5.3 Exponential curves

By plotting points, you can discover that the graph of $y=2^x$ looks like this:

A few points to notice about this graph.

• It goes through the point $(\mathrm{0,1})$ because $2^0=1$ .
• It never dips below the $x$ -axis. The domain is unlimited, but the range is y>0. (*Think about our definitions of exponents: whether $x$ is positive or negative, integer or fraction, $2^x$ is always positive.)
• Every time you move one unit to the right, the graph height doubles. For instance, $2^5$ is twice $2^4$ , because it multiplies by one more 2. So as you move to the right, the $y$ -values start looking like 8, 16, 32, 64, 128, and so on, going up more and more sharply.
• Conversely, every time you move one unit to the left, the graph height drops in half. So as you move to the left, the $y$ -values start looking like $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , and so on, falling closer and closer to 0.

What would the graph of $y=3^x$ look like? Of course, it would also go through $(\mathrm{0,1})$ because $3^0=1$ . With each step to the right, it would triple ; with each step to the left, it would drop in a third . So the overall shape would look similar, but the rise (on the right) and the drop (on the left) would be faster.

As you might guess, graphs such as $5^x$ and ${\text{10}}^x$ all have this same characteristic shape. In fact, any graph $a^x$ where $a>1$ will look basically the same: starting at $(\mathrm{0,1})$ it will rise more and more sharply on the right, and drop toward zero on the left. This type of graph models exponential growth —functions that keep multiplying by the same number. A common example, which you work through in the text, is compound interest from a bank.

The opposite graph is ${\left(\frac{1}{2}\right)}^x$ .

Each time you move to the right on this graph, it multiplies by $\frac{1}{2}$ : in other words, it divides by 2, heading closer to zero the further you go. This kind of equation is used to model functions that keep dividing by the same number; for instance, radioactive decay. You will also be working through examples like this one.

Of course, all the permutations from the first chapter on “functions” apply to these graphs just as they apply to any graph. A particularly interesting example is $2^{-x}$ . Remember that when you replace $x$ with $-x$ , $f(3)$ becomes the old $f(-3)$ and vice-versa; in other words, the graph flips around the $y$ -axis. If you take the graph of $2^x$ and permute it in this way, you get a familiar shape:

Yes, it’s ${\left(\frac{1}{2}\right)}^x$ in a new disguise!

Why did it happen that way? Consider that ${\left(\frac{1}{2}\right)}^x=\frac{1^x}{2^x}$ . But $1^x$ is just 1 (in other words, 1 to the anything is 1), so ${\left(\frac{1}{2}\right)}^x=\frac{1}{2^x}$ . But negative exponents go in the denominator: $\frac{1}{2^x}$ is the same thing as $2^{-x}$ ! So we arrive at: ${\left(\frac{1}{2}\right)}^x=2^{-x}$ . The two functions are the same, so their graphs are of course the same.

Another fun pair of permutations is:

$y=2\cdot 2^x$ Looks just like $y=2^x$ but vertically stretched: all y­-values double

$y=2^{x+1}$ Looks just like $y=2^x$ but horizontally shifted: moves 1 to the left

If you permute $2^x$ in these two ways, you will find that they create the same graph.