Skill: Model periodic phenomena with sinusoidal functions

A sinusoidal model describes a quantity that repeats regularly, such as the height of a Ferris wheel, the temperature during a day, or the position of a swinging object.

Use the form

$$y=A\cos(B(x-C))+D$$

or

$$y=A\sin(B(x-C))+D.$$

The parameters have these meanings:

• Amplitude: $\vert A \vert$, half the distance between the maximum and minimum
• Period: $\dfrac{2\pi}{ \vert B \vert }$, the length of one complete cycle
• Phase shift: $C$, the horizontal shift
• Midline: $y=D$, the average of the maximum and minimum

Worked example

A Ferris wheel has a maximum height of $20$ meters and a minimum height of $4$ meters. It reaches its maximum height at $t=4$ seconds and completes one rotation every $16$ seconds. Model its height and find its height at $t=6$ seconds.

Step 1: Find the midline and amplitude

The midline is the average of the maximum and minimum:

$$D=\frac{20+4}{2}=12$$

The amplitude is half the difference:

$$A=\frac{20-4}{2}=8$$

So the model will have the form

$$h(t)=8\cos(B(t-C))+12.$$

Step 2: Use the period to find $B$

The period is $16$, so

$$\frac{2\pi}{ \vert B \vert }=16.$$

Solving for $B$ gives

$$B=\frac{2\pi}{16}=\frac{\pi}{8}.$$

Step 3: Use the maximum to find the phase shift

A cosine function reaches its maximum when its input is $0$. Since the Ferris wheel reaches its maximum at $t=4$, use $C=4$.

The model is

$$\boxed{h(t)=8\cos\left(\frac{\pi}{8}(t-4)\right)+12}.$$

Here:

• $8$ is the amplitude, in meters.
• $\dfrac{\pi}{8}$ gives a period of $16$ seconds.
• $t-4$ shifts the graph right $4$ seconds.
• $12$ is the midline height.

Step 4: Find the height at $t=6$

Substitute $t=6$:

$$h(6)=8\cos\left(\frac{\pi}{8}(6-4)\right)+12$$ $$h(6)=8\cos\left(\frac{\pi}{4}\right)+12$$

Since $\cos\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$,

$$h(6)=8\left(\frac{\sqrt{2}}{2}\right)+12$$ $$h(6)=4\sqrt{2}+12\approx 17.7.$$

Therefore, the Ferris wheel is approximately

$$\boxed{17.7\text{ meters}}$$

high at $t=6$ seconds.

To build a sinusoidal model, identify the maximum and minimum, calculate the amplitude and midline, use the period to find $B$, and use a known maximum, minimum, or midline crossing to determine the phase shift.