A line segment has endpoints at #(7 ,4 )# and #(3 ,5 )#. If the line segment is rotated about the origin by #(3 pi)/2 #, translated vertically by #2 #, and reflected about the yaxis, what will the line segment's new endpoints be?
Since there are 3 transformations to be performed, name the endpoints A(7 ,4) and B(3 ,5) so that we can 'track' the coordinates of the endpoints after each transformation.
a point (x ,y) → (y ,x)
hence A(7 ,4) → A'(4 ,7) and B(3 ,5) → B'(5 ,3)
a point (x ,y) → (x ,y2)
hence A'(4 ,7) → A''(4 ,9) and B'(5 ,3) → B''(5 ,5)
Third transformation Under a reflection in the yaxis
a point (x ,y) → (x ,y)
hence A''(4 ,9) → A'''(4 ,9) and B''(5 ,5) → B'''(5 ,5)
Thus after all 3 transformations the endpoints are.
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To find the new endpoints of the line segment after the described transformations:

Rotation about the origin by ( \frac{3\pi}{2} ): Applying the rotation matrix: [ \begin{bmatrix} \cos\left(\frac{3\pi}{2}\right) & \sin\left(\frac{3\pi}{2}\right) \ \sin\left(\frac{3\pi}{2}\right) & \cos\left(\frac{3\pi}{2}\right) \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} ] This simplifies to: [ \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} ] For each endpoint, apply this transformation.

Translation vertically by 2: Subtract 2 from the ycoordinate of each endpoint.

Reflection about the yaxis: Negate the xcoordinate of each endpoint.
Now, apply these transformations to the original endpoints:

For the endpoint (7, 4):
 Rotation: (\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 7 \ 4 \end{bmatrix} = \begin{bmatrix} 4 \ 7 \end{bmatrix})
 Translation: (( 4, 7  2) = (4, 5))
 Reflection: (((4), 5) = (4, 5))

For the endpoint (3, 5):
 Rotation: (\begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 3 \ 5 \end{bmatrix} = \begin{bmatrix} 5 \ 3 \end{bmatrix})
 Translation: ((5, 3  2) = (5, 1))
 Reflection: (((5), 1) = (5, 1))
Therefore, the new endpoints of the line segment after the described transformations are ((4, 5)) and ((5, 1)).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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