What is the image point of (2,-1)(2,−1) after the transformation r_{\text{x-axis}}\circ R_{90^{\circ}}r x-axis ​ ∘R 90 ∘ ​ ?

Answer 1

Hi Blake,

Thanks for reaching out with your question. It appears that you are seeking the image of the point (2, -1) after the composite transformation consisting of, in notation order, a reflection over the x-axis and a rotation of 90 degrees counterclockwise. HOWEVER, it is important to note that we will perform these transformations in REVERSE order. Another way to think of it is that we are performing a reflection over the x-axis OF a 90 degree counterclockwise rotation OF the point (2, -1).

I will demonstrate with both graphic and algebraic instructions.

First, let's graph the point (2, -1). Starting at the origin, we can move two units to the right and one unit down, giving us a point in the fourth quadrant of the coordinate grid. To visualize a rotation of 90 degrees counterclockwise, you can rotate your paper one quarter turn (counterclockwise). You can also use the rule for R(90): (x, y) --> (-y, x). Either way, you will end up with the point (1, 2) in the first quadrant.

Next, we must reflect this new point (1, 2) over the x-axis. The x-axis is the horizontal axis; since our point currently sits two units above the x-axis, it will end up two-units directly below the x-axis. We can also follow the rule that r(x-axis): (x, y) --> (x, -y). Ultimately, we will end up at the point (1, -2), back in the fourth quadrant.

So, the final answer is (1, -2). I hope this was helpful, and please feel free to reach out with any follow-up questions.

Best,

Zach Borenstein