周期函数：振幅和中线

##Finding the Amplitude, Midline, and Period/Frequency of Periodic Graphs

Just as a reminder, periodic functions are functions that repeat itself, most notably trigonometric functions like sine and cosine.

Let's look at an example of a $\sin(\theta)$ function:

From this graph, we can see sections of the graph repeating, which are known as periods. However, as mathematicians, we want to quantify these periods, so we use concepts like amplitudes, midlines, and periods to do so.

To better understand the parts of a periodic function, it is useful to think about the standard form of a sine function.

$$y = A\sin(B(x-C))+D$$

For now, we will ignore the $C$ and $D$ because those indicate the phase shift of the graph.

Let's start with amplitudes, which is represented by the $A$ in the equation.

Amplitudes are essentially half the full height of the periodic function. Looking at the $\sin(x)$ graph below, we can see the amplitude labeled. To find the amplitude, you can simply subtract the maximum and minimum, then divide by 2. In other words:

$$Amplitude = \frac{maximum - minimum}{2}$$

For the sine graph, the amplitude equals $\frac{1-(-1)}{2} = 1$.

Now, let's look at the midline. The midline is the middle of the periodic graph. In other words, halfway between the minimum and maximum of the graph. You might notice that the definition of the midline sounds very similar to that of the amplitude, but they're not the same! Essentially, you can add the maximum and minimum, then divide in half.

$$Midline = \frac{maximum + minimum}{2}$$

Notice, the amplitude is subtracting the max and min, while the midline is adding the max and min.

For the sine graph, the midline would be $y = \frac{1+(-1)}{2} = 0$

Lastly, let's find the period of the graph. The period is essentially identifying the subsection of the graph that is being repeated, which is easily identified by finding the distance between two peaks on the graph.

We can find this distance looking at the graph, but sometimes that can be hard, so there is a nifty formula that uses the standard equation of a periodic function to find the period!

Given the function, $y = A\sin(B(x-C))+D$, we can solve for the period by:

$$Period = \frac{2\pi}{|B|}$$

In terms of the sine equation, the period is $\pi$.

Here's the same $y=\sin(x)$ graph as above with the amplitude, midline, and period labeled!

Next, learn how you can use your new skills in finding the amplitude, midline, and period to graph more in-depth sine, cosine, and tangent functions!

Characteristics of Periodic Graphs

Now that we are familiar with periodic data and periodic functions, our next step is to understand the characteristics and main features of periodic graphs.

The amplitude of a periodic function is the distance from the midline to the peak or trough. The peak is the highest point of a function, and the trough is the lowest point of a periodic function.

The period of a function is the distance between two corresponding points on a periodic function. The period can go from the peak to the peak or the trough to the trough. It could also go from an intercept to an intercept.

(Video) Find the Period and Amplitude of the Sine Equation

by Brian McLogan

In this video, Brian McLogan will walk you through how to find the period and amplitude of a sine function. Because a sine function is an example of a periodic function, you can apply these skills to any periodic function! It may be helpful to review what a periodic function is.