A line segment has endpoints at #(3 , 1)# and #(2 ,3)#. If the line segment is rotated about the origin by #(3pi)/2 #, translated horizontally by #7#, and reflected about the x-axis, what will the line segment's new endpoints be?
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To find the new endpoints of the line segment after the described transformations:
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Rotate about the origin by ( \frac{3\pi}{2} ) radians: The rotation of a point ( (x, y) ) about the origin by ( \theta ) radians is given by: [ (x', y') = (x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta)) ] Applying this formula to both endpoints, we get: [ (3, 1) \rightarrow (1, -3) \quad \text{and} \quad (2, 3) \rightarrow (-3, 2) ]
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Translate horizontally by 7: Adding 7 to the x-coordinates of both endpoints, we get: [ (1, -3) \rightarrow (8, -3) \quad \text{and} \quad (-3, 2) \rightarrow (4, 2) ]
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Reflect about the x-axis: The reflection of a point ( (x, y) ) about the x-axis is given by ( (x, -y) ). Applying this to both endpoints, we get: [ (8, -3) \rightarrow (8, 3) \quad \text{and} \quad (4, 2) \rightarrow (4, -2) ]
Therefore, the new endpoints of the line segment are ( (8, 3) ) and ( (4, -2) ).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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