How to move and/or rotate a semi circle or like any equation. Or just add a perimeter cut off on one of the sides.

Question

I'm just some random 6th grader having fun in graphing trying to get some practice in before high-school but my teacher says she doesn't know how to rotate any equation or how to rotate this semi circle graph https://www.desmos.com/calculator/u34ogdt5wd. I was wondering if there's possibly anyone that could help me because im trying to do the desmos marbleslide challenge #30 on Andymath.com and cant get the semi circles to move/rotate/or get a perimeter cut off on the side. Also how do you move any equation on a graph left to right or up and down without any changes to It's shape. Just wondering.

Answer 1

You might want to look at this

Basically, the idea is that a rotation of the graph is basically given by the following:

$$\begin{pmatrix}x_{new} \\\ y_{new} \end{pmatrix} = \begin{pmatrix}x \\\ y \end{pmatrix} \begin{pmatrix}\cos\theta & -\sin\theta\\\ \sin\theta & \cos\theta\end{pmatrix}$$

$(x_{new},y_{new})$ is the new coordinate after the rotation

$(x,y)$ is the old coordinate $$\theta$$ is the angle of rotation.

The proof is a bit long but read up on it when you're free!