Graphing Cosine Functions

How to Graph a Cosine Function

The graph above shows one period of a cosine wave. We know that the period is equal to $2\pi$, since the sine wave will repeat every $2\pi$ units.

Above, we have a sine wave that extends further than $2\pi$. We also have a few more identifiers we need to define.

$\lambda$: Pronounced "lam-duh": This is the wavelength of the sine wave. This is the distance between each crest (or tip) of the wave.

$A$: Amplitude: This is the height of each wave. More technically, it is the measure of deviation from its central value. This value is always positive.

Let's take a look at how we can affect our $A$ and $\lambda$ values!

Pictured above, we have a typical cosine wave. This would be written as $f(t) = \cos(t)$. I used $f(t)$ because cosine is a function of some number $t$. Now, let's look at what happens if we make it $f(t) = 2\cos(t)$.

Notice the change in our $A$ (amplitude) value! By putting a value in front of the $\cos$, we changed the graph by that factor (i.e. multiplied $1 \times 2$ to get the new amplitude)

Great! Now let's look at how we could change $\lambda$ (wavelength).

If we have a function defined as $f(t) = \cos(2t)$, this will look like:

Notice how we will now have $2$ full waves in the same $2\pi$ period! Since the cosine wave is cycling twice as fast, so the period is only half as long! In this case, the period of the graph is $\pi$.

(Video) Graphing a Trig Function with Cosine

by Brian McLogan

In this video by Brian McLogan, you will learn how to graph cosine functions.

Summary

Graph the following trig function: $f(x) = 2 \text{ cos }(2x)$.

We know that the general form of the cosine function is $f(x)=a \text{ cos }(bx-c) +d$

That means that the amplitude of our function is $|2|=2$.

The period is $\frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.

Now, we can graph the cosine function! Since the amplitude is $2$, the distance from the $x$-axis to the highest point is $2$, and the distance from the $x$-axis to the lowest point is $2$.

The period is $\pi$, so halfway is $\frac{\pi}{2}$. Our graph looks like:

Cosine Graphs

In the previous lessons on trigonometric functions, we looked at the sine graph. The sine and cosine functions look very similar, except the cosine function has a horizontal shift (not a vertical shift). The amplitude and period of the sine and cosine functions are the same.

When graphing the parent cosine function graph, there are a few key points to keep in mind that will make it easier to graph the cosine function!

The peak is the highest point of a periodic function, and the trough is the lowest point. The length from peak to peak is the period (or the length from trough to trough).

For a parent cosine graph,

• The amplitude (the distance from the midline to the trough/peak) is $1$.

• The period is $2\pi$.

• The $x$-intercepts are $\frac{\pi}{2}$ and $\frac{3\pi}{2}$.

• The $y$-intercept is at $y=1$.

The equation for the general formula for the cosine function is:

$$\cos(\alpha) = \cos(\alpha + 2k\pi), k \in all \hspace{0.25cm} integers$$

Looking Further

Now that you've mastered the basics of graphing, you are ready to learn about translating cosine functions!