How are functions created/rendered from an observation

I've always wondered in math books how an equation is formulated. For example, "the parabolic motion of a baseball being hit by a bat behaves according to the function -16x^2+5x+3". How do they generate these functions? Do they collect data from it's initial and final locations, or what?

Great question. Sometimes, yes, functions are created by collecting data and then using methods that give equations that best fit the data. There's a branch of mathematics called numerical analysis that deals with how to derive "best fitting" equations. If you do a search on the web for "least squares method", you can see an example.

Other times functions might be derived by making a few assumptions about your system, then basically using logic (with the help of calculus) to derive the rest. This is the case with the example you give. We know (and I'll explain why in a moment) that if you ignore things like air resistance and the curvature of the earth, an object being thrown upwards at an angle will follow a path ("trajectory") that's in the shape of a parabola. Parabolas are in the from of y = ax^2 + bx + c. So if you let "y" be the height and "t" be the time, you have an equation. You just need a few data points so that you can figure out the values of a, b, and c.

As for how we know it follows a parabola, well once you make assumptions like the force due to gravity being a constant, set your variables, etc. you can use calculus to deduce that the result follows an equation that's identical to a parabola in rectangular coordinates.

These pages might help:
http://electron9.phys.utk.edu/phys135d/m…
http://en.wikipedia.org/wiki/Range_of_a_…

The equations are basically models.
In some cases the equations are derived from observations. For example some there is a linear relation between the temperature and the number of chirps from a cricket. You can find the slope and y-intercept of the line by observing the temperature and counting the number of chirps/sec.