# Graphing Calculator Drawing: Rotation

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Setup. Navigate to https://www.desmos.com/calculator. Be sure to sign in so that you can save your graph.

Demonstration - Rotation. Observe the graph as you type each of the following inputs. In general, a graph can be rotated by an angle of $\theta$ about the origin by replacing $x$ and $y$ with the following expressions:

\begin{align*} x &\rightarrow x\cos\theta + y\sin\theta \\[5pt] y &\rightarrow y\cos\theta - x\sin\theta \end{align*}

Note that $\theta$ should be given in radians, and one can convert degrees to radians by multiplying by the conversion factor $\frac{\pi}{180}.$

\begin{align*} &y\cos \frac{\pi}{6} - x \sin \frac{\pi}{6} = \left( x \cos \frac{\pi}{6} + y\sin\frac{\pi}{6} \right)^2 \\[10pt] &\left( \frac{x \cos \frac{\pi}{4} + y \sin \frac{\pi}{4} }{4} \right)^{2}+\left( \frac{ y \cos \frac{\pi}{4} - x \sin \frac{\pi}{4} }{2} \right)^{2}=1 \end{align*}

Exercise. Reproduce the graph below by drawing an absolute value function and then rotating it a fifth of a circle counterclockwise.

Exercise. Continue drawing rotated absolute value functions to form a star.

Exercise. Draw a circle that passes through the sharp points of the star.

Exercise. Add a background layer by drawing rotated parabolas.

Exercise. Finally, add non-Euclidean ellipses to the background.

Challenge. Create your own emblem.

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