Sine functions and Sinusoids

by Hwa Young Lee

The curve in the graph of the sine function is called a sinusoid. To graph a sinusoid, we take the basic graph of the sine function and expand or compress it vertically or horizontally. We can also shift its position relative to the x or y axes.

When these changes are made to the graphs of these functions, corresponding changes occur in the equations.

Let's examine graphs of and investigate how the changes in the equations are related to the changes in the graph of the sine function.

The graph of looks like this:

There are a few noticeable properties of the graph above.

First, the period is .

Second, the domain is all real numbers and the range is [-1, 1].

Third, the x-intercepts are at , with n an integer. The y-intercept is 0.

Finally, the graph is symmetric about the origin.

Now, what happens to the graph of when we change the values of a, b, and c?

Let's examine what the graph of looks like for a few different values of a.

What does the graph look like as the value of a varies in between 1 and 10?

We can see that multiplying sinx by a positive constant greater than 1 causes a vertical expansion of the graph. This varies the amplitude of the function and thus, the range of the function.

Let's find out what happens if a is negative. Let's examine a few graphs for negative a graphs.

Examine what the graph of looks like as the value of a varies in between -10 and 10.

We can see that multiplying sinx by a constant from -10 to 10 causes a vertical expansion of the graph, only the graph is reflected by the x-axis when a negative constant is multiplied. Thus, we can see that the absolute value of a causes a vertical expansion and effects the amplitude and range of the graph.

What if the absolute value of a is less than 1? To find out, let's examine what the graph of looks like for a few values of a with absolute values less than 1.

Now, let's examine what the graph of looks like as the value of a varies in between -1 and 1.

We can see that multiplying sinx by a constant with an absolute value less than 1 causes a vertical compression of the graph. Likewise, this varies the amplitude of the function and thus, the range of the function.

From the above exploration, the absolute value of parameter 'a' causes a vertical expansion or compression and affects the range of the graph.

Second, let's go on to b. What does the parameter 'b' do?

To find this out, we investigate the graph of when b is positive and say, varies from 1 to 10. What happens?

We can see that the graph is compressed horizontally.

How is it compressed? Let's take a close look and compare the three graphs of y=sinx, y=sin2x, and y=sin4x.

We can see that the original graph of is compressed such that the new periods of the graphs are (a half of the original) and (one fourth of the original), for y=sin2x and y=sin4x respectively.

In the original graph of , the period is 2 which means that the graph is repeating every 2's of x. Likewise, in , the graph should repeat every 2's of bx. Thus, the graph repeats every of x. Let's check this with our two graphs above. The period of y=sin2x is and y=sin4x is !

This gives us an idea of what the graph of would look like when b is positive and less than 1. Can you guess? Let's check out the graph of with 0<b<1.

Since the period of the graph is and 0<b<1, the period is greater than 2so the graph would be expanded horizontally.

Now what happens if b is negative? We can check this by observing the graph of with b varying from -10 to 10.

We can see that the graph has the same change when b is negative, and the only difference is that the graph is flipped over the y-axis. Thus, the absolute value of b determines the period and the period is

Multiplying the variable in the sine function by a constant affects the period and causes a horizontal expansion or compression of its graph. If |b|<1, there is an expansion; if |b|>1, there is a compression.

What does the parameter 'c' do? To find this out, we start with a few c values: c=-1, c=0, c=1.

Now, let's investigate the graph of , when c varies from -10 to 10. What happens?

We can see that the graph shifts horizontally, to the right when c is negative; to the left when c is positive.

Adding a constant to the variable in the sine function causes a horizontal translation of the graph. A positive constant causes a translation to the left; a negative constant causes a translation to the right.

Now, note that in our function , the constant b and c are in the same parantheses. Rewriting our function as , we can see that the actual horizontal translation is determined by the value of .

Furthering our exploration, what happens if we add a constant d to the function?

In other words, what would the graph of look like? It is pretty obvious but let's investigate the graph of  when d varies from -10 to 10. What happens?

We can see that the graph shifts vertically, upwards when d is positive; downwards when d is negative.

Adding a constant to the sine function causes a vertical translation of the graph. The sign of the constant indicates whether the graph is translated up or down. A positive constant causes a translation up; a negative constant causes a translation down.

☑ Question: Which of the constants a, b, c and d do you think determine the period, range, x- and y-intercepts in the graph of ? More specifically, what is the period and what is the range? Use this information and try to guess what the graph of  would look like. Click here for answers.