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Sometimes patterns that repeat over time seem to emerge when using functions to model real-life situations. Knowing the properties of these functions is important to better understand their behavior. This lesson will introduce the concept of *periodic function*, as well as its key features. ### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Consider the following graphs.

- What do these graphs have in common?
- What are their differences?
- Try determining if there is a repeating pattern in each graph.

Discussion

A periodic function is a function that repeats its outputs at regular intervals, forming a definite pattern. The cycle of a periodic function is the shortest repeating portion of the graph, and the period is the horizontal length of one cycle.

More precisely, a function $f(x)$ is said to be periodic if a non-zero constant $P$ exists such that $f(x+P)$ and $f(x)$ have the same value if and only if both $x$ and $x+P$ are in the domain.

$f(x+P)=f(x)$

Example

Kriz went to a hospital to with their cousin, who went there to donate blood. While waiting for their cousin, Kriz noticed that a nurse was setting up some device. They took a look at the screen and found a cool graph.

The nurse said that it was an electrocardiogram, a device that is used to record the electrical activity of a heart. Kriz noticed that the graph in the electrocardiogram was periodic, so they wondered if they could use the screen's grid to find the period. What is the period of the graph shown in the electrocardiogram?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"units","answer":{"text":["4"]}}

Identify and find the horizontal length of a cycle in the graph.

Begin by identifying a repeating portion of the graph. The peaks can be used as a reference in this case.

Note that any other portion of the graph could be used as well. This portion was chosen because the peaks fall on vertical gridlines and their heights stand out from the rest of the graph. Now that one cycle has been identified, the horizontal length of the cycle can be measured.

The horizontal length of the cycle is $4$ units. This is the period of the function. Note that any other two points on the graph that are $4$ units apart have the same $y-$value.

Pop Quiz

Consider the periodic function in the applet.

Example

Still waiting for their cousin, Kriz picked up a pamphlet in the hospital waiting room about a diabetes awareness campaign. Kriz was interested in one particular graph in the pamphlet.

This graph represents the glucose level in blood during one day. Be aware that the graph for another day may be completely different. The pamphlet explains that a person's blood glucose level increases after every meal, and after a while it goes back down. Kriz noticed that each hump of the graph corresponds to a meal.

There appears to be a pattern in the blood glucose level graph, and Kriz thinks that this is a periodic function. Is Kriz correct?{"type":"choice","form":{"alts":["Yes","No"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":1}

Recall that a periodic function is a function that repeats its outputs at regular intervals, forming a distinct pattern. The humps of the graph appear to be a repeating pattern, so their heights will be inspected.

The heights of the humps are not equal, which means that the function does not repeat its outputs. Keep in mind that the graph for a different day may be completely different. Therefore, the given function is not a periodic function. This means that Kriz is not correct.

Discussion

The main characteristic of periodic functions is their repeating behavior. For example, a periodic function alternates between its maximum and minimum values at a regular pace. For this reason, it is important to find the mean between the maximum and minimum values.

Concept

The midline of a periodic function is the horizontal line located in the middle of the minimum and maximum values.

The value of the midline can be found by taking the average of the maximum and the minimum values of the function.

$y_{mid}=2y_{max}+y_{min} $

Just like we can find the mean between the maximum and minimum values of a periodic function, we can also find their difference — or half their difference.

Concept

The amplitude is half the difference of the maximum and minimum values of a periodic function.

$A=2y_{max}−y_{min} $

The amplitude of a periodic function does not depend on the period of the function. Drag the glider in the following applet to change the period of the function.

Additionally, because translations move all the points from a graph in the same way, even if a graph is translated, its amplitude does not change.

The *peak-to-peak amplitude* is defined as the distance between the highest value and the lowest value of a periodic function.

All in all, the amplitude of a periodic function is half the difference between the maximum and minimum values of the function. The midline is the horizontal line that passes right between these maximum and minimum values.

Example

Once Kriz's cousin finished donating blood, Kriz went back with the nurse to thank them for explaining the functionality of the electrocardiogram. The nurse smiled and told Kriz that they still need to wait a little more to see if Kriz's cousin would pass out due to the blood extraction.

The nurse decided to show them another device. This time it is a capnometer, which is a device used to monitor the concentration of carbon dioxide as a person breathes. The capnometer draws a capnogram.

Kriz noticed how periodic functions are present even in breathing! Help Kriz study the properties of the graph shown in the capnogram.

a Find the amplitude of the graph shown in the capnogram.

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b Find the midline of the graph shown in the capnogram.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.625em;vertical-align:-0.19444em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.03588em;\">y<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["1.5"]}}

a Use the grid to find the maximum and the minimum values of the function's graph. Use these values to calculate the amplitude.

b Use the maximum and minimum values found in Part A to calculate the midline.

a Begin by recalling the formula for the amplitude of a periodic function.

$A=2y_{max}−y_{min} $

Looking at the graph, notice how the $y-$values all lie between $0$ and $3.$
This means that $y_{max}=3$ and $y_{min}=0.$ Substitute these values into the formula to find the amplitude.
Therefore, the amplitude is $23 ,$ or $1.5.$
b Recall the formula for the midline of a periodic function.

$y_{mid}=2y_{max}+y_{min} $

In Part A it was found that the values of $y_{max}$ and $y_{min}$ are $3$ and $0,$ respectively. Substitute them into the formula to find the midline.
$y_{mid}=2y_{max}+y_{min} $

SubstituteII

$y_{max}=3$, $y_{min}=0$

$y_{mid}=23+0 $

IdPropAdd

Identity Property of Addition

$y_{mid}=23 $

Pop Quiz

Consider the periodic function in the applet.

Closure

In this lesson, the concept of a periodic function was introduced. A few real-life applications of this type of function were also presented. The main properties of periodic functions — the period, the amplitude, and the midline — were also discussed.

The function shown above is a special periodic function called a trigonometric function. This function takes an angle as an input and relates it to a trigonometric ratio of its corresponding reference angle. More details about this type of function will be covered in later lessons.