Sinusoidal Models from Period and Extremes

From extreme values to a sinusoidal model

A repeating quantity may rise and fall in a smooth cycle. Examples include an idealized daily temperature pattern, the height of a point on a rotating wheel, and the position of a simple oscillation. If the pattern is sinusoidal, its period and extreme values tell us much of its shape.

There is an important distinction between determining the shape and placing the model on the horizontal axis. The maximum and minimum determine the amplitude and midline; the period determines how quickly the cycle repeats. But these values alone do not say when a maximum or minimum occurs. To determine the phase shift, we also need the horizontal location of at least one extreme, or other equivalent timing information.

Amplitude and midline from the extremes

Let the maximum output be \(M\) and the minimum output be \(m\), with \(M>m\). The amplitude measures the distance from the midline to either extreme. The midline is halfway between the maximum and minimum. Thus the vertical distance between the extremes is twice the amplitude.

These formulas apply whether the outputs are positive, negative, or on both sides of zero. The amplitude is nonnegative; when the maximum and minimum are distinct, it is positive. The midline can be negative, zero, or positive.

How the period determines the input coefficient

The basic sine and cosine functions repeat when their input increases by \(2\pi\). For a sinusoid with period \(P\), the input angle must therefore increase by \(2\pi\) when \(x\) increases by \(P\). This is achieved by multiplying the horizontal displacement by \(2\pi/P\).

Here \(h\) is a horizontal location used to align the cycle. The factor \(2\pi/P\) makes one full cycle correspond to a change of \(P\) in \(x\). The angle is in radians, consistent with the standard sine and cosine periods.

For a cosine model with positive amplitude, the cosine reaches its maximum value \(1\) when its input is \(0\), and its minimum value \(-1\) when its input is \(\pi\). Therefore, choosing \(h\) as the location of a maximum gives a direct model:

If instead the known anchor is a minimum at \(x=q\), use a negative cosine term, or equivalently shift the maximum by half a period:

Why the maximum-anchored cosine works

We can verify the model directly. Suppose \(A=(M-m)/2\), \(D=(M+m)/2\), and the maximum occurs at \(x=h\). At that point, the cosine input is zero, so the output is \(D+A\). At a point half a period later, the input is \(\pi\), so the output is \(D-A\).

After another half-period, the input has increased by another \(\pi\), so the cosine returns to \(1\) and the output returns to \(M\). A complete cycle therefore takes \(P\). This verifies the extremes and the period, not just the shape of the expression.

Building the model from a maximum

When the time or position of a maximum is known, a positive-amplitude cosine model is often the simplest choice. In a model written as \(D+A\cos\!\left(\frac{2\pi}{P}(x-h)\right)\), the phase shift is \(h\): the graph is shifted horizontally so its maximum occurs at \(x=h\).

Writing a sine model

Sine can describe the same cycle, but its maximum occurs when its input is \(\pi/2\), rather than \(0\). If a maximum occurs at \(x=h\), the sine input must equal \(\pi/2\) there. With the same period \(P\), the corresponding model is

Indeed, at \(x=h\), the input is \(\frac{2\pi}{P}\cdot\frac{P}{4}=\pi/2\), giving the maximum. In this form the horizontal shift is \(h-P/4\), not \(h\). The sine and cosine models describe the same cycle; their phase shifts differ because their unshifted graphs start at different positions in the cycle.

When the minimum location is given

A minimum can serve as the phase anchor just as well as a maximum. With positive amplitude, subtracting a cosine term makes the output smallest when the cosine is \(1\). If the minimum occurs at \(x=q\), then \(y=D-A\cos\!\left(\frac{2\pi}{P}(x-q)\right)\) reaches \(D-A=m\) at that location.

Phase information and common pitfalls

A period and a pair of extreme values do not determine a unique horizontal placement. For instance, shifting a sinusoid horizontally leaves its maximum, minimum, and period unchanged. To identify a particular model as a function of \(x\), you need an anchor such as “the maximum occurs at \(x=h\)” or “the minimum occurs at \(x=q\).” If no timing or horizontal location is supplied, you can find the amplitude, midline, and period coefficient, but not a unique phase shift.

A frequent mistake is to call the full vertical range the amplitude. For the maximum \(18\) and minimum \(6\), the incorrect calculation \(A=18-6=12\) would give a range of \(24\) if used with the correct midline \(12\): the predicted extremes would be \(12+12=24\) and \(12-12=0\), not \(18\) and \(6\). The correction is to halve the range: \(A=(18-6)/2=6\). Another common error is to set the cosine shift equal to the maximum’s location in a sine model. A sine reaches its maximum one quarter-period after its shift, so for a maximum at \(h\), the sine shift is \(h-P/4\).

Interpreting the parameters

Each parameter answers a different question. The amplitude is half the distance between the extremes, so it measures the size of the oscillation about the midline. The midline is the average of the extremes, so it gives the central output level. The period is the horizontal length of one complete cycle. The phase shift places that cycle relative to the horizontal coordinate \(x\).

Keep the units meaningful: if \(x\) is measured in hours, then \(P\) and the phase shift are also in hours. The amplitude and midline have the units of the output. For example, in the temperature model, \(24\) and \(14\) are hours, while \(10\) and \(20\) are degrees.

These steps construct a model from specified cycle information. When a model must instead be inferred from a collection of repeating observations, the additional task is to choose parameters that represent those data.