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Functions and Graphing

Now, we can move onto the final topic, graphing. For this, we have some pre-information we need to cover. For starters, the functions we are working with are graphed on what's called the xy-plane (shown below). Another thing is that we will continuously flip flop notations using f(x) and y. For all intents and purposes, y = f(x). This is understood whenever you graph functions. Finally, we need a way to show specific values on the xy-plane. We write these down as an ordered pair, known as coordinates. We use the notation (x, y) = (x, f(x)) for our coordinates in the xy-plane.

The point where the two axes meet is called the origin. This is just a fancy way of saying the coordinate pair (0, 0). In higher-level math, and some physics classes, you'll find that a change in the coordinate system sometimes leads to a change in the origin. 

For now, we will stick to recognizing functions and graphing basic functions. To recognize whether a graph is showing a function or not, we use the vertical line test. What this test says is to put a vertical line anywhere on the graph. If there is a location you can place the line such that the function crosses the line twice or more, then the graph is not showing the graph of a function (NOTE: We are only looking at functions of x, not functions of y). A simple way to do this is to use either your finger or a pencil as the vertical line. Below, I just drew dotted lines as the vertical line so the pencil doesn't take up the entire image.

There are many different types of functions, and different ways functions can be manipulated as well. In due time, you'll learn about how graphs transform under certain changes to the function, but for now, we'll stick to figuring out how to graph the base functions first.

Our first case is simple, f(x) = x. The way we graph it is very simple, wherever x is, the y-value is going to be the same value as x. Again, I always plot a couple of points to get an idea of how the graph works, and then I sketch the rest of the graph. Below is what the graph for f(x) = x looks like.

This is the first type of graph that you've probably already learned about. The next type of graph I'll show you is the next step up from the linear graph above. The function we'll take a look at is f(x) = x^2. The first thing we want to do is set up our coordinate system (i.e. the x and y axes). After this, we'll then plot a few points. I'd normally choose small values for this kind of graph. Below, I've plotted the points from x = -2 to x = 2. 

As we can clearly see, this graph is no longer linear. The points do not increase at a constant rate. So, how do we draw the line to connect these points? Well, we create a curved line that runs through all the points. Below I've completed this step so you can see what it looks like. (Forgive my not very good drawing. I'm not an artist.)

Normally, most functions are not linear. So It's safe to assume that if your function is not linear, your graph won't be linear either. I'll use a computer to show you a better drawing of what a parabola looks like below. (Thank god for Desmos)

This should give you a better idea of what the curvature of the y=x^2 graph should look like. This shape is known as a parabola.  You can see this kind of shape in a lot of random places. In fact, you could probably walk outside and find a parabola somewhere.

This concludes everything on functions and graphing. Remember, on exams you normally won't have a calculator, so be sure to plot a few points that are easy to calculate (normally 0, 1, and 2). Then also sketch the line going through each of the points. It's a good idea to use a computer graphing calculator to get an idea of what basic functions look like. That way you don't need to do as much work when trying to sketch the rest of the graph. If you have any questions or if anything confuses you, feel free to ask me for help. I'm happy to answer any questions you have.

Sources

desmos.com/calculator