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.
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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).
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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 \( (x, y + 7) \).
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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: \[ T_{(0, 7)}(x - 6, y) = (x - 6, y + 7) \]
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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.