Tip:
Highlight text to annotate it
X
In this lesson, we will be modeling data with a periodic function.
Whenever we model data with a periodic function, there is no specific test for us to use. With linear data, we tested to see if it had a
constant average rate of change. With exponential data, we check to see if the ratios were constant. Instead, with periodic data we
just look at the data to see if it has a repetitive pattern. Does is oscillate up and down between a maximum and minimum value.
A periodic model has the form y = A*sin(B(x – C)) + D.
The A was the amplitude, which was how tall the graph was.
The B was related to the period, so how wide it is.
The C was the horizontal shift, moving it left and right.
The D was the median, which was the middle value or a vertical shift.
Let’s look at an example. We have a population based on the generation, with an initial population of 410.
The next generation decreased down to 177. Then the next few generations increased, until the 5th generation had 1123.
It decreases until the 8th generation of 177. It again increases until the 12th generation of 1123.
In the 13th generation, it starts decreasing again.
So we see this pattern where it goes from 177 up to 1123 and back to 177 and back up to 1123, and so we are seeing a repetitive pattern.
The numbers aren’t all the same, and they don’t have to be. But we are still seeing this oscillation, this going up and down.
To find a function for this data, we first need to find the period. Then we can find B.
To find the period, we need to find the distance between 2 maximums, or 2 minimums.
The first maximum was in the 5th generation and the second maximum occurred in the 12th generation.
Period = 12 – 5 = 7 generations B = 360 / Period = 360 / 7 = 51.43
We could have computed the period using the minimums, in the 1st generation and 8th generation.
Once we have the period and B, we can find the median D and amplitude A. They are related to each other by the maximum and minimum
values. The minimum value was 177 and the maximum value was 1123.
A = (Max – Min) / 2 = (1123 – 177) / 2 = 473
D = (Max + Min) / 2 = (1123 + 177) / 2 = 650
The last thing we need to do is figure out the horizontal shift C, which is the hardest part.
In order to get C, we need to find where the data will pass through the median the first time while it is increasing.
If we remember, we have the median is D = 650. So where is the data going to pass through 650?
And the 650 doesn’t need to be a data value, just where the data would pass through if we filled in all the gaps.
The initial generation was 410 and it decreases.
From the 1st to 5th generation it increases from 177 to 1123, which means it has to pass through 650.
That happened in the 3rd generation. So C = 3. This time it was pretty easy because the 650 was an actual data value.
That is not always the case. What if, for example, the median had instead been 400? The 400 is not a data value. So where would 400
appear while the generation is increasing? In the 2nd generation the population was 282 and in the 3rd generation it was 650. So somewhere
between 282 and 650 is where it would pass through 400, so somewhere between the 2nd and 3rd generation.
We can make a guess that C would be somewhere between 2 and 3, so an educated guess would be c = 2.5.
Once we have the constants, we can write the function. Remember the general formual y = A*sin(B(x – C)) + D.
A = 473, B = 51.42, C = 3, and D = 650 Putting that together gives the population as a function of the generation.
P(G) = 473*sin(51.43(G – 3)) + 650