1, -5)\), \(B"(4, -1)\), and \(C"(6, -6)\), and the transformed triangle is located in the **fourth quadrant**.