Exponential Functions at a Glance

The exponential functions we'll deal with here are functions of the form

y = ab^{(linear function of x)} + c

where a and c are real numbers, and b is greater than 1. Really, this just means we have a number greater than 1 getting raised to the x. Numbers less than 1, you can catch the next train to Outtahereville.

The simplest kind of exponential function would be something like:

y = 2^x

Instead of just raising to the x, we could also raise 2 to a linear function of x, such as:

y = 2^{(0.5x + 1)}

We could multiply by something:

y = 4(2)^{(0.5x + 1)}

We could also add a constant: 

y = 4(2)^{(0.5x + 1)} + 11

All these things are exponential functions. Some are uglier than others, but luckily for them, we find beauty within.

How does this translate to graphs?

Sample Problem

Graph the function y = 2^x.

If we plug in x = 0 we get y = 2⁰ = 1, so that gives us the point (0, 1), which takes care of the y-intercept.

This function won't have any x-intercepts, since no value of x will satisfy the equation 2^x = 0.

Ugh, so hard to please.

For the general shape of the graph, let's find a few more points and see what happens.

Okay, so as x goes to the right, y gets bigger (and does so quickly). What happens as x goes to the left?

Hm. As x goes to the left, the values of y get really tiny. Maybe they just stole a shrink ray from a top-secret government lab. They'll never hit zero, so this function has no x-intercepts, but they get very, very close. So close that when we try to draw the graph, it'll look like the function is hitting the x-axis. It's like when your little brother used to put his hand real close to your face without actually brushing your skin and taunt you by shouting, "I'm not touching you!"

Now we connect the dots in a nice, curvy shape:

This picture shows the general shape of an exponential function. All exponential functions will look like this. They may be turned upside-down or shifted around, but they'll all have roughly this same kind of curve. Any exponential function will also have an asymptote—a value that the function gets really close to, but never quite hits. In our previous example with your little brother, your face would be the asymptote.

The asymptote for the function y = 2^x is the line y = 0. We can draw an asymptote in a graph by drawing a dashed line:

The dashed line indicates that the function never quite reaches that value. It also indicates that vehicles are allowed to pass freely across it, as long as they check first .

Sample Problem

Graph the function y = 2^{(0.5x + 1)}.

This function still won't have any x-intercepts, since 2^{(0.5x + 1)} is not 0 for any value(s) of x. Instead of being 1 when x = 0, this function will be 1 when x = -2, since then we'll get:

2^{(0.5(-2) + 1)} = 2⁰ = 1

Let's keep point-hunting.

This graph looks very similar to the previous graph. Separated at birth much?

Sample Problem

Graph the function y = 2^-x.

Let's find some points:

We seem to have turned the graph around. Next up, we'll be attempting to turn the beat around.

As x gets bigger, y is now getting smaller. As x gets smaller and heads off the page to the left, y is getting bigger. The graph looks like this:

A fancy way to say this is that the function y = 2^-x is what we get when we reflect y = 2^x across the y-axis.

Sample Problem

Graph the function y = -3(2)^x.

What will happen now? Let's think about it before we find points. Maybe we can pull a Nostradamus and predict this sucker.

For starters, all the y-values of the function will be negative, as there's still no way to get 0. When x is positive, 2^x will be "far away'' from 0, so -3(2)^x will also be far away from 0. It's okay, they can still write or send a postcard. If we find a couple of points, we see this is exactly what happens:

How about when x is negative? In that case, 2^x will be "close" to 0, so -3(2)^x will also be close to 0. It's not surprising. After all, they did grow up together.

If we find some points with negative values of x, we see that as x becomes smaller (more negative), y becomes closer to 0:

If we graph all the points we have so far and connect the dots with a curve, we find the same general shape as the graph of y = 2^x, only upside-down:

Let's review some of the things we've figured out from the examples so far. In general, adding a constant to the end of a function moves the graph of that function up or down. Hopefully no bratty kid makes his way into the function, or he may hit all the buttons so the graph stops at every floor.

The graph of y = 3^x + 5 will be 5 higher than the graph of y = 3^x, and the graph of y = 3^x – 5 will be 5 lower than the graph of y = 3^x.

The asymptote of an exponential function is given by the constant term. As we can see from the graph, the asymptote of y = 3^x + 5 is y = 5, while the asymptote of y = 3^x – 5 is y = -5.

A function such as y = 3^x, when graphed, makes a nice curve:

If we switch the sign on the exponent, the graph turns around, as if startled:

And if instead of switching the sign on the exponent we multiply the whole function by -1, the graph turns upside-down, as if incredibly startled:

One big question at this point is, "Why should I care? The calculator can draw it for me. I bow to and worship the electronic mini-god." Well, okay, but sometimes, particularly on an exam, it will save valuable time to be able to look at a graph like:

...and recognize it as an exponential graph where the exponent is negative. Because it'll be saving you time, and because time is money, being able to recognize an exponential graph makes you money. Or something like that.