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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.

Solution

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 $y\text{-}$value.

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.

Solution

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.

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.

Solution

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

$$\begin{array}{c}A=\frac{y_{\text{max}}-y_{\text{min}}}{2}\end{array}$$

Looking at the graph, notice how the $y\text{-}$values all lie between $0$ and $3.$

This means that $y_{\text{max}}=3$ and $y_{\text{min}}=0.$ Substitute these values into the formula to find the amplitude.

$A=\frac{y_{\text{max}}-y_{\text{min}}}{2}$

SubstituteII

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

$A=\frac{3-0}{2}$

SubTerm

Subtract term

$A=\frac{3}{2}$

Therefore, the amplitude is $\frac{3}{2},$ or $1.5.$

b Recall the formula for the midline of a periodic function.

$$\begin{array}{c}{y}_{\text{mid}}=\frac{y_{\text{max}}+y_{\text{min}}}{2}\end{array}$$

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

$y_{\text{mid}}=\frac{y_{\text{max}}+y_{\text{min}}}{2}$

SubstituteII

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

$y_{\text{mid}}=\frac{3+0}{2}$

IdPropAdd

Identity Property of Addition

$y_{\text{mid}}=\frac{3}{2}$

Therefore, the midline is also $y=\frac{3}{2},$ or $y=1.5.$