What is the image point of (4,−9) after the transformation R_{90^{\circ}}\circ r_{\text{y-axis}}R 90 ∘ ​ ∘r y-axis ​ ?

Question

What is the image point of (4,−9) after the transformation R90∘​∘ry-axis​?

Answer 1

Hi Jeanine,

Thanks for the question. I will break it down and explain how you could solve this both graphically and algebraically.

It appears to be that you are starting with the point (4, -9) on the coordinate plane and performing a composition of two transformations. You can start by graphing that point. Start at the origin. Since the point has an x-coordinate of positive 4, move 4 units to the right. As the y-coordinate is -9, move nine units down from that point. Your point will be in the fourth quadrant of the coordinate plane.

It is important to note that with composite transformations written in the notation you provided, they are performed in reverse order. Another way to think of this is that you are performing a 90 degree counterclockwise rotation OF a reflection over the y-axis OF the original point (4, -9).

So, we will first reflect the original point over the y-axis, which is the vertical axis. As you may notice, the original point lies four units directly to the right of the y-axis. The image will then lie four units directly to the left. The algebraic rule of r: y-axis is (x, y) --> (-x, y). As we started with the point (4, -9), this takes us to the point (-4, -9). You can plot this point in the third quadrant.

The final step is to rotate this new point, (-4, -9), 90 degrees in the counterclockwise direction. You could visualize this by literally turning your graph paper one quarter turn (90 degrees) counterclockwise. You can also use the rule for R(90): (x, y) --> (-y, x). Either way, you will arrive at the same point of (9, -4).

So, the final answer to this question is (9, -4). If you have any follow-up questions, please feel free to reach out. Thanks, take care, and I hope this will be helpful!

Best,

Zach Borenstein

P.S. Full disclosure, in my original answer I confused counterclockwise and clockwise! However, another problem helped me recognize my mistake and correct it here. Let me know if you have any questions or concerns!