How Do You Model Periodic Data With a Sinusoidal Function (Tides; Ferris Wheel)?

Use a sinusoidal model when data repeats in a regular cycle. For IB Mathematics: Applications and Interpretation SL 2.5, determine the amplitude, principal axis, and period, then choose sine or cosine to match the starting position.

A suitable SL model is

$$y=a\sin(bx)+d \quad \text{or} \quad y=a\cos(bx)+d.$$

The parameters are found as follows:

$$|a|=\frac{\text{maximum}-\text{minimum}}{2}, \qquad d=\frac{\text{maximum}+\text{minimum}}{2}, \qquad b=\frac{360}{\text{period}}.$$

Here, amplitude (|a|) is the maximum displacement from the middle, the principal axis is $y=d$, and the period is the time for one complete cycle. At SL, angles are measured in degrees.

First, identify one maximum and one minimum from the data or context. Their vertical separation determines the amplitude, while their average locates the midline. Then measure the horizontal distance between matching points, such as consecutive maxima, to obtain the period. The coefficient $b$ converts that period into one full 360-degree cycle.

For a Ferris wheel with radius $12\text{ m}$, centre height $14\text{ m}$, and rotation time $40\text{ s}$, starting at its lowest point:

$$h(t)=-12\cos(9t)+14, \qquad b=\frac{360}{40}=9.$$

The height ranges from $2\text{ m}$ to $26\text{ m}$, confirming amplitude $12\text{ m}$ and principal axis $14\text{ m}$. Substituting $t=0$ gives , so the model correctly represents the lowest starting position. Keep full precision when evaluating later times.

For tides with maximum depth $5.8\text{ m}$, minimum depth $1.4\text{ m}$, and period $12.4\text{ h}$, beginning at high tide:

$$d(t)=2.2\cos\left(\frac{360}{12.4}t\right)+3.6.$$

A common misconception is to swap the amplitude and principal axis. Remember: amplitude uses half the difference, while the principal axis uses the average.

Exam technique: This is SL 2.5 and may appear in either Paper 1 or Paper 2; both are GDC-active. Show how each parameter was obtained, write the complete equation, use degree mode, and interpret the variables and units in context.