Find Periodic Function

harpazo

harpazo

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You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form
y = A sin(Bt - C) + D whose values approximate the monthly work.

1. Find A

A = (M - n)/2

A = (105.1 - 65.5)/2

A = 39.6/2

A = 19.8

2. Find B

2pi/B = 12 months

2pi = 12B

2pi/12 = B

pi/6 = B

3. Find C

I am having problems finding C.
Example 8 tells me to set C/B = phase shift.

I can pretty much do the rest. How is C found? Example 8 is not too clearly explained by Cohen. I know the values of A and B. When C is found, D should not be a big deal. See picture below.

20200524_103448.jpg

Note: If you need to see Example 8, I can try to upload the picture as given on pages 450-451.
 
Were you given a table of values, or are you to estimate the specifics?

Please make an attempt at finding C, and show at least your basic thinking and where you are unsure. You'll be looking for how far the graph is shifted from the basic sine, that is, how far the first crossing of the midline is from 0, or how far the first maximum is shifted from 1/4 cycle (3 hours) past 0. That has to be an estimate, but I think they've tried to arrange things so there is a reasonable point to use.

The trouble is, this doesn't look exactly like a sine wave, as it lacks some of the required symmetry, so that you will get different answers if you go by the midline than by the max or the min, and if you check points against your formula, they won't work exactly. So don't worry too much if your result doesn't quite work.

By the way, I was able to find enough of the book by searching to have an idea of how they do it in example 8; they use the maximum.
 
Dr.Peterson said:
Were you given a table of values, or are you to estimate the specifics?

Please make an attempt at finding C, and show at least your basic thinking and where you are unsure. You'll be looking for how far the graph is shifted from the basic sine, that is, how far the first crossing of the midline is from 0, or how far the first maximum is shifted from 1/4 cycle (3 hours) past 0. That has to be an estimate, but I think they've tried to arrange things so there is a reasonable point to use.

The trouble is, this doesn't look exactly like a sine wave, as it lacks some of the required symmetry, so that you will get different answers if you go by the midline than by the max or the min, and if you check points against your formula, they won't work exactly. So don't worry too much if your result doesn't quite work.

By the way, I was able to find enough of the book by searching to have an idea of how they do it in example 8; they use the maximum.
Click to expand...

1. Yes, a table is given. I did not upload a picture of the table.

2. I will work on C and come back here to show where I get stuck.

3. Would you like for me to upload Example 8 here?
 
harpazo said:
… find …
y = A sin(Bt - C) + D whose values approximate the monthly work …
Click to expand...
Technically speaking, it's the results of monthly work (done by the Earth and Sun) that we're trying to approximate. ;)

A = (105.1 - 65.5)/2
A = 19.8
Click to expand...
Perfect.

… When C is found, D should not be a big deal …
Click to expand...
D is easy to calculate at any stage in the exercise. I generally find D right after A because the calculation is similar.

D is the average of the maximum and minimum values of y.

Let's go over A and D, a bit. Recall the shape of the graph of y=sin(x). It oscillates above and below the x-axis. The wave shape is symmetrical, and the x-axis runs through the middle of the wave, dividing it into an upper half and lower half. We call such a dividing line the 'midline'. Therefore, the peaks are the same distance above the midline as the valleys are below it. That distance is A (the amplitude parameter).

That's why we take the total difference from the top of the peak to the bottom of the valley (105.1-65.5) and divide it by 2; half that distance is above the midline, and half is below. That's how A is related to the midline.

D is related to the midline, also. D is the vertical-shift parameter. Thinking about y=sin(x) again, the midline lies on the x-axis, and the wave bounces up and down between y=-1 and y=1. In this thread, the midline has been shifted vertically upwards, and the wave bounces up and down between y=65.5 and y=105.1. The midline is in the middle; therefore, D needs to be halfway between 65.5 and 105.1, and the value halfway between two given numbers is always their average.

D is the average of the maximum and minimum values of y.

2pi/B = 12
pi/6 = B
Click to expand...
Perfect.

… I am having problems finding C.
Example 8 tells me to set C/B = phase shift …
Click to expand...

The phase shift is the horizontal shift. The following is review, from a previous thread, where we talked about the interesting points of y=sin(x), over the first period. Those five points are the Origin (left end of the period), the maximum (1/4th of the way through the period), the axis intercept (1/2 way through the period), the minimum (3/4ths of the way through the period) and the next axis intercept (right end of the period). This pattern is the same for all sine waves that do not have a phase shift (i.e., have not been shifted horizontally).

So, here's how we determine C/B (the distance to shift the graph). We know from the data that the first maximum occurs when t=7 (July). We know from the pattern just discussed that the first maximum (before a function is shifted) occurs 1/4th of the way through the period. We compare these two locations (the unshifted and shifted maximums), to see how far apart they are (the shift amount).

The period is 12, and 1/4th the distance from t=0 to t=12 is t=3. In other words, without a phase shift, the first maximum will occur at t=3 instead of t=7. Therefore, the required shift (from 3 to 7) is 4 units to the right.

The phase shift (C/B) needs to be 4, and you already know B. Solve for C.

C/B = 4

After you substitute all parameter values in the form y=A∙sin(B∙t-C)+D, try substituting some t-values from the chart, to see how close the function approximates the corresponding temperatures.

?
 
Otis said:
Technically speaking, it's the results of monthly work (done by the Earth and Sun) that we're trying to approximate. ;)


Perfect.


D is easy to calculate at any stage in the exercise. I generally find D right after A because the calculation is similar.

D is the average of the maximum and minimum values of y.

Let's go over A and D, a bit. Recall the shape of the graph of y=sin(x). It oscillates above and below the x-axis. The wave shape is symmetrical, and the x-axis runs through the middle of the wave, dividing it into an upper half and lower half. We call such a dividing line the 'midline'. Therefore, the peaks are the same distance above the midline as the valleys are below it. That distance is A (the amplitude parameter).

That's why we take the total difference from the top of the peak to the bottom of the valley (105.1-65.5) and divide it by 2; half that distance is above the midline, and half is below. That's how A is related to the midline.

D is related to the midline, also. D is the vertical-shift parameter. Thinking about y=sin(x) again, the midline lies on the x-axis, and the wave bounces up and down between y=-1 and y=1. In this thread, the midline has been shifted vertically upwards, and the wave bounces up and down between y=65.5 and y=105.1. The midline is in the middle; therefore, D needs to be halfway between 65.5 and 105.1, and the value halfway between two given numbers is always their average.

D is the average of the maximum and minimum values of y.


Perfect.



The phase shift is the horizontal shift. The following is review, from a previous thread, where we talked about the interesting points of y=sin(x), over the first period. Those five points are the Origin (left end of the period), the maximum (1/4th of the way through the period), the axis intercept (1/2 way through the period), the minimum (3/4ths of the way through the period) and the next axis intercept (right end of the period). This pattern is the same for all sine waves that do not have a phase shift (i.e., have not been shifted horizontally).

So, here's how we determine C/B (the distance to shift the graph). We know from the data that the first maximum occurs when t=7 (July). We know from the pattern just discussed that the first maximum (before a function is shifted) occurs 1/4th of the way through the period. We compare these two locations (the unshifted and shifted maximums), to see how far apart they are (the shift amount).

The period is 12, and 1/4th the distance from t=0 to t=12 is t=3. In other words, without a phase shift, the first maximum will occur at t=3 instead of t=7. Therefore, the required shift (from 3 to 7) is 4 units to the right.

The phase shift (C/B) needs to be 4, and you already know B. Solve for C.

C/B = 4

After you substitute all parameter values in the form y=A∙sin(B∙t-C)+D, try substituting some t-values from the chart, to see how close the function approximates the corresponding temperatures.

?
Click to expand...

Thank you so much. By the way, I never said A, B, AND D are hard to find. I had problems finding C to complete forming the needed equation or function. Nicely done!! I'll take it from here.
 
Last edited:
harpazo said:
… I never said A, B, AND D are hard to find …
Click to expand...
Yes, that's correct. I misinterpreted your comment that "D should not be a big deal". We may calculate the parameters in any order, really, as long as we have B to find C.

Here are plots of the first year. The green curve is the function you found; the red curve is the best-fit solution from regression software (Least-Squares method).

hsinu.JPG
 
Otis said:
Yes, that's correct. I misinterpreted your comment that "D should not be a big deal". We may calculate the parameters in any order, really, as long as we have B to find C.

Here are plots of the first year. The green curve is the function you found; the red curve is the best-fit solution from regression software (Least-Squares method).

View attachment 19247
Click to expand...

The Least-Squares Method is statistics, right?
 
harpazo said:
… A, B, C, and D are parameters, right?
Click to expand...
Right.

Parameters are constants that change from exercise to exercise, application to application. Within a specific exercise or application, parameter values remain fixed (constant). In other words, parameters are like variable constants!

h(t) = At2 + Bt + C

Here, symbols A,B,C are parameters; their purpose is to express a "generic" quadratic polynomial. The parameters represent constants; symbols h(t) and t are variables.

For example, let's write a height function (feet) in terms of time (seconds) that describes an ice cream cone's position above the ground when dropped from the observation deck of the Space Needle at time t=0. To do that, we assign specific values for each parameter.

h(t) = -16t2 + 605

A = -16
B = 0
C = 605

Those values are fixed, for that function.

To write another height function (meters) for a different application (for example, the height of a toy rocket that's spring-launched vertically upwards at 10 meters per second), the parameters need to change. They become new constants.

H(t) = -4.9t2 + 10t + 2

A = -4.9
B = 10
C = 2

That's how parameters are related to variables and constants.

?
 
harpazo said:
The Least-Squares Method is statistics, right?
Click to expand...
The method is a tool for minimizing error (using regression). It may be used more often by statisticians than by others (eg: algebra teachers, physicists, economists). I'm not sure how to label it, but I tend to think it has broad application.

I first learned of it in precalculus, and I've seen it discussed in various courses since (including high-school algebra).

?
 
Otis said:
Right.

Parameters are constants that change from exercise to exercise, application to application. Within a specific exercise or application, parameter values remain fixed (constant). In other words, parameters are like variable constants!

h(t) = At2 + Bt + C

Here, symbols A,B,C are parameters; their purpose is to express a "generic" quadratic polynomial. The parameters represent constants; symbols h(t) and t are variables.

For example, let's write a height function (feet) in terms of time (seconds) that describes an ice cream cone's position above the ground when dropped from the observation deck of the Space Needle at time t=0. To do that, we assign specific values for each parameter.

h(t) = -16t2 + 605

A = -16
B = 0
C = 605

Those values are fixed, for that function.

To write another height function (meters) for a different application (for example, the height of a toy rocket that's spring-launched vertically upwards at 10 meters per second), the parameters need to change. They become new constants.

H(t) = -4.9t2 + 10t + 2

A = -4.9
B = 10
C = 2

That's how parameters are related to variables and constants.

?
Click to expand...

Excellently stated.
 
Otis said:
The method is a tool for minimizing error (using regression). It may be used more often by statisticians than by others (eg: algebra teachers, physicists, economists). I'm not sure how to label it, but I tend to think it has broad application.

I first learned of it in precalculus, and I've seen it discussed in various courses since (including high-school algebra).

?
Click to expand...

I had a feeling that this method was connected to statistics.
 
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