Model periodic phenomena with sinusoidal functions

Periodic phenomena are represented with sine or cosine functions of the form $y=A\sin(B(x-C))+D$ or $y=A\cos(B(x-C))+D$ , where the amplitude $|A|$ , period $2\pi/|B|$ , phase shift $C$ , and midline $D$ describe measurable features of the phenomenon. Modeling connects these parameters to quantities such as time, height, temperature, or distance, while interpreting graphs, tables, equations, and units and distinguishing periodic repetition from linear change; more advanced models involving damping, multiple frequencies, or arbitrary periodic functions are not included.

Detailed Explanation: 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

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.

Learn by doing: Model periodic phenomena with sinusoidal functions

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Sinusoidal Function Parameters (3 Params) - Parameters to Function

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