I can interpret functions in the formula form but I simply can’t understand how it can become a graph. How would you transform one into the other?
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When you have some function y=f(x) (like y=3x+2, an easy one because it is a line), you have to assign values to x and figure out what y value matches it (when x=2, y=8), and then you go onto your handy-dandy graph paper and put a dot at the place that has that specific x and y value. Do this for a number of different x values and see what sort of pattern develops.
The reverse idea, getting the applicable formula from looking at a graph, is partly recognition of the pattern (certain functions make certain shapes) and learning how constants in a function affect where the graph sits in x-y space. That is a bit more complicated but most of the common functions like lines, parabolas, circles, ellipses, can be taken from a graph and written out as a formula.
In our case, if we had plotted that example line, we would have a line rising from left to right (a positive slope, the y value gets bigger as x gets bigger). We could get the slope pretty easy (change in x and change in y between two points) and see that it is 3. We get the y intercept=2 by looking where the line crosses the y axis at x=0 (when X=0, y=2).
It turns out that a lot of functions, even the higher order ones, have inflection points (places where the graph changes direction or hits a minimum or maximum) and these locations are defined by the constants in the function, so you have a pretty good idea where you want to graph the function in x-y space to show its overall shape (not much point in plotting the function y=x^2 +3 way out where x=500, for example, unless that is the region that really interests you for other reasons; it won’t tell you that the minimum y value happens when x=0).
EDIT: it gets a lot more difficult when you are dealing with more than two variables (higher dimensions than x versus y) but the same general rules apply. Drawing it can become difficult.
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