Well, given
only what you've posted here, it's impossible to get "the" equation for the function that passes through all seven points, as there are infinitely many such functions. Now, unless there's some additional instructional text you were given, I'd not recommend using a quartic function for this problem. There may well be a quartic function that passes through all those points, but it's not guaranteed. To guarantee a polynomial passes through
n points, it must be of degree
n-1 or greater. Accordingly, I'd use a sixth-degree polynomial. As you (should) know, the general form of a sixth degree polynomial looks like: f(x) = Ax
6 + Bx
5 + Cx
4 + Dx
3 + Ex
2 + Fx + G. What, then, does it mean that f(1) = 29? What does it mean that f(2) = 41.8? After interpreting all the information from all seven points, what are you left with? How does that help you solve the problem?
This problem involves a lot of grunt work, but can definitely be made easier by using matrices. I don't know how familiar you are with that subject, but
this pagehttp://mth.bz/fit/polyfit/findcurv.htm contains a nice overview of the process.
