How do I use the graph of a function to predict future behavior?
Well, this is a difficult one! I am not sure this is going to help, but I try anyway.
When I have to analyze a phenomenon (I am a physicist...!) I collect data that characterize the phenomenon (for example, I measure the height of a kid each week) and then I plot them.
The result is a graph with points on it that hopefully shows a "tendency".
This can be a linear tendency, for example, so that I can use a line to represent them (the line that "best fit" all the data points).
Next I evaluate the equation of the line in the general form:
Now, even if I do not know what happens for a certain value of the variable
In the example of the height of a kid, I can measure it along, say, a period of 5 weeks, get my graph and evaluate the linear equation as:
If I want the height of the kid in the 7th week I put time=7 and find the projected height.
You can have different curves that fit your data (parabolas. hyperbolae, sinusoidals...);
The fit depends upon the "goodness" of you experiment in the first place. If you try to correlate the height of a kid to the number of red cars you observe ...it won't probably work very well!
(hope it helps!)