What is the image point of (0,-5)(0,−5) after the transformation D_{\frac{1}{2}}\circ T_{2,-1}D 2 1 ​ ​ ∘T 2,−1 ​ ?

Answer 1

FIRST - What is being transformed? a point located at (0,-5)

SECOND - How is it being transformed?

In 2 ways: first by DILATION with a scale factor of 1/2

second by TRANSLATION of (2,-1) which means 2 units right and 1 unit down

When you plot the given point (0,-5) in the coordinate plane, the point is in line with the Origin (vertically) - it is 5 units below the origin. So, automatically, the distance between the origin and the given point is a no-brainer. They are 5 units apart.

When we dilate this point using the scale factor of 1/2, the distance between the origin and the point will decrease. It will be cut in half. Just think of it like this:

(0, -5) -----> (0 × 1/2, -5 × 1/2) ----> (0, -2.5) is the new point after the dilation. (1st transformation

Now, onto the 2nd transformation....

We take the location of the point after the dilation [(0,-2.5)] and TRANSLATE IT as stated above (2 units to the right and 1 unit down)

Moving 2 units to the right (in the POSITIVE direction) from (0,-2.5), we land on (2,-2.5)

Moving 1 unit down, we land on (2,-3.5)

That is the final location of the original point after both transformations.