Section 3.1: Powers and Exponential Functions

In this section, we want to:

• Introduce the concept of an exponential function;
• Explain how, even though the formulas for exponential functions are deceptively simple, these functions require us to learn to use their algebraic properties (i.e., the so-called rules of exponents);
• Discover the basic characteristics of an exponential graph and discuss how to graph a general exponential function; and
• Show how the rules of exponents correspond to geometrical properties of their graphs.

By the end of this section, you should have a deeper understanding and appreciation for the rules of exponents and be confident in your ability to graph an exponential function. 

Introduction to Exponential Functions

We have already used powers, that is, expressions such as 2³.  By rules of exponents, we know that this stands for 2 multiplied by itself three times  Note: When writing such expressions, it is very important that we raise the second number high enough (and typically, we also write it a bit smaller), or we may confuse this with the number 23 or the product 2 3 = 6.  We refer to the number written up and to the right (in this case, 3) as the exponent, while the number on the baseline (in this case, 2) is called the base.  Notice that these two numbers play very different roles, so that functions, like s(x) = x³ (i.e., where the independent variable is in the base) are very different from exponential functions, such as f(x) = 2^x (i.e., where the independent variable is in the exponent).  In particular, while the computation of s is easy to describe and carry out (e.g., "multiply the input three times"), even though, from a notational standpoint, f does not look any more complicated, it is actually much more difficult to compute.

In fact, there are many functions involving powers that are quite difficult to compute.  This becomes evident only after carefully considering the rules of exponents.  For example, when the exponent is a positive integer, such as, g(x) = x², rules of exponents give us explicit instructions on how to compute the output from the input: "Take the input and multiply it with itself".  Since we know how to multiply any two numbers, this is a clear rule.  This is not much more difficult for negative integer exponents, since rules of exponents tell us that h(x) = x^-2 = 1/x², that is, "Multiply by itself and take the reciprocal."  

However, consider the case when the exponent is a fraction, such as k(x) = x^1/2.  While rules of exponents say that we may rewrite this as, , which may seem more familiar, in general, this function is very difficult to compute.  That is because, by definition, this function is the inverse of, .  That means that to compute y = k(x), we must use the defining property of inverse functions and look for the positive solution to x = r(y) = y².  For some inputs, like x = 4, we can solve this equation in our heads to obtain the value y = 2.  However, for most inputs, such as x = 2, we cannot give the precise value of k.  While we may write the "value" as , this is simply notation which represents the value (no better than writing 2^1/2).  To compute this value we must solve, 2 = y², which we can only do with a calculator (i.e., we cannot do it by hand).  Even when the calculator gives us the value of 1.4142135623730950488016887242097¼ (to 32 digits), since the actual value has infinitely many digits, we can never know the exact value!  

The situation is even worse for the exponential function f(x) = 2^x, since this is supposed to give a rule for every possible real number input.  Not only are we required to compute 2^1/2 (which we have just seen is quite difficult), we must also be able to compute values such as  .  We cannot even describe this value as the solution to some arithmetic equation.  While our calculator can compute the correct value, it must use a very difficult formula from Calculus to do so.  This means that, at this point, the only ways for us to understand exponential functions is through their properties, namely, the rules of exponents, or by thinking of it as a function given by the black-box of our calculator.

Graphing Exponential Functions

While the general formula for computing a^b is too messy for us to use for computations, it is "reasonable", in that small changes in either the base, a, or the exponent, b, correspond to small changes in the resulting value.  In Calculus, this phenomena, is known as "continuity".  Practically speaking, this implies that we may make reasonable estimates of its value.  For example, since  , we can estimate  to be a bit less than  , or 3.  Since the true value is 2.6651441426902251886502972498731... (to 32 digits), we can see that "a bit less than 3" is a pretty decent estimate.  

More importantly, this property of continuity implies that the graph is a nice, smooth unbroken curve, so that we can make a reasonable sketch by "connecting-the-dots".  Note: We have, in fact, taken this property for granted in all of the functions that we have graphed to this point (except for the greatest integer function).  For example, y = 2^x has a typical looking exponential graph.  We can easily compute seven integer points on this graph:

This displays two general trends in the graph:

• The outputs double every time we move one unit to the right; in particular, the values quickly grow quite large to the right of the origin.
• Conversely, the outputs halve every time we move one unit to the left; in particular, the values shrink quickly as we move to the left of the origin, always staying positive (i.e., above the horizontal axis).

Both of these trends are quite visible when we create a plot in XFunctions:

Select the function, f, to see a plot of y = 2^x.  If you zoom out a few times, you can really see how quickly the graph increases; from far enough out, the graph looks almost like a right-angle.  This also emphasizes the shape of the graph on the left, how it comes to look more and more like the horizontal axis.  We say that the graph "approaches" the horizontal axis "asymptotically"; that is, as we move to the left (towards x = -¥), the graph gets closer and closer to the horizontal axis, without crossing it.  In this case, we refer to this axis, y = 0, as a horizontal asymptote for the graph.

As before, once we know the basic shape of the graph, we only need to remember a few key points on the graph in order to sketch a reasonable graph.  As usual, the three points nearest the origin, x = -1, 0, 1, are generally sufficient.  In order to plot the horizontal asymptote, it is helpful to include one more "point" at x = -¥, giving the table of points:

We would then plot these points, including the asymptote as a dotted line, like this:

Now we can use our transformational graphing technique to graph any function built from an exponential "core", such as h(x) = -4·3^{x - 1} + 2.  This decomposes as:

• subtract 1 from the input,
• take this as a power of 3,
• multiply the result by 4,
• negate this, and
• add 2.

Note: Remember that the usual order of operations says that we should take the power before we multiply.  This leads to the following analysis:

We can imagine roughly how the graph will look:

• Shifting right and stretching make the graph look a bit steeper, but the general shape remains the same.
• Once we flip the graph, it will start from an asymptote of y = 0 and go down, instead of up.
• Shifting the graph up will move the horizontal asymptote up to y = 2.

Drawing in the asymptote, plotting the points and "connecting-the-dots" gives:

Note: Although the graph never actually touches the asymptote, it quickly gets so close that it looks like it does.

You can see how this graph is "built" out of the graph of 3^x by selecting the Examples in the following applet:

Practice graphing exponential functions by completing the following Exercises.  

Symmetries of Exponential Functions

Many of the rules of exponents correspond to "symmetries" among the set of exponential functions.  For example, one rule implies that 8^x = (2³)^x = 2^(3x).  We recognize 2^(3x) as graphing like 2^x, but shrinking horizontally by a factor of 3.  By our transformational graphing technique:

This will make the graph look steeper and flatter:

Note: While we could have simply plotted this by plugging in the points x = -1, 0, 1, to obtain (-1, 1/8), (0, 1), and (1, 8), which are on our graph, but our analysis shows how this graph corresponds to the graph of y = 2^x that we have already made.

In general, we can say that all exponential graphs, with base greater than 1, are related in this way:

Increasing the size of the base (greater than 1), corresponds to a horizontal shrink, which makes the graph look steeper and flatter.

We can see this by looking at the Example in the following applet; this is an animation which shows the effect of increasing the base (in this case, 2) of the exponential:

You can see that the larger the base, the more extremely the two trends are displayed; that is, the larger the base, the faster the graph grows the right and shrinks to the left.  Note: If the animation moves too fast for you to see, hit the "Stop" button, and then step through the animation, one step at a time, with the "Next" button.

Another rule of exponents says that a^-x = 1/a^x = (1/a)^x.  This relates the exponential graphs with base greater than 1 that we have just examined to exponential graphs with base less than 1.  For example, the graph of y = g(x) =  0.5^x = (1/2)^x = 1/2^x = 2^-x looks like that of y = 2^x reflected horizontally: 

We know that the graph of y = 2^x starts on the left near the asymptote of y = 0, and increases slowly until it crosses the vertical axis at 1 and then increases quickly.  When we flip the graph horizontally, y = 0.5^x decreases quickly, crossing the vertical axis at 1, and approaches an asymptote of  y = 0, as we move to the right.  You can see this by selecting g from the following applet:

If you select the Example, you can see in the animation what happens as we make the base even smaller: as the base gets smaller, the graph gets steeper and flatter.  That is, the two trends that we saw before become more pronounced, but on the opposite sides of the origin.  Since a very small base has a very large reciprocal, this is simply the reflection of the animation we saw earlier.  In general:

Taking the reciprocal of the base corresponds to a horizontal reflection.

Putting these two observations together, we now know what happens when either the base gets large (i.e.,  base > 1) or small (i.e.,  base < 1).  When the base is equal to 1, we have the constant function y = 1^x = 1, which graphs as a horizontal line.  You can quickly view a sample of all types of exponential graphs in the Example of the following applet: 

One very important consequence of these two symmetries is that, up to horizontal scaling and/or flip, there is only one exponential function.  That is, we can obtain the graph of any exponential function by taking an appropriate horizontal scale factor, ±r, in the graph of y = 2^±rx.  We will return to discuss this point in more detail when we consider logarithms.

One final symmetry of exponential graphs is that:

A horizontal shift corresponds to a vertical scaling.

That is because rules of exponents imply that a^{x + c} = a^ca^x.  For example, if we shift the graph of y = 2^x left by 1, to obtain y = 2^{x + 1} = 2^x2¹ = 2·2^x, the graph simply looks like it has been stretched vertically by 2.  We can compute some points on "both" graphs:

and:

We can see that these points all lie on the same graph:

Note: In fact, this "symmetry" completely determines the shape of all exponential graphs.

To summarize, these three symmetries highlight the key features of all exponential graphs (except when the base is 1); any exponential graph:

• has a horizontal asymptote of y = 0,
• crosses the vertical axis at y = 1, 
• the graph curves upwards, and
• as we move from left to right, either the graph increases from the asymptote or decreases to the asymptote, depending on whether the base is greater or less than 1.

Since we know how these graphs are shaped, once we plot values at x = -1, 0, 1, and the asymptote at x = ±¥, we can "connect-the-dots" to obtain a decent sketch of any exponential graph.

Note: You will notice that we have not considered the case when the base is negative.  That is because, for example, the function y = (-2)^x is undefined for "most" values of the input.  For example, while we can compute this for integer inputs like (-2)² = 4 or (-2)³ = -8, when the input is not an integer, such as  and , the result cannot generally be defined.  As we will see in the next section, exponentials only arise in applications with positive bases anyway, so this is not a big problem.

Practice graphing exponential functions by completing the following Exercises.  

Go to Applications of Exponential Functions

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Send questions or comments to jwicks@northpark.edu

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