Building sinusoidal models for AP Precalculus

The four numbers in a sinusoidal model

A sinusoidal model has the form $f(t) = a\cos(b(t - c)) + d$ or $f(t) = a\sin(b(t - c)) + d$. Each letter comes from one fact about the situation.

• Midline, $d$: halfway between the highest and lowest values, $d = \frac{\text{max} + \text{min}}{2}$.
• Amplitude, $|a|$: half the distance from the lowest value to the highest, $|a| = \frac{\text{max} - \text{min}}{2}$.
• Period: the time for one full cycle. Then $b = \frac{2\pi}{\text{period}}$.
• Phase shift, $c$: where the cycle starts. Cosine starts at a maximum, $-\cos$ starts at a minimum, and sine starts on the midline going up.

A worked example from a high and a low

Here's the whole process on one context. The same steps work for any description that gives a highest value, a lowest value, and when they happen.

On a calculator question you'd find the times in (c) by graphing $y = D(x)$ and $y = 11.5$ in Desmos and clicking the intersections. A sinusoid crosses most horizontal lines twice per period, so check the interval for every crossing.

Sine or cosine, and the sign of the shift

The same curve has many correct formulas. In the harbor model, the depth crosses the midline going up a quarter period before high tide, at $t = 4 - 3 = 1$. So $D(t) = 5\sin\left(\frac{\pi}{6}(t - 1)\right) + 9$ is the same function. Either one is correct, as long as the maximum lands at the right time.

• Start at a maximum: $a\cos(b(t - c)) + d$ with $a > 0$ and $c$ the time of the maximum.
• Start at a minimum: $-a\cos(b(t - c)) + d$, with $c$ the time of the minimum.
• Start on the midline going up: $a\sin(b(t - c)) + d$, with $c$ the time of the crossing.
• Check your model by plugging in the time of the maximum. You should get the highest value.

How a sinusoid increases, decreases, and bends

• Increasing from each minimum to the next maximum, and decreasing from each maximum to the next minimum.
• Concave down above the midline, and concave up below it.
• Changing fastest as it crosses the midline, and slowest near a maximum or minimum.

Free-response questions usually ask you to back these up with average rates of change. Over consecutive intervals of the same length, increasing rates of change mean concave up, and decreasing rates of change mean concave down.

Fitting a sinusoid to data in Desmos

When you're given a table instead of a description, fit the model in Desmos. This comes up on calculator multiple-choice questions.

1. 1Enter the data in a table, or as lists $x_1 = [\ldots]$ and $y_1 = [\ldots]$.
2. 2If the period is known, type $b$ in. For a 12-month cycle, type $y_1 \sim a\sin\left(\frac{\pi}{6}(x_1 - c)\right) + d$.
3. 3Read $a$, $c$, and $d$ from the regression. A negative $a$, or a $c$ that differs from yours by a full period, describes the same curve.
4. 4Graph the model over the points to check the fit, then evaluate it or intersect it with a horizontal line.

Typing the known period keeps Desmos from settling on a curve that wiggles through the points with a period that makes no sense. More on choosing and checking models is in the regression guide.

A quick check

For how this fits into the third free-response question, see the free-response guide.

Questions

How do I find the period from a maximum and a minimum?

The time from a maximum to the next minimum is half the period, so double it. Then b is 2π divided by the period.

Should I use sine or cosine?

Either works. Cosine is easiest when you know when the maximum happens, negative cosine when you know the minimum, and sine when you know a midline crossing going up.

Where is a sinusoidal function changing fastest?

As it crosses the midline. It changes slowest near its maximum and minimum values.

Is the sinusoidal free-response question done with a calculator?

No. FRQ 3, Modeling a Periodic Context, is in the no-calculator part, so values come from the unit circle.