Model periodic phenomena with trigonometric functions

Periodic quantities such as tides, daylight, or rotating motion can be represented by sinusoidal functions of the form $y=A\sin(B(x-h))+D$ or cosine equivalents, with amplitude $|A|$ , period $2\pi/|B|$ , horizontal shift $h$ , and midline $y=D$ interpreted in the context’s units; the model’s range and repeated behavior must agree with the phenomenon and data or graph. The focus is on constructing, comparing, and interpreting these models, not on Fourier analysis, differential-equation models, or other more advanced periodic-function theory.

Detailed Explanation: Model periodic phenomena with trigonometric functions

Periodic phenomena repeat in regular cycles, so they can be modeled with a sine or cosine function:

y=A\sin(B(x-h))+D

y=A\cos(B(x-h))+D.

The important features are:

Amplitude: $\lvert A\rvert$ , half the distance between the maximum and minimum
Midline: $y=D$ , the average of the maximum and minimum
Period: $\dfrac{2\pi}{\lvert B\rvert}$
Horizontal shift: $h$
Range: from $D-\lvert A\rvert$ to $D+\lvert A\rvert$

Worked example

The depth of water at a harbour varies periodically. At low tide, the depth is $2$ m at 3:00 a.m. At high tide, the depth is $8$ m at 9:00 a.m. The pattern repeats every $12$ hours. Create a model for the water depth.

Let $x$ be the number of hours after midnight, and let $y$ be the water depth in metres.

1. Find the amplitude

The maximum is $8$ and the minimum is $2$ , so

A=\frac{8-2}{2}=3.

The water depth varies $3$ m above and below its midline.

2. Find the midline

The midline is the average of the maximum and minimum:

D=\frac{8+2}{2}=5.

Thus, the average water depth is $5$ m.

3. Find $B$ using the period

The period is $12$ hours. Since

\text{Period}=\frac{2\pi}{\lvert B\rvert},

we have

12=\frac{2\pi}{B}.

Solving gives

B=\frac{\pi}{6}.

4. Choose sine or cosine and determine the shift

At $x=9$ , the water is at its maximum of $8$ m. Cosine starts at a maximum, so use a cosine model shifted to $x=9$ :

y=3\cos\left(\frac{\pi}{6}(x-9)\right)+5.

This is a model for the water depth.

5. Check the model

At high tide, $x=9$ :

y=3\cos(0)+5 =3(1)+5 =8.

At low tide, $x=3$ :

y=3\cos\left(\frac{\pi}{6}(3-9)\right)+5

=3\cos(-\pi)+5 =3(-1)+5 =2.

The model gives the correct high and low tides.

Therefore,

\boxed{y=3\cos\left(\frac{\pi}{6}(x-9)\right)+5}

where $y$ is the water depth in metres and $x$ is the number of hours after midnight.

Its range is

2\le y\le 8,

and it repeats every $12$ hours, matching the situation.

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Sinusoidal Function Parameters (4 Params) - Graph to Function

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