Question
Write a composition of translations that is equivalent to Upper T Subscript left angle negative 6 comma 7 right angle Baseline left parenthesis x comma y right parenthesis.
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Part 1
Upper T Subscript left angle negative 6 comma 7 right angle Baseline left parenthesis x comma y right parenthesisequalsleft parenthesis Upper T Subscript left angle 0 comma nothing right angle Baseline circle Upper T Subscript left angle nothing comma 0 right angle right parenthesis left parenthesis x comma y right parenthesis
Question content area bottom
Part 1
Upper T Subscript left angle negative 6 comma 7 right angle Baseline left parenthesis x comma y right parenthesisequalsleft parenthesis Upper T Subscript left angle 0 comma nothing right angle Baseline circle Upper T Subscript left angle nothing comma 0 right angle right parenthesis left parenthesis x comma y right parenthesis
Answers
Answered by
GPT-4o mini
To express the translation \( T_{(-6, 7)}(x, y) \) as a composition of translations, we can break it down into two simpler translations that affect the \( x \)-coordinate and the \( y \)-coordinate separately.
1. **Interpret the translation**: The notation \( T_{(-6, 7)}(x, y) \) indicates that we are translating the point \( (x, y) \) by moving it \( -6 \) units in the \( x \)-direction (to the left) and \( +7 \) units in the \( y \)-direction (upwards).
2. **Defining the individual translations**:
- The translation \( T_{(0, 0)}(x, y) \) refers to not moving the point at all, which can be expressed as simply \( (x, y) \).
- The translation \( T_{(-6, 0)}(x, y) \) translates any point \( (x, y) \) by moving it 6 units left, yielding \( (x - 6, y) \).
- The translation \( T_{(0, 7)}(x, y) \) translates any point \( (x, y) \) by moving it 7 units up, yielding \( (x, y + 7) \).
3. **Combining translations**:
- To translate \( (x, y) \) to \( (x - 6, y) \) first, we apply \( T_{(-6, 0)} \):
\[
T_{(-6, 0)}(x, y) = (x - 6, y)
\]
- Next, we can apply the second translation \( T_{(0, 7)} \) to the result:
\[
T_{(0, 7)}(x - 6, y) = (x - 6, y + 7)
\]
4. **Final Composition**:
- Hence, the entire translation \( T_{(-6, 7)}(x, y) \) can be written as a composition of these two translations:
\[
T_{(-6, 7)}(x, y) = T_{(0, 7)} \circ T_{(-6, 0)}(x, y)
\]
- This means we first apply the \( T_{(-6, 0)} \) translation to \( (x, y) \), followed by the \( T_{(0, 7)} \) translation to the result.
### Conclusion
The final expression illustrating the composition of translations is:
\[
T_{(-6, 7)}(x, y) = T_{(0, 7)} \circ T_{(-6, 0)}(x, y)
\]
This expresses the desired translation as a sequential application of two simpler translations.
1. **Interpret the translation**: The notation \( T_{(-6, 7)}(x, y) \) indicates that we are translating the point \( (x, y) \) by moving it \( -6 \) units in the \( x \)-direction (to the left) and \( +7 \) units in the \( y \)-direction (upwards).
2. **Defining the individual translations**:
- The translation \( T_{(0, 0)}(x, y) \) refers to not moving the point at all, which can be expressed as simply \( (x, y) \).
- The translation \( T_{(-6, 0)}(x, y) \) translates any point \( (x, y) \) by moving it 6 units left, yielding \( (x - 6, y) \).
- The translation \( T_{(0, 7)}(x, y) \) translates any point \( (x, y) \) by moving it 7 units up, yielding \( (x, y + 7) \).
3. **Combining translations**:
- To translate \( (x, y) \) to \( (x - 6, y) \) first, we apply \( T_{(-6, 0)} \):
\[
T_{(-6, 0)}(x, y) = (x - 6, y)
\]
- Next, we can apply the second translation \( T_{(0, 7)} \) to the result:
\[
T_{(0, 7)}(x - 6, y) = (x - 6, y + 7)
\]
4. **Final Composition**:
- Hence, the entire translation \( T_{(-6, 7)}(x, y) \) can be written as a composition of these two translations:
\[
T_{(-6, 7)}(x, y) = T_{(0, 7)} \circ T_{(-6, 0)}(x, y)
\]
- This means we first apply the \( T_{(-6, 0)} \) translation to \( (x, y) \), followed by the \( T_{(0, 7)} \) translation to the result.
### Conclusion
The final expression illustrating the composition of translations is:
\[
T_{(-6, 7)}(x, y) = T_{(0, 7)} \circ T_{(-6, 0)}(x, y)
\]
This expresses the desired translation as a sequential application of two simpler translations.
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