# Yahoo Answers 4-29-14

- Author:
- Michael Brown

Trig - The average temperature monthly (in F) in Phoenix, Arizona is shown on the table.
https://s.yimg.com/hd/answers/i/720b4171558747ecb9c43243bfc46c0b_A.png?a=answers&mr=0&x=1398816521&s=110d0b705c665f6489b7641e3ec5533a
a. Predict the average yearly temperature.
b. Plot the average monthly temperature over a two year period, letting x = 1 correspond to January of the first year.
c. Determine a function of the forms f(x) = aCOS(b(x-d)) + c, where a, b, c, and d are constants, that models the data.
d. Graph f together with the data on the data on the same coordinate axes. How well does f model the data?
e. Use the sine regression capability of a graphing calculator to find the equation of a sine curve that fits these data (two years).
a. The average will be 1/2 between the high and the low. (or 73.5)
b. See chart
c. aCOS(b(x-d))+c
The average is what we need to add to the overall Cos function to shift it up (+c)
the difference between high and low is 40 degrees (approx) so we need to multiply the whole function by 20 to account for the range (a)
b affects the period, as b gets smaller the period gets longer. The distance from low point to low point (1 cycle) is 12 months. A normal period is 2*pi (call it 6). So we need to double the period. b=1/2
d moves the curve left or right. It needs to move a little more than half way to the left. Call it 5pi/4.
f=20COS(1/2(x-5pi/2))+73.5
d and e See chart