Precalculus : Graphing the Sine and Cosine Functions

Example Questions

Example Question #1 : Find The Period Of A Sine Or Cosine Function

Given , what is the period for the function?

Explanation:

The formula for the period of a sine/cosine function is .

With the standard form being:

Since , the formula becomes .

Simplified, the period is .

Example Question #2 : Find The Period Of A Sine Or Cosine Function

What is the period of this graph?

Explanation:

One wave of the graph goes exactly from 0 to before repeating itself. This means that the period is .

Example Question #3 : Find The Period Of A Sine Or Cosine Function

Find the period of the following function in radians:

Explanation:

If you look at a graph, you can see that the period (length of one wave) is . Without the graph, you can divide  with the frequency, which in this case, is 1.

Example Question #4 : Find The Period Of A Sine Or Cosine Function

Find the period of the following function.

Explanation:

The period is defined as the length of one wave of the function. In this case, one full wave is 180 degrees or  radians. You can figure this out without looking at a graph by dividing  with the frequency, which in this case, is 2.

Example Question #5 : Find The Period Of A Sine Or Cosine Function

What is the period of this sine graph?

Explanation:

The graph has 3 waves between 0 and , meaning that the length of each of the waves is divided by 3, or .

Example Question #6 : Find The Period Of A Sine Or Cosine Function

Write the equation for a cosine graph with a minimum at  and a maximum at .

Explanation:

The equation for this graph will be in the form where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

To write this equation, it is helpful to sketch a graph:

From sketching the maximum and the minimum, we can see that the graph is centered at and has an amplitude of 2.

The distance between the maximum and the minimum is half the wavelength. Here, it is . That means that the full wavelength is , so the frequency is 1.

The minimum occurs in the middle of the graph, so to figure out where it starts, subtract from the minimum's x-coordinate:

This graph's equation is

.

Example Question #7 : Find The Period Of A Sine Or Cosine Function

Give the period and frequency for the equation .

Period: , Frequency:

Period: , Frequency:

Period: , Frequency:

Period: , Frequency:

Period: , Frequency:

Period: , Frequency:

Explanation:

Our equation is in the form

where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

We can look at the equation and see that the frequency, , is .

The period is , so in this case .

Example Question #1 : Find The Period Of A Sine Or Cosine Function

What is the period of the graph ?

Explanation:

The equation for this function is in the form

where A is the amplitude, f is the frequency, h is the horizontal shift, and k is the vertical shift.

By looking at the equation, we can see that the frequency, , is .

The period is , so in this case .

Example Question #9 : Find The Period Of A Sine Or Cosine Function

What could be the function for the following graph?

Explanation:

What could be the function for the following graph?

Begin by realizing we are dealing with a periodic function, so sine and cosine are your best bet.

Next, note that the range of the function is and that the function goes through the point .

From this information, we can find the amplitude:

So our function must have a  out in front.

Also, from the point , we can deduce that the function has a vertical translation of positive two.

The only remaining obstacle, is whether the function is sine or cosine. Recall that sine passes through , while cosine passes through . this means that our function must be a sine function, because in order to be a cosien graph, we would need a horizontal translation as well.

Example Question #1 : Find The Amplitude Of A Sine Or Cosine Function

What is the amplitude of ?