Actually, only analytic functions can be approximated by polynomials that way, and just from the graph, we have no way of telling whether the solution is even continuous. Also, you're correct with your second point, as there are infinitely many continuous functions (in fact, more than there are real numbers) that approximate the given graph arbitrarily well (in a sense of having an arbitrarily small difference in value globally).

As for the original problem, you could
A) try to approximate the function with a polynomial of a high degree, which will give you a good estimate of the values inside the interval you're looking at, but will behave horribly outside of it
or
B) do something similar to #29 and find out about the theoretical background that determined the graph. This seems like a much more sensible solution.

What do you need the formula for?

Good point. From my understanding, most functions explaining physical phenomena are going to be analytic until you get to quantum scales. I think it's a safe assumption in this case.

Find a few (x, y) pairs and interpolate from that. Type "polynomial {{x1, y1}, {x2, y2}, {x3, y3}, ...}" into Wolframalpha, it should give you an approximation polynomial that hopefully describes your function to the desired accuracy.

This is what I referred to in an earlier post. If you plug in a couple data points (the more points the better the interpolation), then wolfram alpha will give you the function of the line.

However, if you really want to get cheeky, this function looks like it represents the position of an object as it breaks the speed of sound based on what an earlier poster said. You could simply look up the function of that phenomenon.

This. Why exactly do you need to know the exact algebraic function?

To put it in the words of your course, this is definitely a polynomial, since none of the other methods are even close to this that your calculator supports.

It's at least a cubic since it has two visible points where negative acceleration turns to positive acceleration and vice versa, but it being a cubic is doubtful due to how much it flattens when y is less than 1. Because of that flattening, there's probably a repeated zero or two zeroes very close to each other that isn't visible, so I'm pretty sure quartic regression is what they want.

Use quartic regression, be sure to use at least 5 data points.

That graph is obtained by:

Solve this for v:

http://blog.wolfram.com/data/uploads/2012/10/Speed-Equation31.png

Then divide by the speed of sound at that altitude:

http://blog.wolfram.com/data/uploads/2012/10/Speed-Out5.png

You're not going to be able to find the "correct" equation, but the graph somewhat resembles a (negative) quartic graph so you could stick coordinates on the graph into a system of equations to get coefficients. Maybe.

While this method might work. I think he needs to find a way that works for his 12th grade functions class.

Not really feasible for him to submit something like that on a test.

I have no idea what 12th Grade functions is like, being from Scotland. I was just explaining where the graph actually came from and why he wouldn't be able to get a perfect answer to it. I see now where some misunderstanding might have come from - I meant for coordinates from his original graph to be substituted.

Is the coordinate-substitution method (don't know what it's actually called) beyond 12th Grade?

youre right, it has to be simple enough so that i can present in a way that my class can understand it, im going to consult with my teacher and see if i can model it as a combination of functions and if not ill just model it as a cubic function even if its not perfect.

What you have is the same as what I posted but just stretched

Edit: Here

It's not the exact same graph but all you wanted was the type of graph. It's a line raised to an exponential. I don't know what all the fuss is about