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| 12 Theory slides |
| 8 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
During a math lesson, Ali realized many real-life situations that repeat at regular intervals could be modeled by periodic functions. His friends do not believe him. To prove to his friends that trigonometric functions are used in real life, he scoured the internet and found several interesting examples of these functions.
Function rule | What does the function model? |
---|---|
R(t)=32900sin2t | The horizontal distance from a player to a basketball thrown with an initial velocity of 30 feet per second at an angle with measure t. Negative distance means that the direction of the throw changes. |
s(t)=3.5sin180π(t+4)+15.7 | The average wind speed in a certain city, measured in miles per hour. |
P(t)=100−20cos38πt | The blood pressure of a person at rest, measured in millimeters of mercury. |
N(t)=3.7sin(54t−0.7)+20 | The number of thousands of employees at a sports company. |
Ali wanted to share the examples with his class, so he printed out the graphs of the functions.
Unfortunately, he forgot to label the graphs. Graph the given functions using a graphing calculator and help Ali match the graphs with their corresponding function rules.
Ali's brother Davontay is a volunteer firefighter. At the end of each year, he likes to analyze certain data related to fire-fighting and rescue services for his department. This year, he received a special assignment from the Fire Chief.
Davontay was asked to collect data about the average monthly water temperature D, in degrees Fahrenheit, as a function of the month of the year t. The table's data then needs to be analyzed. Later, this analysis will test for a relationship between the temperatures and number of rescue boat operations. By analyzing this table, the Fire Chief will let Davontay drive the rescue boat!
Average Monthly Water Temperatures | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
D (∘F) | 70 | 73 | 75 | 78 | 82 | 85 | 86 | 85 | 84 | 81 | 75 | 73 |
The Fire Chief outlines the mission in more detail, here.
Davontay explains that he and his squad frequently cooperate with medical rescue teams during road accidents. When providing first aid to victims, paramedics measure their vital signs — the electrical activity of a heart in millivolts — using an electrocardiogram (ECG).
A pulse rate of a person is the number of times their heart beats in one minute. Each cycle in the given graph represents one heartbeat.
Write as a fraction
LHS⋅35=RHS⋅35
a⋅cb=ca⋅b
Calculate quotient
Despite analogue radio receivers, SMS, and phone calls, electronic sirens are still used in many fire stations. They create a loud modulated sound that can be heard within a few miles.
Davontay's fire station has a siren that makes 3 rotations per minute and whose sound beam can be heard within a 4 mile radius. It so happens that Davontay's house is exactly 4 miles away from the fire station. Let D(t) be the periodic function that represents the distance from the end of the sound beam to Davontay's house in terms of the time t, measured in seconds.
Number of Rotations | Time |
---|---|
3 | 1 minute=60 seconds |
1 | 360=20 seconds |
Therefore, the siren makes one cycle in 20 seconds, which is the period of the function.
Although it was sufficient to make an example graph of the distance function, the function rule can also be found. Begin by recalling the coordinates of the points on a unit circle.
These coordinates can be generalized to the coordinates of the points on a circle of different radius. They are known as polar coordinates. Let r be the radius of a circle. The coordinates of a generalized circle are multiplied by r.
In the given situation r is equal to 4. The angle of rotation depends on the time t. Recall that the period of the function is 20 seconds. This means that the siren makes a full turn — a rotation of 2π radians — in 20 seconds. Next, the angle θ1 by which the siren rotates in 1 second can be found.Substitute (4,0) & (4cos10πt,4sin10πt)
(a−b)2=a2−2ab+b2
Subtract term
(ab)m=ambm
Calculate power and product
Commutative Property of Addition
Factor out 16
sin2(θ)+cos2(θ)=1
Identity Property of Multiplication
Add terms
Split into factors
Factor out 16
a⋅b=a⋅b
Davontay originally volunteered as a firefighter because they live near a volcano along the coast of a seismic zone — an area where earthquakes usually occur. A few years ago, on a dark cloudy day there was danger of a looming earthquake. Davontay's fire department had no choice but to organize the evacuation of all the town's homes.
Davontay was told that the predicted earthquake wave had a frequency of 61 hertz and an amplitude — displacement of the ground — of 5 millimeters.
Frequency=61
b/ca=ba⋅c
Identity Property of Multiplication
1a=a
Period=6
LHS⋅∣b∣=RHS⋅∣b∣
LHS/6=RHS/6
ba=b/2a/2
b≥0: b=3πb<0: b=-3π(I)(II)
Finally, by connecting the points with a smooth curve and continuing it periodically in both directions, the sinusoid can be drawn.
t=28
ca⋅b=ca⋅b
Write as a sum
Write as a sum of fractions
Calculate quotient
Split into factors
sin(θ)=sin(θ+n⋅2π)
a(-b)=-a⋅b
a⋅cb=ca⋅b
On the weekends, Ali likes to help Davontay do some cleaning at the fire station. This particular time, Davontay asks his brother to download a mobile app to tune a guitar left at the station by a firefighter who plays it during some downtime. Not everything at a station is related to an emergency!
The standard tuning of a guitar is A–440, which means that the same note A on a keyboard will vibrate 440 times per second. After downloading the app, the guitar player showed up and excitedly came over to talk about sound waves with Ali.
LHS⋅∣b∣=RHS⋅∣b∣
b1⋅a=ba
LHS⋅440=RHS⋅440
b≥0: b=880πb<0: b=-880π(I)(II)
It is given that the amplitude is 60 decibels. In Part A, it was also obtained that the period is 4401 seconds. Next, since the cosine function is not translated, the midline of this function is y=0. Graph this line on a coordinate plane.
Now, key points ranging over one cycle will be plotted. These points are the maximums, minimums, and intersections with the midline. The maximums of y=cosx occur once every cycle at the following x-coordinates.The minimum of the function is located halfway between the maximums at x=8801. The minimum value is 0−60=-60. Plot the minimum in the same coordinate plane.
Next, between every neighboring maximum and minimum the sinusoid intersects with the midline.Finally, by connecting the points with a smooth curve, the sinusoid can be drawn. Although the graph may not represent the whole sound wave, keep in mind that only one cycle of each graph is given in the options.
This graph corresponds to option B.
The amplitude of a periodic function is half the difference between the maximum and minimum values, while the period is the length of one cycle.
The amplitude is related to neither period nor frequency. Therefore, when the frequency decreases 2 times, the amplitude does not change. This means that the pitch of a sound does not influence its volume, which makes perfect sense. The guitar player is thankful for this understanding.
One of the most important and valuable fire brigade operations is qualified first aid. Ali admires his brother for conducting first aid training for people that do not have professional knowledge or experience in this field. The basic emergency procedure taught during training is cardiopulmonary resuscitation (CPR) — chest compressions often combined with artificial breaths.
Trainees practice CPR on a medical-training mannequin. CPR for an adult person consists of cycles of 30 chest compressions and two breaths. Compressions should be 5 to 6 centimeters into the mannequin's or person's chest at a rate of 100 to 120 compressions per minute.
a−(-b)=a+b
Identity Property of Addition
∣3∣=3
a≥0: a=3a<0: a=-3(I)(II)
LHS⋅∣b∣=RHS⋅∣b∣
b1⋅a=ba
LHS⋅2=RHS⋅2
Rearrange equation
b≥0: b=4πb<0: b=-4π(I)(II)
Transformations of y=cosx | |
---|---|
Vertical Stretch or Shrink | Vertical stretch, a>1y=acosx
|
Vertical shrink, 0<a<1y=acosx
| |
Horizontal Stretch or Shrink | Horizontal shrink, b>1y=cos(bx)
|
Horizontal stretch, 0<b<1y=cos(bx)
|
According to the previous examples, wavelike and periodic data sometimes can be modeled by sinusoids. The function rule of a sine function that is the curve of best fit can be found by performing a sinusoidal regression in a graphing calculator.
Most graphing calculators have a functionality called sinusoidal regression, which can be used to find a sine model that gives the curve of best fit. A sinusoidal regression fits data that fluctuate over time and have the look of wavelike curves. For example, consider the following data set.
x | y |
---|---|
1 | 7.61 |
2 | 7.52 |
3 | 6.86 |
4 | 5.99 |
5 | 5.18 |
6 | 4.67 |
7 | 4.68 |
8 | 5.11 |
9 | 5.67 |
10 | 6.23 |
11 | 6.82 |
12 | 7.41 |
On a graphing calculator, begin by entering the data points. To do so, press the STAT button and select the option Edit.
This gives a number of columns, labeled L1, L2, L3, and so on. These columns will correspond to the given variables.
Use the arrow keys to choose where to fill in the data values of each variable. Enter the x-values of the data points in L1 and press ENTER after each value. The same can be done for the the corresponding y-values in column L2.
Having entered the values, the corresponding points can be plotted in a scatter plot to check whether they can be modeled by a wavelike curve. To do so, push 2nd and Y=. Then, turn the plot On,
choose a scatter plot from the list, and assign L1
and L2
as XList
and YList,
respectively.
Then, the plot can be made by pressing the button ZOOM and then choosing the ninth option ZoomStat.
The scatter plot of the data appears.
Looking at the plot, a sinusoidal curve may fit the data. Therefore, a sinusoidal regression can now be performed.
To use the sinusoidal regression functionality, push STAT, go to CALC
and choose option C, SinReg.
Before performing the regression, the function needs to be stored in order to draw it later. Push VARS and go to Y-VARS.
Then, choose the first option, Function,
and choose one of the functions to store the equation.
Finally, push ENTER to obtain the regression function.
The obtained sine function can be graphed in the same coordinate plane as the scatter plot by pushing GRAPH.
The model seems to be a good approximation of the data points.
Every year, Davontay collects atmospheric data for a yearly report to compare local weather to the number of rescue operations. The table shows the monthly precipitation P — rain or snow — for his city in inches. In this data, t=1 represents January, t=2 representes February, and so on.
Average Monthly Precipitation | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
t (months) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
P (inches) | 0.5 | 0.6 | 0.9 | 1.5 | 2.2 | 2.6 | 2.7 | 2.3 | 1.6 | 1.2 | 0.7 | - |
Since the year is not over yet, the precipitation in the last month is missing. Davontay wants to predict the precipitation in December based on the collected data.
Editand enter the values. The first column represents the number of the month and the second number represents the precipitation.
After entering the values, draw a scatter plot by pushing 2nd and Y=. Then, turn the plot On,
choose a scatter plot from the list, and assign L1
and L2
as XList
and YList,
respectively.
Now the data points can be finally plotted by pressing the ZOOM button and then choosing the ninth option ZoomStat.
The scatter plot of the will data appear.
Looking at the plot, it appears that the sinusoidal curve fits the data. Therefore, a sinusoidal regression can be performed. To use this functionality, push STAT, go to CALC
and choose option C, SinReg.
Now, push ENTER to obtain the regression function.
t=12
Round to 1 decimal place(s)
In the given challenge, the data set on average monthly water temperatures was given.
Average Monthly Water Temperatures | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
t | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
D (∘F) | 70 | 73 | 75 | 78 | 82 | 85 | 86 | 85 | 84 | 81 | 75 | 73 |
Scatter Plot and Function:
Example Model: D=7.91sin(0.48t−1.9)+78.18
Period: ≈13 months
Use the sinusoidal regression feature in a graphing calculator. Start by plotting the data in a scatter plot.
Generally, temperatures appear to be periodic and repetitive over years. This means that a periodic function can be used to model the given data. Start with a scatter plot.
The data points can be plotted in a graphing calculator. To do so, push the STAT button, choose the first option Edit
and enter the values. The first column represents the number of the month and the second column represents the water temperature.
Having entered the values, draw a scatter plot by pushing 2nd and Y=. Then, turn the plot On,
choose a scatter plot from the list, and assign L1
and L2
as XList
and YList,
respectively.
Then, the data points can be finally plotted by pressing the ZOOM button and then choosing the ninth option ZoomStat.
The scatter plot of the data appears.
CALCand choose option C,
SinReg.
Before performing the regression, the function needs to be stored in order to draw it later. Push VARS and go to Y-VARS.
Then, choose the first option, Function,
and choose one of the functions to store the equation.
Now, push ENTER to obtain the regression function.
The obtained sine function can be graphed in the same coordinate plane as the scatter plot by pushing GRAPH.
The model seems to be a good approximation of the data points.
b=0.48
∣0.48∣=0.48
Use a calculator
Round to nearest integer
Graph the given function and consider the following graphs.
Choose the graph that corresponds to the given function.We are given a cosine function that models the motion of a mathematical pendulum. d(t) = 10 cos 4π t In this function, d(t) is the horizontal displacement of the pendulum relative to its resting position in time t. We will find and interpret its period. To do so, let's recall the general form of a cosine function. d(t) = a cos bt In this function rule, a and b are nonzero real numbers. Amplitude&= | a| [0.5em] Period&= 2π/| b| We can see that in our case, a= 10, which means that the amplitude is equal to 10. Also, b= 4π, so we can find the period by substituting this value into the formula.
The period is 12. Recall that the period is the horizontal length of one cycle. Therefore, it takes half a second for the pendulum to make one complete cycle — go back and forth and return to the same position.
Now, we want to graph the given function. To begin, let's examine the first second of the motion! Recall that the period of the pendulum is 12 or 0.5 second.
To graph the function that models the horizontal displacement d, we should first identify the points of the function where the x-intercepts, maximum values, and minimum values occur. To do so, we will substitute a= 10, b= 4π into the known formulas for the cosine function.
Formula | Substitute | Simplify | |
---|---|---|---|
x-intercepts | (1/4*2π/b,0), (3/4*2π/b,0) | (1/4*2π/4π,0), (3/4*2π/4π,0) | (0.125,0), (0.375, 0) |
Maximum | (0, a), (2π/b, a ) | (0, 10), (2π/4π, 10 ) | (0, 10), (0.5, 10 ) |
Minimum | (1/2*2π/b, - a) | (1/2*2π/4π, - 10) | (0.25, - 10) |
We can graph one cycle of the function by plotting the points that we found and connecting them with a smooth curve. Let's do it!
Finally, we can obtain the desired graph by extending the pattern for the next values of t.
The obtained graph corresponds to option B.
Dominika went with her friend to ride a Ferris Wheel. It turns for 300 seconds.
When riding the Ferris Wheel, Dominika's height h in feet above the ground at any time t in seconds can be modeled by the following equation.What is the graph of the function?
We are told that the Ferris wheel turns for 300 seconds and we are given an equation that models the height h above the ground at any time t. h=80sin π/25(t-9)+84 To graph this function, we will start by comparing it with the general form of a transformed sine function. General Form y= a sin b(x- h)+ k Given Function h= 80 sin π/25 (x- 9)+ 84 As we can see, a= 80, b= π25, h= 9, and k= 84. Now, we can use the formulas for the x -intercepts, maximum value, and minimum value of a sine function of the form y= a sin bx.
Formula | |
---|---|
x-intercepts | (0,0), (1/2*2π/b,0), (2π/b,0) |
Maximum | (1/4 * 2 π/b, a) |
Minimum | (3/4*2π/b, - a) |
Our function is in the form y= a sin b(x- h)+ k. The values of h and k tell us about the vertical and horizontal translations, respectively. Here, it is a translation 84 units up and 9 units right. Therefore, to find a few points on our function, we will also add 9 to the x-coordinates and 84 to the y -coordinates of the points from the table.
Formula | Substitute | Simplify |
---|---|---|
( h, k), (1/2*2π/b+ h, k), (2π/b+ h, k) | ( 9, 84),(1/2*2π/π25+ 9, 84), (2π/π25+ 9, 84) | (9,84),(34,84), (59,84) |
(1/4* 2π/b+ h, a+ k) | (1/4 * 2π/π25+ 9, 80+ 84 ) | (21.5,164) |
(3/4*2π/b+ h, - a+ k ) | (3/4*2π/π25+ 9, - 80+ 84) | (46.5,4) |
Now we know five points that the given function passes through. Let's plot and connect them with a smooth curve.
Finally, let's extend the pattern along the x-axis to obtain the desired graph.
The obtained graph matches option B.
We want to find the number of cycles the Ferris wheel makes in 300 seconds. To do it, let's begin by finding the period of the given function. It will tell us the time in which the Ferris wheel makes one cycle. Period: 2π/| b| In our case, b= π25. Let's substitute it into the formula for the period.
The period of the Ferris wheel is 50 seconds. This means that it completes one cycle in 50 seconds. Therefore, to find the number of cycles it makes in 300 seconds, let's divide 300 by 50. 300/50 = 6 The Ferris wheel makes 6 cycles in 300 seconds.
In Part A, we found the maximum and minimum points of the given function. Maximum:& (21.5,164) Minimum:& (46.5,4) By looking at their y-coordinates, we can conclude that the maximum height of a seat on the Ferris wheel is 164 feet and the minimum height is 4 feet.
When paramedics treat their patients, they sometimes need to measure the electrical activity of a heart measured in millivolts by using an electrocardiogram (ECG).
A pulse rate is the number of times a heart beats per minute. Each cycle in the given graph represents one heartbeat.
To find the pulse rate, we need to calculate the number of heart beats in 60 seconds. Let's start by finding the period of the given periodic function. First, we will analyze the graph to identify one cycle — the shortest repeating portion of the graph. We can use the peaks as the reference points.
This is only one example portion of the graph that could be chosen as one cycle. As we can see, the horizontal length of one cycle is 3 units. We are told that 1 unit corresponds to 0.25 seconds. Therefore, we can find the period of the function by multiplying the length of one cycle in units by the time corresponding to one unit. 3 * 0.25 = 0.75seconds The period is 0.75 seconds. This means that one heartbeat lasts 0.75 seconds. Recall that the pulse rate is the number of times a heart beats per minute. We can find it by dividing 1 minute or 60 seconds by the time of one heartbeat, 0.75 seconds. Pulse=60 sec/0.75sec=80 beats We obtained that the pulse rate of the patient is 80 beats per minute.
The amplitude is half the difference of the maximum and minimum values of a periodic function. \begin{gathered} A = \dfrac{y_\text{max} - y_\text{min}}{2} \end{gathered} Since each repeating portion of the graph is identical, we can analyze only one cycle. Let's find the points of maximum and minimum electrical activity in the graph and determine the vertical distance between them.
We know that one vertical unit corresponds to 1 millivolt, so the difference y_\text{max} - y_\text{min} is 4 millivolts. By dividing this value by 2, we can calculate the amplitude of the function. A = 4/2 mV ⇔ A= 2 mV
Consider the data that shows daily average temperatures T in degrees Fahrenheit in Boise, Idaho, on several days d of the year.
Day of Year | 14 | 45 | 75 | 105 | 134 | 165 | 200 | 228 | 256 | 286 | 317 | 348 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Temperature (F∘) | 36 | 40 | 50 | 64 | 78 | 90 | 96 | 92 | 83 | 70 | 54 | 42 |
We are given a table that illustrates daily average temperature on several days of the year in Boise, Idaho.
Day of Year | 14 | 45 | 75 | 105 | 134 | 165 | 200 | 228 | 256 | 286 | 317 | 348 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Temperature (F^(∘)) | 36 | 40 | 50 | 64 | 78 | 90 | 96 | 92 | 83 | 70 | 54 | 42 |
Let's plot the ordered pairs on a coordinate plane. Let the x-axis represent days of the year and the y-axis represent the temperature in Fahrenheit.
Now, we need to form a cosine function that models the average daily temperature as a function of d. To do so, let's first recall the general form of a transformed cosine function. y= a cos b(x- h)+ k We will first calculate | a|, which is the amplitude of the function. It is equal to half the difference between the maximum and minimum values of the function. | a|=1/2(Max-Min) Let's examine the graph to determine the maximum and minimum values.
Now, we will substitute Max= 96 and Min= 36 into the equation of the amplitude and calculate a.
Note that a can be 30 or -30. We are told to write the function with positive coefficients, so we will choose the positive value a=30. Second, we will calculate b using the formula for the period of a function. Period=2π/| b| Since one complete cycle takes 365 days, the period is 365. Let's substitute the value of the period into the formula and solve it for b.
Once again, we will choose the positive value b= 2π365. Third, we will find the value of h, which is the amount of the horizontal shift. Recall that the maximum of the parent function y=cos x occurs at x=0. In our graph, the maximum occurs at x=200, so we can conclude that the graph is shifted 200 units to the right. h= 200 Next, we will find k, which represents the vertical shift. The value of k is the value of the midline. We can find it by calculating the average of the maximum and minimum values of the function. k=Max+Min/2 Let's substitute Max= 96 and Min= 36 into this equation and solve it for k.
Finally, we will write our cosine model by substituting a= 30, b= 2π365, h= 200, and k= 66 into the general form of a transformed cosine function. Remember also that in this case y-variable is represented by T and x-variable is represented by d. y= a cos b(x- h)+ k ⇕ T= 30 cos 2π/365(d- 200)+ 66 By graphing the function on the coordinate plane with the plotted points, we can see that the function indeed models the data.