Question

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 a new rule that combines the two transformation rules. Use the function rules to find the coordinates of ΔA"B"C".(6 points)

Answer 1

To combine the two transformations applied to the triangle \( ΔABC \), we need to analyze each transformation step-by-step and then combine them into one rule.

### First Transformation
The first transformation is given by the rule:

\[
(x, y) \rightarrow (x + 4, y - 6)
\]

Let's apply this transformation to each vertex of triangle \( ABC \):

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

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

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

Thus, after the first transformation, the coordinates of triangle \( A'B'C' \) are:
\[
A'(-4, -2), \quad B'(-1, 2), \quad C'(1, -3)
\]

### Second Transformation
The second transformation is given by the rule:

\[
(x, y) \rightarrow (x + 5, y - 3)
\]

Now we apply this transformation to each vertex of triangle \( A'B'C' \):

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

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

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

Thus, after the second transformation, the coordinates of triangle \( A''B''C'' \) are:
\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]

### Combined Transformation Rule
We can combine the two transformations first. We know that:
- The first transformation moves any point \((x, y)\) to \((x + 4, y - 6)\).
- The second transformation moves \((x, y)\) to \((x + 5, y - 3)\).

If we consider the combination directly, we add both transformations together:

1. For the \(x\)-coordinate:
\[
x \rightarrow x + 4 \rightarrow (x + 4) + 5 = x + 9
\]

2. For the \(y\)-coordinate:
\[
y \rightarrow y - 6 \rightarrow (y - 6) - 3 = y - 9
\]

Thus, the new combined rule is:

\[
(x, y) \rightarrow (x + 9, y - 9)
\]

### Final Coordinates
Using the final combined rule, if we were to apply it to the original triangle \( ΔABC \) or any point \((x, y)\):

For the original vertices:
1. A(-8, 4):
\[
A'' = (-8 + 9, 4 - 9) = (1, -5)
\]

2. B(-5, 8):
\[
B'' = (-5 + 9, 8 - 9) = (4, -1)
\]

3. C(-3, 3):
\[
C'' = (-3 + 9, 3 - 9) = (6, -6)
\]

### Conclusion
The coordinates of triangle \( ΔA''B''C'' \) are:

\[
A''(1, -5), \quad B''(4, -1), \quad C''(6, -6)
\]