Geometry of graphs: Odd functions, asymptotes and translation

We continue our study of graphs. Let’s look at the function x cubed– that is, x to the power 3. I call it h here. We can trace its graph, as we’ve done for other functions, and you get something like the following curve. Does that curve look odd to you? Well, the function h is odd. h of minus x is the same as minus h of x, as you can check. This property of a function gives you a kind of symmetry. Different from the symmetry about an axis that we saw earlier, this gives you symmetry with respect to the origin. The meaning of that is the following.

If you take a point x, y on the graph, if you proceed to draw a straight line, from x, y through the origin, you will meet the point minus x, minus y. And that point will also be on the graph.

Now we know, from an earlier proposition, that the function h is strictly increasing on r plus. But, given this symmetry, we can go on to deduce that h is actually strictly increasing globally on the real line. Another function of interest is the function 1/x. It’s clear here that we have to restrict the domain. Right? We have to avoid the value x equals 0. When we draw the graph of the function, we get something like this. And this curve, the red curve, has the property that, as x moves out far to the right, the red curve gets closer and closer to the x-axis without actually meeting it. We say that the x-axis is a “horizontal asymptote” of the curve.

You notice that the function capital F here is an odd function. Therefore, in order to have the left-hand branch of the graph, we can apply symmetry with respect to the origin. And we know it’s going to look like this. Look at the behaviour of this function at x equals 0, where it’s not defined. As x approaches 0, from the right, and getting very, very small, you can see the value of the function gets bigger and bigger and goes up to plus infinity, asymptotically. Similarly, as x approaches 0 but from the left, the value goes down to minus infinity, asymptotically. We say that the y-axis is a “vertical asymptote,” in this case. Sometimes functions are defined in a piecewise manner.

They have two different defining formulas, depending where the arguments are. For example, we could have a function f that agrees with g, if x is less than k, but then switches over to a different function h when x is greater than k. You know, we already have an example of such a function. The absolute value function is of this type. It’s given by minus x when x is negative and by plus x if x is positive. How do find the graph of absolute x? Well, we could look at the graphs of the two components, the functions minus x and x. Those are affine functions, so we know how to graph them.

And then we take the left-hand branch of g and the right-hand branch of h, put them together, and you have the graph of the function absolute x. Let’s speak now about dilation. A “dilation” means multiplication by a constant. What happens to the graph of a function when you multiply a function by a constant c? Let’s take our familiar example, the x-squared function, with the parabola being the graph. What happens if we multiply the function by a constant bigger than 1? Well, the answer is, the graph will sort of be expanded. It’ll be dilated upwards, because c is bigger than 1. If we multiplied by a positive c that was smaller than 1, then the graph would be dilated down.

It would get squished. What about negative constants? What if the graph of this function were multiplied by minus 1? In that case, the graph of the new function would be rotated, relative to the x-axis. Whatever was positive becomes negative. That would be the graph of the function minus x squared. Another way to alter a function is by translation. For example, one can just add a constant k, a fixed real number, to the values of f. In that case, the graph of f will be vertically translated– that means vertically shifted– up or down, depending on the sign of k. Here’s an example. What is the graph of the function x squared plus 1?

Well, we know the graph of the function x squared. It’s our parabola. And if you want to know the graph of the new function, you just take the old one and shift it up one unit, and that’ll be the graph of the new function, x squared plus 1. Another way to apply translation is inside the function. Given an argument x, we’re going to apply f not at x but at x plus k, where k is a fixed real number. In that case, it turns out the graph of f will be horizontally translated, left or right. Let’s look at this example. What is the graph of x plus 1 squared?

Well, if the x plus 1 were the real variable, let’s make a change of variables and call x plus 1 capital X. In that case, we would know exactly what the graph looks like, in terms of capital X. That would be it. How do we relate that to what the graph should be, in terms of our real variable, a lowercase x? Well, the key is to note that capital X equals 0, in the change of variables, corresponds to little x equals minus 1. That’s where the origin should be shifted to. So we take our graph, we shift it to minus 1, and that is the graph of the new function x plus 1 squared.