How Do You Model Periodic Data With a Sinusoidal Function Including a Phase Shift?

Model periodic data by identifying its amplitude, period, principal axis, and phase shift, then substituting these parameters into $y=a\sin(b(x-c))+d$. This is AHL 2.9 Further modelling functions and is therefore HL-only content.

The parameters control different features of the model:

Choose the base function that matches the starting condition. Use sine when a principal-axis crossing is convenient, and cosine when the cycle begins at a maximum or minimum. These forms can model the same data; the phase shift changes accordingly.

For example, suppose periodic data has a maximum of $8$, a minimum of $2$, a period of $12$ units, and reaches a maximum at $x=5$.

First calculate

$$a=\frac{8-2}{2}=3,\qquad d=\frac{8+2}{2}=5,$$

and

$$b=\frac{2\pi}{12}=\frac{\pi}{6}.$$

A positive sine curve reaches its maximum one quarter-period after its upward principal-axis crossing. Since $T/4=3$, the crossing occurs at $x=5-3=2$. Therefore $c=2$, giving

$$y=3\sin\left(\frac{\pi}{6}(x-2)\right)+5.$$

Check the model against the data: its principal axis is $y=5$, its range is $2\leq y\leq8$, and its maximum occurs when $x=5$.

The modelling process should also include a check against a known data point. Substituting $x=5$ into the model gives $y=8$, confirming the stated maximum. Over one full period, the key points occur every quarter-period, so they are separated by $3$ units. This provides a quick graphing and GDC check.

A common misconception is that (x-c) shifts the graph left. It shifts the graph right by $c$; the horizontal sign appears reversed.

In an IB response, show how each parameter comes from the data, state a reasonable contextual domain, and use a GDC to verify the fit. Examiners may also expect interpretation of the phase shift and a comment on whether extrapolation is valid.