The average monthly temperatures in New Orleans, Louisiana, are given in the following: http://postimage.org/image/q4bcfm79f/ Show all steps.
We are given a table of values: (1,16.5),(2,18.3),(3,21.8),(4,25.6),(5,29.2),(6,32.6),(7,32.6),(8,32.4),(9,30.3),(10,26.4),(11,21.3),(12,18.0) where the independent variable is the month and the dependent variable is the temperature in degrees Celsius.
(a) The range of the function is `32.6-16.5=16.1` (Highest minus lowest value)
(b) The average yearly temperature is `(sum"Temps")/12~~25.36`
(c) The function appears to be a sinusoid, it looks like a sine wave. It appears to have a repetition (When you get back to month 1 it repeats the same data). The values rise and fall smoothly.
(d) the graph (showing the graph of the function -- you would plot the points given)
(e)(f) Given `T(t)=asin[b(t-c)]+d`
T is the function
t is time measured in months
a gives the amplitude -- this is 1/2 of the range. If a<0, the function is reflected over the x-axis
b gives the period -- `p=(2pi)/b`
c gives the horizontal translation of the base function; this is usually called a phase shift for periodic functions
d gives the vertical translation -- this usually determines the midline
(g) The amplitude will be `16.1/2=8.05` so `a=+-8.05` (Depending on the base function we choose and the phase shift)
The period is 12, so `b=(2pi)/12=pi/6`
It appears that, if we choose the sin function as the base function, the graph has been shifted right about 3.8 units so c=3.8
The midline is `y=(32.6+16.5)/2=24.55` so d=24.55
The graph of `T(t)=8.05sin(pi/6(x-3.8))+24.55:`