We are given the population of a fictional animal at different years:
$$\begin{array}{l|r} \textrm{Year} & \textrm{Population}\\\hline 1945 & 347,0000\\ 1955 & 76,000\\ 1965 & 295,000\\ 1975 & 84,000\\ 1985 & 243,000\\ 1995 & 92,000 \end{array}$$
We are asked to come up with a formula for the population over time. I am so lost. I can get a graph to go through the first 2 points, or I can get it to go through all of the maximums or all of the minimums, but I can't get it to go through all of the points in one go.
On the drawing of the given data, one can see that the maximums and minimums are respectively linear functions of the years. It is easy to express both linear équations, which are significant of the evolution of the population.
Of course, it is possible to find a "damped" sinusoidal function which accurately fits the six given points (with a 10 years half-period) The coefficients $C_0, C_1, C_2, C_3$are computed thanks to linear regression. But what is the meaning of such a function ? This is questionable ! Why not a one year period, ore else ? Are you sure that the problem requires such a sinusoidal model ? If statistical computations have to be carried out, why not directly using the given data instead of an intermediate function with, may be, no real significance!