Exponential Functions

Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power, rather than the base. Previously, you have dealt with such functions as f(x) = x², where the variable x was the base and the number 2 was the power. In the case of ex

ponentials, however, you will be dealing with functions such as g(x) = 2^x, where the base is the fixed number, and the power is the variable.

Let's look more closely at the function g(x) = 2^x. To evaluate this function, we operate as usual,

picking values ofx, plugging them in, and simplifying for the answers. But to evaluate 2^x, we need to remember how exponents work. In particular, we need to remember that negative exponents mean "put the base on

the other side of the fraction line".

So, while positive x-values give us values like these:

...negative x-values give us values like these:

Putting together the "reasonable" (nicely graphable) points, this is our T-chart:

...and this is our graph:

You should expect exponentials to look like this. That is, they start small —very small, so small that they're practically indistinguishable from "y = 0", which is the x-axis— and then, once they start growing, they grow faster and faster, so fast that they shoot right up through the top of your graph.

You should also expect that your T-chart will not have many useful plot points. For instance, for x = 4and x = 5, the y-values were too big, and for just about all the negative x-values, the y-values were too small to see, so you would just draw the line right along the top of the x-axis.

Note also that my axis scales do not match. The scale on the x-axis is much wider than the scale on the y-axis; the scale on the y-axis is compressed, compared with that of the x-axis. You will probably find this technique useful when graphing exponentials, because of the way that they grow so quickly. You will find a few T-chart points, and then, with your knowledge of the general appearance of exponentials, you'll do your graph, with the left-hand portion of the graph usually running right along the x-axis.