The following is a table for the maximum temperature per month, for Boulder, CO
Month|° F
Jan:45.1
Feb:48.2
Mar:53.8
Apr:62.8
June:71.6
July:81.5
Aug:85.1
Sept:76.6
Oct:66.7
Nov:52.9
Dec45.5
Find the sinusoidal function that models this data.
How do I do this ?
you might want to look at the equation you yourself posted in
http://www.jiskha.com/display.cgi?id=1295020805
and see where the numbers in that equation come from and how they affect the answer.
e.g.
the period would certainly be the same, one year
Look at your data above and determine the difference between the highest and the lowest, that would be twice the amplitude in front of the sine curve.
looks like the hightest temp is in Aug at 85.1 and the lowest in Jan at 45.1
wouldn't the median be 65.1 ?
let me know what you come up with?
Is the answer y = 21.15sin(pi/6 t + 3)+66.25?
To find the sinusoidal function that models the given data, you can use the following steps:
Step 1: Identify the period (P)
The period is the total length of one complete cycle. In this case, the data spans across 12 months, so the period is 12.
Step 2: Calculate the horizontal shift (h)
The horizontal shift is the number of units the graph is shifted to the right or left. In this case, since the data starts in January (month 1), there is no horizontal shift, so h = 0.
Step 3: Calculate the vertical shift (k)
The vertical shift is the number of units the graph is shifted up or down. In this case, the data represents the maximum temperature, so the average temperature can be considered the vertical shift. To calculate k, sum up all the temperature values and divide by the number of months:
k = (45.1 + 48.2 + 53.8 + 62.8 + 71.6 + 81.5 + 85.1 + 76.6 + 66.7 + 52.9 + 45.5) / 12
Step 4: Find the amplitude (A)
The amplitude is half the range of the function, which represents the maximum distance the function deviates from the vertical shift. In this case, the range is the difference between the highest and lowest temperature values:
Range = 85.1 - 45.1
Amplitude = Range / 2
Step 5: Write the sinusoidal function
The general form of a sinusoidal function is:
f(x) = A * sin((2π / P) * (x - h)) + k
Substituting the values we have found:
f(x) = (Amplitude) * sin((2π / Period) * (x - Horizontal Shift)) + Vertical Shift
f(x) = (Amplitude) * sin((2π / 12) * (x - 0)) + k
f(x) = (Amplitude) * sin(π/ 6 * x) + k
So the sinusoidal function that models this data would be:
f(x) = (A value calculated in Step 4) * sin(π/6 * x) + (k value calculated in Step 3)
Remember to substitute the calculated values for amplitude (A) and vertical shift (k) for the final equation.