# Graphing Calculator Drawing: Sine Waves

*This post is part of a series.*

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

** Demonstration - Equilibrium.** Observe the graph as you type each of the following inputs. In general, the graph of $y=\sin x$ looks like an infinite wavy squiggle oscillating up and down around an equilibrium at $y=0.$ The graph $y=\sin(x)+b$ shifts the equilibrium of the wavy squiggle to the line $y=b.$

** Demonstration - Frequency.** Observe the graph as you type each of the following inputs. The “frequency” of a sine wave refers to how quickly or “frequently” it oscillates. For a sine wave $y=\sin(vx),$ the frequency is controlled by $v.$ If you double $v,$ then the sine wave will oscillate twice as frequently; if you halve $v,$ then the sine wave will oscillate half as frequently. If you set $v=0,$ then the sine wave will not oscillate at all.

** Demonstration - Amplitude.** Observe the graph as you type each of the following inputs. The “amplitude” of a sine wave refers to how high/low its peaks/valleys are in relation to its equilibrium. For a sine wave $y=A\sin(x),$ the amplitude is controlled by $A.$ The peaks of the sine wave reach a height of $A,$ and the valleys of the sine wave reach a depth of $-A.$

** Demonstration - Horizontal Shift.** Observe the graph as you type each of the following inputs. The sine graph $y=\sin(x-a)$ is shifted right $a$ units, meaning that each peak and each valley occurs $a$ units right of its original location.

** Demonstration - Composition with Absolute Value.** Observe the graph as you type each of the following inputs.

** Exercise.** Previously, you drew a fish using parabolas. Now, create a layer of scales on it, using a function of the form $y=-A \vert \sin(x) \vert +b.$

** Exercise.** Now, create a second layer of scales, using a function of the form $y=-A \vert \sin(x-a) \vert +b.$ The peaks of the first layer should line up with the valleys of the second layer.

** Exercise.** Continue making layers of scales until the fish is completely scaled.

** Exercise.** Lastly, use lines to create spines in the tail of the fish.

** Challenge.** Try to draw other scaled creatures, such as a snake!

*This post is part of a series.*