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Real-World Periodic Functions
23
Real-World Periodic Functions
By: Maira Inayat
Periodic Object
An object that can be represented as a periodic function is the minute hand of a clock. The reason why the minute hand of a clock is periodic is because after it rotates and completes one revolution, it repeats the same cycle continuously. The minute hand of a clock reaches the same values at regular intervals.
Periodic Object
Point of Reference and Hand Displacement
Reference Point:
The reference point is going to be at "12" on the clock because this is when t=0 where "t" is the number of minutes. I chose this reference point because the starting value will be at t=0 which will simplify the process of determining a cosine function. This is because if the cycle begins at t=0 there will be no phase shift for the cosine function. From the reference point "12" on the clock I will use a ruler to measure the distance from the tip of the minute hand to the new positions "3," "6," and "9".
Hand Displacement:
Calculations
When the minute hand is at 0 minutes, the distance is 0cm which gives the point (0, 0).
When the minute hand is at 45 minutes, the distance is 12cm downwards which gives the point (45, -12).
When the minute hand is at 15 minutes, the distance is 12cm downwards which gives the point (15, -12).
When the minute hand is at 30 minutes, the distance is 24cm downwards which gives the point (30, -24).
Period of the Object and "k" Value
We are modeling the displacement of the minute hand in terms of the amount of time elapsed in minutes. Since it takes 60 minutes for the minute hand to return back to its initial position, the period of the clock (minute hand) will be 60 minutes.
By knowing the period we can determine the value of "k" for the cosine and sine functions we need to create:
Period of the object
Therefore, the value of "k" will be k= Pi/30.
Maximum and Minimum Values
Maximum and Minimum Values
Maximum Value:
The maximum value is located at D= 0cm
The maximum value will be at the position "12" on the clock which is the point (0,0) as we are measuring the displacement of the minute hand up to "12" so the maximum value will be at the top when D= 0cm.
Minimum Value:
The minimum value is located at D= -24cm
The minimum value will be at the position "6" on the clock which is the point (30, -24). We are measuring the displacement of the minute hand downwards to "6" so the minimum value will be at the bottom when D= -24cm.
Amplitude and Vertical Translation
Amplitude:
Vertical Translation:
Amplitude and Vertical Shift
Therefore, the amplitude is a= 12.
Therefore, the vertical translation is c= -12.
Cosine Function
There is no phase shift because the function begins at the maximum value like the basic cosine function.
a= 12
k= Pi/30
d= 0
c= -12
Cosine Function
"Let "D" represent the displacement of the minute hand in centimeters."
"Let "t" represent the amount of time elapsed in minutes."
Graph of Cosine
This is the graph of the cosine function with the domain D: {t|t ≥ 0, tER} since time cannot be negative.
Graph
The graph is graphed within one period or 0≥ t ≥ 60.
Sine Function
The basic sine function begins at the central axis which is the midpoint between the maximum and minimum values. In this case, on the clock, (15, -12) is the midpoint between the maximum and minimum values. This tells us that our sine function needs to have a phase shift of 15 units to the right which gives us d= 15.
The basic sine function begins at the central axis and moves in an upward direction from the beginning to reach the maximum value. However, in this case, the sine function will be moving in a downwards direction from the central axis to reach the minimum value because the distance is in negative values. This shows that the function will be reflected in the x-axis and we will use -sin(x).
Sine Function
a= -12
k= Pi/30
d= 15
c= -12
"Let "D" represent the displacement of the minute hand in centimeters."
"Let "t" represent the amount of time elapsed in minutes."
Graph of Sine
This is the graph of the sine function with the domain D: {t|t ≥ 0, tER} since time cannot be negative.
Graph
The graph is graphed within one period or 0≥ t ≥ 60.
x-Values
t= Period/3
t= 5(Period)/6
x and y values
Therefore, t=20 minutes.
Therefore, t=50 minutes.
y-Value 1
y-value for t=20 minutes:
Sine
Cosine
y-value
1
Therefore, when t=20 minutes , D= -18cm which gives the point (20, -18).
y-Value 2
y-value for t=50 minutes:
Cosine
Sine
y-value
2
Therefore, when t=50 minutes , D= -6cm which gives the point (50, -6).
Verify Using Graphing Technology
D(t)= 12cos(Pi/30 (t))-12
D(t)
t
Verification Method
To verify the trigonometric functions we created, we can use graphing technology as a simple and accurate method. I used "desmos" to graph the functions I created by inputting the functions into the software. The graphing technology automatically generates an accurate representation of the graph of the functions. We can know if the functions are accurate by identifying the key points on the graph, for example, the maximum and minimum values and the midpoints between the maximum and minimum values to see if they are the same as the values I had measured. In this case, the values on the graph and the values I had collected are the same. Therefore, these functions are accurate.
D(t)= -12sin(Pi/30(t-15))-12
D(t)
t
Reflection
In this assignment, the most easiest part was calculating some parameters for the sine and cosine functions, graphing the sine and cosine functions, and finding and substituting the x-values to find their y-value. Calculating some of the parameters was easy because there were basic formulas that were used to find the amplitude, “k” value, and vertical translation. Graphing the sine and cosine function was also very easy because I just needed to make a table of values and plot the points in the table. Finding the x-values in question #1 part 9) and substituting them into the function for their “y” values was easy because there was no critical thinking involved since it was just basic substitution and calculations to find the “x” and y-values.
Reflection
The most difficult parts in the assignment was to determine the maximum and minimum values, determining the phase shifts, and figuring out what object could be modeled periodically. The most difficult part in the assignment was to determine what the maximum and minimum values were because I did not know clearly if the distance values were negative or not so I could not easily determine which value was the maximum and which was the minimum. Determining the phase shifts for the sine and cosine function was also difficult for me because I did not have a solid understanding of how to determine the phase shift by comparing the data with a basic sine and cosine function. Deciding what object was periodic was also difficult for me because I was not familiar with looking at the world around me with that sort of thinking.
In this assignment, I learned how to better compare data to basic sine and cosine functions and find parameters based on that instead of using long and complicated calculations. I also learned how to determine whether the data will be graphed below the x-axis or above by learning to see whether the y-values are negative or not. I also learned where my strengths and weaknesses are in terms of interpreting periodic functions and figuring out how to construct a sine or cosine function from scratch. This assignment opened my eyes to the fact that there are many objects around us that are periodic that we do not pay attention to. If we choose to see these objects this way in the real-world, we can have a greater understanding of periodic functions.