4.1 Graphs of Sine and Cosine Functions

In this post, we are going to get into graphing the functions of sine and cosine. While this may seem daunting at first, it’s actually quite simple one you understand what the base graphs of these functions look like.

The graphs of sine and cosine are periodic functions, which means that repeat over a certain interval. A period in a function is two points along the x-axis in which one full repetition of the function takes place.

It’s also important to note the domain, range, and period of these functions.

\text{Domain: } (-\infty,\infty) \\ \text{Range: } [-1,1] \\ \text{Period: } 2\pi

Another important point to make is that the amplitude of these base graphs is 1.

Amplitude when referring to functions like sine and cosine is a graphs maximum distance from its midline, or where its peaks and troughs will lie. You can find the amplitude with this equation:

A = \frac{\text{maximum-minimum}}{2}

Basic Graph of Sine

You can see the amplitude is denoted by the number 1, but if there is no number in front of sinx, you can assume the amplitude is 1.

f(x) = 1\text{sin}x \text{ OR } f(x) = \text{sin}x

Here below is my best rendition of the graph of sine over one period. The graph technically keeps going in both directions across the x-axis and does not end after this one period, but I wanted to show what one specific interval of this graph is doing to break it down and make it obvious.

Typically when you are graphing for homework or a test, you will probably only need to graph one to two periods of a graph to show you understand what is happening in the graph.

As you can see, the graph of sine crosses though the point (0,0), comes up to 1, crosses the x-axis, then goes back down on the other side to 1 before coming up to conclude one period of the graph at the x-axis.

The period of the graph of sine is 2π. As you can see, the first period of this graph ends just after 6 (or around 2π), and half the period of the function is π, or just past 3.

Manipulating the Sine Graph

Now, what if we take that same graph and make it negative?

f(x) = -\text{sin}x

Now, instead of starting at (0,0) and increasing from there, we are reversing the function and decreasing from (0,0), all the way until y equals -1 and x equals π/2, where we start to increase the function, cross through π, and then peak at 1 and 2π. This graph shows one whole period of f(x) = -sinx.

Now let’s play with the amplitude of our sine graph.

f(x) = 2\text{sin}x

This means that we are going to have a vertical stretch of two. Let’s take a look at how our graph is affected.

At first glance, the graph appears exactly identical to the graph of f(x) = sinx; however, we can see on closer look that the peaks and troughs for our graph are occurring at y=2 and y=-2. This has changed the range of our function, but the domain has remained the same (as the function continues on forever and ever in either direction along the x-axis).

Graph of Cosine

Below, we have one period of the graph of the cosine function. Similar to the graph of sine, the period of the graph of cosine is 2π. Instead of using whole integers to demonstrate the units on the graph, I decided to use units of π to show that you can graph either way.

One of the first immediate differences we can note between the graph of sine and cosine is the fact that the graph crosses through the point (0,1) and then decreases through 1. Also, instead of crossing the x-axis at the points of π and 2π, our graph is doing the opposite and reaching its peaks and troughs at these points.

Now, what if we take the inverse of the cosine graph?

f(x) = -\text{cos}x

Now, instead of starting at (0,1), we are starting at the point (0,-1) and then increasing until we hit the peak at π and 1, and then decreasing.

Horizontal Shifts

f(x) = \text{asinbx} \\ f(x) = \text{acosbx} \\ \text{a} = \text{amplitude} \\ \text{period} = \frac{2\pi}{\text{b}}

With these equations, we can see how the different numbers in the function will affect our output. For example, the number in front of sine or cosine (a) will determine our vertical stretch or shrink, and the number after sine or cosine and before our x will determine the period of the graph (horizontal stretch or shrink).

Let’s take a look at an example of a horizontal shift.

Let’s say we have the equation:

f(x) = \text{sin}2x

This is where graphing can get a little bit trickier because we might have to do a little bit of math to get our points on our graph, but it’s nothing a little basic math can’t cover.

**ALWAYS PLOT AT LEAST 5 POINTS ON YOUR GRAPH!

So, because we have a horizontal shrink of 1/2, we know our ending point for the period is going to be at x = π. We know that our first point is going to be (0,0). Now let’s take half π to find our middle point. Half π is π/2. Now to find the next point, we can add π to π/2 to get 3π/2, and then divide it by 2 to get 3π/4. Divide π/2 by 2 to get π/4. Now we have all of our points.

Dealing with horizontal stretches and shrinks is really that easy.

More Examples

y = 6\text{sin}(-x) 

Let’s extract the negative from the parentheses:

y=-6\text{sin}x

This graph is going to look exactly like our y=-sinx graph, except we are going to have an amplitude of 6 instead of 1.

y= \text{sin}\frac{1}{4}x

y = -4\text{cos}(\frac{1}{2}x)

Find the equation for the following graph:

See if you can find the equation for this graph yourself before continuing on.

Solution

So the graph we have above looks a lot like a sine graph as it passes through (0,0), but it is the inverse of the sine graph, so we know that our equation is going to be negative.

The next thing that we can notice is that the amplitude of the graph is 2.

We can also see that one whole period of the graph goes until π, so we know that there’s a vertical shrink happening of 1/2 as the period is normally 2π.

This is enough information to give us the equation of:

y= -2\text{sin}2x

Conclusion

So, in this post, we have learned about the basic sine and cosine graphs and how they can be manipulated. What happens when we make the function negative, how we can vertically stretch and shrink the graph by playing with the amplitude, and how we can horizontally stretch and shrink our graph by changing the period.

I hope that this post has been enough to make graphing sine and cosine a much simpler task for you! Expect a video soon to accompany this post.

Thanks!