Exponential Functions

Hi guys Its Suhyeon :-)
I am going to show you some information about Exponential Functions


Exponential Functions: Introduction

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 exponentials, 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 of x, 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: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

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 = 4 and 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.

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You may have heard of the term "exponential growth". This "starting slow, but then growing faster and faster all the time" growth is what they are referring to. Specifically, our function g(x) above doubled each time we incremented x. That is, when x was increased by 1 over what it had been, y increased to twice what it had been. This is the definition of exponential growth: that there is a consistent fixed period over which the function will double (or triple, or quadruple, etc; the point is that the change is always a fixed proportion). So if you hear somebody claiming that the world population is doubling every thirty years, you know he is claiming exponential growth.

Exponential growth is "bigger" and "faster" than polynomial growth. This means that, no matter what the degree is on a given polynomial, a given exponential function will eventually be bigger than the polynomial. Even though the exponential function may start out really, really small, it will eventually overtake the growth of the polynomial, since it doubles all the time.

For instance, x10 seems much "bigger" than 10x, and initially it is:

But eventually 10x (in blue below) catches up and overtakes x10 (at the red circle below, where x is ten and y is ten billion), and it's "bigger" than x10 forever after:

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Exponential functions always have some positive number other than 1 as the base. If you think about it, having a negative number (such as –2) as the base wouldn't be very useful, since the even powers would give you positive answers (such as "(–2)² = 4") and the odd powers would give you negative answers (such as "(–2)³ = –8"), and what would you even do with the powers that aren't whole numbers? Also, having 0 or 1 as the base would be kind of dumb, since 0 and 1 to any power are just 0 and 1, respectively; what would be the point? This is why exponentials always have something positive and other than 1 as the base.





Exponential Functions: Evaluation

The first thing you will probably do with exponential functions is evaluate them.

• Evaluate 3^x at x = –2, –1, 0, 1, and 2.

To find the answer, I need to plug in the given values for x, and simplify:

 

• Given f(x) = 3^–x, evaluate f(–2), f(–1), f(0), f(1), and f(2).

To find the answer, I need to plug in the given values for x, and simplify

Take another look at the values I came up with: they were precisely reversed between the two T-charts. Remember that negative exponents mean that you have to flip the base to the other side of the fraction line. This means that 3^–x may also be written as ( ¹/₃ )^x, by taking the "minus" in the exponent and using it to flip the base "3". With this in mind, you should be able to predict the values for the following problem: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

• Given g(x) = ( ¹/₃ )^x, evaluate for x = –2, –1, 0, 1, and 2.
  
Plug in the given values for x, and simplify:


 
 

This exercise points out two things. First, you really do need to be good with exponents in order to do exponentials (so review the topic, if necessary), and, second, exponential decay (getting smaller and smaller by half (or a third, or...) at each step) is just like exponential growth, except that either the exponent is "negative" (the "–x" in "3^–x") or else the base is between 0 and 1 (the "¹/₃" in "( ¹/₃ )^x").

It will likely be necessary for you to be able to just look at an equation or an expression or a graph and correctly identify which type of change it represents, growth or decay, so go back and study the above examples, if you're not sure of what is going on here.

To be thorough, since 3x models growth:
...and since 3–x and ( 1/3 )x model decay:

...it seems reasonable that ( ¹/₃ )^–x (small base and a "negative" exponent) should model growth. Let's check....


Exponential Functions: Graphing

• Graph h(x) = ( ¹/₃ )^–x.

I will compute the same points as previously:
Then the graph is:

So y = ( ¹/₃ )^–x does indeed model growth. For the record, however, the base for exponential functions is usually greater than 1, so growth is usually in the form "3^x" (that is, with a "positive" exponent) and decay is usually in the form "3^–x" (that is, with a "negative" exponent).

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Every once in a while they'll give you a more-complicated exponential function to deal with:

• Graph y = e^x2.

I will compute some plot-points, as usual:

Note that, for graphing, the decimal approximations are more useful than the "exact" forms. For instance, it is hard to know where "e^2.25" should be plotted, but it's easy to find where "9.488" goes. Note also that I calculated more than just whole-number points. The exponential function grows way too fast for me to use a wide range of x-values (I mean, look how big y got when x was only 2). Instead, I had to pick some in-between points in order to have enough reasonable dots for my graph.

Now I plot the points, and draw my graph: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

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The above function did involve an exponential, but was not in the "usual" exponential form (since the power was not linear, but quadratic). However, usually you'll get the more-standard form, with a greater-than-one base, perhaps multiplied by some constant, and a linear exponent. Note that the graphs all look roughly the same; they may have been moved up or down, flipped upside-down, shifted left or right, etc, etc, but they all pretty much have the same shape: