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: #y=ax+b#

Now, even if I do not know what happens for a certain value of the variable #x# (because I didn't measure it directly) I can put it in the equation of the line and get the corresponding value for #y#.

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:
#height=2*time+40# in cm for the eight and week for the time (2 will have units of cm/week and 40 of cm).

If I want the height of the kid in the 7th week I put time=7 and find the projected height.

Remember:

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!)