Question

The mean for average temperature in Buffalo ,New york , is 47.5°. The temperture fluctuates 23.5° above and below the mean temperture . If t=1 present January ,the phase shift of the sine function is 4
a) Write the funtion for the average monthly tempperature in Buffalo
b) According to your modle ,what is the average tenperature in March ?
c) According to your modle ,what is the average tenperature in August ?

Answer 1

\[a \sin( bx +c ) + d \]
a influences amplitude ( a = 1 => -1 < sin < 1 )
b influences frequency ( b = 1 => i cycle takes 2 pi )
c phase shift
d vertical translation

So what you need is the following
mean is 47.5 so d = 47.5
23.5 fluctiation means a = 23.5

The last bit is a bit confusing, but i'll give it a try:
phase shift is 4 which makes c 4
now we want it to have a frequency of 12 months (at least i think that's what's needed here) we need b to be pi/6

Conclusion:
\[T(x) = 23.5 \sin( { \pi \over 6}x +4) +47.5 (x = months)\]
for quetions b and c simply insert the monthnumber in this function

Answer 2

http://www.wolframalpha.com/input/?i=23.5sin%28%28%CF%80%2F6%29x%2B4%29%2B47.5+from+0+to+12

Answer 3

What's tricky is the monthnumber, 1 represents the 31st of jan in this formula, not the entire month of jan. since it's a continuous function.