A periodic function is a function whose output values repeat on a regular interval. In mathematical terms, we say that a periodic function is a function for which a specific horizontal shift, \(P\text{,}\) results in the original function: \(f(x+P)=f(x)\) for all values of \(x\text{.}\)
The period of a periodic function is how long it takes for the output values to begin repeating, or the smallest horizontal shift with \(P>0\text{,}\) such that \(f(x+P)=f(x)\) for all values of \(x\text{.}\)
In addition to the period we also define the midline and amplitude of a periodic function.
The midline of a periodic function is the horizontal line halfway, or midway, between the function's maximum and minimum output values.
The amplitude of a periodic function is the distance between the function's maximum (or minimum) output value and the midline.
The midline often represents the average value of a wave-like periodic function and should always be written as the equation of a horizontal line. The amplitude of a periodic function represents how "large" the function's oscillations are.
The London Eye is a huge Ferris wheel in London, England. It completes one full rotation every 30 minutes, and the diameter of its passenger capsules is 130 meters. Riders board the passenger capsules from a platform that is 5 meters above the ground. At time \(t=0\text{,}\) an individual boards the Ferris wheel.
Sketch a graph of \(h=f(t)\text{,}\) where \(h\) is the height of the individual above ground (in meters) after \(t\) minutes.
Let's start by drawing a picture of the Ferris wheel and labeling the known information.
We know that at time \(t=0\) a rider boards the Ferris wheel from the boarding platform, which is 5 meters above the ground. We also know that the Ferris wheel has a diameter of 130 meters and it takes 30 minutes to complete one full revolution around the Ferris wheel.
Since it takes 30 minutes to complete a trip around the Ferris wheel, a rider will reach the top of the Ferris wheel after 15 minutes (assuming that the wheel rotates at a constant speed). Similarly, the rider will reach the three o'clock and nine o'clock positions on the Ferris wheel at 7.5 minutes and 22.5 minutes.
Next, we can make a table of the rider's height above ground vs. time.
At time \(t=0\text{,}\) the rider is at the boarding platform, which corresponds to a height of 5 meters above ground. After 15 minutes, the rider reaches the top of the Ferris wheel, which corresponds to a height of 135 meters above ground. We can find this height by taking the height of the boarding platform plus the diameter of the Ferris wheel.
We can also find an intermediate height corresponding to when the rider has been on the Ferris wheel for 7.5 minutes and 22.5 minutes. At these two positions, the rider is halfway between the top of the Ferris wheel and the boarding platform. We can find this height by adding the height of the boarding platform to the radius of the Ferris wheel. Therefore, we get a height of \(5+65 = 70\) meters above ground at these two times.
After 30 minutes, the rider is back at the boarding platform, which corresponds to a height of 5 meters above ground. If the individual were to continue riding the Ferris wheel, this pattern of heights would continue and we could extend our table with additional times and heights as shown below.
| time, \(t\) (minutes) | 0 | 7.5 | 15 | 22.5 | 30 | 37.5 | 45 |
| height, \(h=f(t)\) (meters) | 5 | 70 | 135 | 70 | 5 | 70 | 135 |
Now that we have a table of times and corresponding heights, we can plot these points and draw a smooth curve between them to create a height vs. time graph.
Since we are graphing height as a function of time, time is our independent variable and should be plotted along the horizontal axis while height is the dependent variable and should be plotted along the vertical axis.
Note that in this example, we assumed that the Ferris wheel was rotating counterclockwise, as shown in our sketch above. For this problem, it did not matter what direction the Ferris wheel was rotating because both clockwise and counterclockwise rotation would have produced the same height vs. time graph. However, this will not always be the case. If the rider were to start in the three o'clock or nine o'clock position, we would need to know whether the Ferris wheel was turning clockwise or counterclockwise in order to sketch a graph of the rider's height above ground.
Find the period, midline, and amplitude of the function shown below.
By definition, the period of the function is how long it takes for the function to start repeating. If we think about the function "starting" at \(x=0\text{,}\) or when it crosses the \(y\)-axis, we can see that it begins repeating when \(x=4\text{.}\) Therefore, the period of the function is 4.
The midline of the function is the horizontal line halfway between the function's maximum and minimum values. Here, the maximum value of the function is 5 and the minimum value is 1. To find the number halfway between 5 and 1, we can take the average of the two numbers. The average of 5 and 1 is
Thus, the midline of the function is the line \(y=3\text{.}\)
The amplitude of the function is the distance between the function's maximum value and the midline. We can find this value by subtracting the midline from the maximum value. Therefore, the amplitude of this function is \(5 - 3 = 2\text{.}\)
While measuring angles in degrees may be familiar to many students, doing so often complicates matters since the units of measure can get in the way of calculations. For this reason, another measure of angles is commonly used. This measure is based on the distance around the unit circle.
The unit circle is a circle of radius 1, centered at the origin of the \(xy\)-plane. When measuring an angle around the unit circle, we travel in the counterclockwise direction, starting from the positive \(x\)-axis. A negative angle is measured in the opposite, or clockwise, direction. A complete trip around the unit circle amounts to a total of 360 degrees.
A radian is a measurement of an angle that arises from looking at angles as a fraction of the circumference of the unit circle. A complete trip around the unit circle amounts to a total of \(2\pi\) radians.
Radians are a unitless measure. Therefore, it is not necessary to write the label "radians" after a radian measure, and if you see an angle that is not labeled with "degrees" or the degree symbol, you should assume that it is a radian measure.
Radians and degrees both measure angles. Thus, it is possible to convert between the two. Since one rotation around the unit circle equals 360 degrees or \(2\pi\) radians, we can use this as a conversion factor.
Since \(360 \text{ degrees} = 2\pi \text{ radians}\text{,}\) we can divide each side by 360 and conclude that
So, to convert from degrees to radians, we can multiply by \(\displaystyle \ \frac{\pi \text{ radians}}{180^\circ}\)
Similarly, we can conclude that
So, to convert from radians to degrees, we can multiply by \(\displaystyle \ \frac{180^\circ}{\pi \text{ radians}}\)
Convert \(\displaystyle \frac{\pi}{6}\) radians to degrees.
Since we are given an angle in radians and we want to convert it to degrees, we multiply the angle by \(180^\circ\) and then divide by \(\pi\) radians.
Convert \(15^\circ\) to radians.
In this example, we start with an angle in degrees and want to convert it to radians. We multiply by \(\pi\) and divide by \(180^\circ\) so that the units of degrees cancel and we are left with the unitless measure of radians.
