For the first part, sounds like you did the right thing. If you have a value in your table where x is at or near 0, you can always evaluate your model at x=0 to make sure your model is reasonable. For the other two parts, the process is similar. Just keep in mind the generic forms of quadratic and exponential models and what the constants represent and you'll reason it out in no time.
For the quadratic, in general these models look like y=a(x-h)^2 +k. First things first, the vertex of such a graph is at the point (h,-k). So, plot your table of values using a graphing calculator and look for a good place to for the vertex. You've already got 2/3 of your necessary variables accounted for. Now you can trial-and-error to find your a-value, or use a point from the table to solve for your a-value (keep in mind that a positive a-value will make the parabola open up, a negative a-value will make it open down). Don't forget to graph your model and check that it is a reasonable representation of your data! Also, unless the table of values is very close to a parabola to begin with, the model will only be reasonable for a certain range of x-values, but this is okay as long as the vast majority of your data points are reasonably accounted for.
For the exponential, use the generic form y=a*e^(bx). If you have an x-value in your table at or near 0, whatever the corresponding y-value is will be your a-value. Smaller "b" (between 0 and 1) will essentially horizontally stretch your graph, bigger "b" will make it increase faster (with respect to x). Again, guess-and-check, or pick a point and solve for b-value.