Introduction

Hello! Welcome to topic 3.7: Sinusoidal Function Context and Data Modeling. In this article, we will discuss how to determine the period, frequency, amplitude, translations, and dilatations of a sinusoidal function. In addition, you will learn how you can use Desmos to find the equation of a function using just data points. Let’s get started!

Period / Frequency

The period of a function $f (x)$ is the minimum positive value $p$ such that $f (x) = f (x + p)$ for all values of $x$ . Note that $f (x)$ must not be a constant.

Let’s take a look at the graph of the function $sin (x)$ .

Notice that the function repeats every so often, specifically every $2 π$ units.

For each, which are spaced $2 π$ units apart, the outputs of the function in between the black lines are the exact same. This means that the period of the function $sin (x)$ is $2 π$ .

An additional property of periodicity is that if $f (x)$ has a period of $p$ , $f (x + h)$ also has a period of $p$ for all values of $h$ , or in other words, a horizontal shift of a function doesn’t change its periodicity. This means that, since $cos (x) = s in (x + \frac{π}{2})$ , the periodicity of $cos (x)$ is also $2 π$ .

Frequency

The frequency of a sinusoidal function is the number of times a sinusoidal function repeats itself over an interval of $2 π$ . The frequency of a sinusoidal function is $\frac{2 π}{p}$ , where $p$ is the period of the function. In the general equation for a sinusoidal function ( $a sin (b (x + c)) + d)$ , $b$ is equal to the frequency.

Amplitude / Vertical Shift

The amplitude of a sinusoidal function is equal to the height that one of its waves makes.

To find the amplitude of a sinusoidal function, we can use the formula $\frac{ma x ( f ) - min ( f )}{2}$ . Since the height of the waves of a sinusoidal function is the same both when the function is negative and positive, the total height of a wave, or the amplitude, is equal to half of the total height of the function.

So far, we have only discussed sinusoidal functions that have a midline, or center, at $y = 0$ . That won’t always be the case, as seen in the graph below:

This graph has a midline at $y = 3$ . To find the midline, we can take advantage of the fact that the waves below the midline have the same height as the waves above the midline, so the height of the midline is $3$ . The height of the midline is also called the vertical shift.

Finding the equation for a sinusoidal function given a set of points

With a Calculator

We will be using Desmos for this, as you are provided this calculator for the AP Exam. For usage of a personal calculator, please consult your calculator’s manual: do note that your calculator might not have the ability to do custom regressions, in which case you will have to settle for what you can.

Click on the plus icon and add a table, and fill in your values

Without deleting your table, type in the following equation. To type the subscript, or the small text after $x$ and $y$ , use the underscore: _. Note that an equals sign is not used, rather, a tilde ( $\sim$ ) is used. $y_{1} \sim a \cdot sin (b (x_{1} + c)) + d$

Desmos will calculate the values of $a$ , $b$ , $c$ , and $d$ that best fit your dataset. It will also graph it for you.

Phase Shift calculations

As you may recall, a phase shift is simply the horizontal translation for sinusoidal functions. One way to determine the phase shift of a function is to compare your actual values with the values the Desmos calculated and modeled. Substitute the actual input value into the modeled equation and solve for $c$ , which is the variable for determining the phase shift. There you go, here’s your phase shift!

Quick Note: In the real world, these sinusoidal functions only occur over a period of time. However, from a mathematical context, these functions go on FOREVER, so keep in mind that just because a real world situation models a sinusoidal function does not mean that it will continue to go on forever.

Solving without a calculator

To find the sinusoidal function that matches a specific set of points given the general equation $a \cdot s in (b (x_{1} + c)) + d$ , follow the following steps.

To find your amplitude: subtract the minimum $y$ -coordinate from the maximum $y$ -coordinate and divide by $2$ , this is your $a$ .
To find the midline / vertical shift, take the average of the maximum and minimum $y$ -coordinate: this is your $d$
To find our frequency, let’s first find our period. We can do this in one of two ways:
1. Find two coordinates with the same y-coordinate where the $y$ -coordinate is equal to the maximum or minimum output of the function. The distance between those two points is the period.
2. Find three coordinates with the same y-coordinate where the $y$ -coordinate is not equal to the maximum or minimum output of the function. The distance between the first and third point is the period.
Our frequency, or $b$ , is just $\frac{2 π}{p}$ , where $p$ is the period.
To find the horizontal shift, we can take advantage of the fact that $sin (0) = 0$ and use our period, $\frac{2 π}{∣ b ∣}$ , and our vertical shift, $d$ . Because $sin (x)$ hits every value twice in a period, it hits the midline at exactly $\frac{1}{2} \cdot period$ , so the $x$ -value within $[0, \frac{period}{2}]$ that results in the vertical shift of $- c$ .
To find missing input or output coordinates, simply plug in the $x$ or $y$ coordinate in its place and solve for the unknown value
1. To find the $x$ -coordinate when given a $y$ -coordinate, inverse trig functions will be used. (aka, $sin^{- 1}$ , $cos^{- 1}$ ). This is a later topic.

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