Graphing Calculator Drawing: Shading with Sine

by Justin Skycak on

High-frequency sine waves can be used to draw shaded regions.

This post is a chapter in the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Shading with Sine. Graphing Calculator Drawing Exercises. https://justinmath.com/graphing-calculator-drawing-shading-with-sine/

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

Demonstration - High Frequency. Observe the graph as you type each of the following inputs. In general, the graph of $y=\sin(vx)$ looks more and more solid as $v$ increases.

$\begin{align*} y&=\sin(x) \\[10pt] y&=\sin(10x) \\[10pt] y&=\sin(100x) \\[10pt] y&=\sin(1000x) \end{align*}$


Demonstration - Thickening a Curve. Observe the graph as you type each of the following inputs. In general, for large $v,$ the graph of $y=f(x)+A\sin(vx)$ thickens the curve $y=f(x)$ to have a vertical thickness of $2A.$

$\begin{align*} y&=x+0.1\sin(1000x) \\[10pt] y&=x^2+0.1\sin(1000x) \\[10pt] y&=x^2+0.5\sin(1000x) \\[10pt] y&=\sin(x)+0.1\sin(1000x) \\[10pt] y&=\sin(x)+0.5\sin(1000x) \end{align*}$


Demonstration - Varying Amplitude. Observe the graph as you type each of the following inputs. In general, when $v$ is large, the graph of $y=f(x)\sin(vx)$ shades the area between the graphs of $y=-f(x)$ and $y=f(x).$

$\begin{align*} y&=x\sin(1000x) \\[10pt] y&=x^2\sin(1000x) \\[10pt] y&=\left( \sin x \right) \sin (1000x) \end{align*}$


Demonstration - Shifts. Observe the graph as you type each of the following inputs. In general, the graph of $y=f(x-a)\sin(vx)+b$ is the graph of $y=f(x)\sin(vx)$ shifted right by $a$ units and up by $b$ units.

$\begin{align*} y&=x^2 \sin(1000x) \\[10pt] y&=(x-2)\sin(1000x)+5 \end{align*}$


Demonstration - Limitations. Observe the graph as you type each of the following inputs.

$\begin{align*} y&=\sin(1000x) \lbrace 5 < x < 10 \rbrace \\[10pt] y&=(x-2)^2 \sin(1000x) \lbrace 2 < x < 4 \rbrace \\[10pt] y&=\left(\sqrt{x-3}\right)\sin(1000x)+5\left\{3<x<4\right\} \end{align*}$


Exercise. Reproduce the downward parabola shown below.

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Exercise. Use the parabola as the amplitude of a high-frequency sine function to create a shaded area.

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Exercise. Shift the shaded area up and right.

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Exercise. Draw parabolas around the shaded area to create an eye.

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Exercise. Create another eye.

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Exercise. Create a parabola in the shape of a mouth.

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Exercise. Thicken the parabola that forms the shape of the mouth.

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Challenge. Make other kinds of emoji faces, such as a sad face or a laughing face.

This post is a chapter in the book Graphing Calculator Drawing Exercises. Suggested citation: Skycak, J. (2019). Graphing Calculator Drawing: Shading with Sine. Graphing Calculator Drawing Exercises. https://justinmath.com/graphing-calculator-drawing-shading-with-sine/