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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At its basic level, the graph is plotting the relationship between the input (x) and the outputs (f(x)) for the function. To start with, the best thing is to make a table of values for x and then calculate the corresponding f(x) values and plot them.
Unfortunately, that’s tedious, so we don’t de it every time. Luckily, there are function families with specific shapes that are easy to start to recognize. You can see a cat and identify it as a “cat” without all cats needing to look identical, because you understand some basic properties of cats and how they’re different from, say, raccoons or dogs. Same thing with functions. A function in the form of f(x) = x^(2) will look different than f(x) = 1/x, which looks different than f(x) = sin (x), but functions in the form f(x) = ax^(2) + c will have some very similar characteristics.
Once you learn about the various function families, you can learn about some common transformations. Adding a constant term “outside” the function will shift the graph up/down (f(x) = sin(x) + 2). Adding a constant term “inside” the function (f(x) = sin(x-3) ) will shift the function left/right. Multiplying “outside” the function (g(x) = -3sin(x)) stretches and/or reflects the function vertically. Multiplying “inside” the function ( g(x) = sin(1/2 x) ) stretches it horizontally.
Once you become familiar with the common transformations and the common function families, you can predict what a function looks like pretty easily without graphing technology. This is sort of like someone telling you about a cat that is an overweight, long haired, orange tabby with a white belly. That’s a pretty specific cat, but you can most likely picture it in your mind if you’re familiar with cats. With practice, algebraic functions become similar.
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