In all living things, there is a process where new cells are constantly being made to create growth and to replace old cells. One common way these new cells are made is by cell division:
We have generated the number pattern that is produced by doubling: $1,2,4,8,16,\ldots$1,2,4,8,16,…
We can generalize this elegantly, and say that if $x$x represents the number of days that have passed, and $y$y represents the number of cells, then $y=2^x$y=2x.
What we immediately notice is how quickly the number of cells increases. On top of this, the more cells there are the more rapidly the number of cells increases.
There are other phenomena that behave in the same way. Bacteria increase in number very quickly, and populations of some species increase very rapidly over time.
When a quantity increases in this way, we say that it experiences exponential growth, and we can use some basic equations to model this growth. We can also use exponentials to model quantities that decrease in a particular way over time.
A basic exponential function has the form:
$y=a^x$y=ax or $y=a^{-x}$y=a−x
where $a$a can be any number greater than $1$1.
In exponential equations, the variable $x$x is in the power.
Let's take the example of $y=2^x$y=2x.
By looking at the equation of the exponential function, we can create a table of values and determine a few features of its graph:
$x$x | $-4$−4 | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|---|---|---|---|---|
$y$y | $\frac{1}{16}$116 | $\frac{1}{8}$18 | $\frac{1}{4}$14 | $\frac{1}{2}$12 | $1$1 | $2$2 | $4$4 | $8$8 | $16$16 |
This last point is important. Notice that even for negative values of $x$x, the function value is greater than zero. That is, the exponential function is always positive and so never crosses the $x$x-axis.
The graph below shows the exponential function $y=2^x$y=2x.
We can observe all of the features that we noted above:
The horizontal line (the $x$x-axis) that the curve approaches but never crosses is called an asymptote.
This next graph shows the exponential function $y=2^{-x}$y=2−x:
Notice that this graph is a reflection of the graph of $y=2^x$y=2x about the $y$y-axis. This means that:
Basic exponential functions have the form $y=a^x$y=ax or $y=a^{-x}$y=a−x, where $a$a can be any number greater than $1$1.
Their graphs have certain features:
Consider the function $y=3^x$y=3x.
Complete the table of values:
$x$x | $-4$−4 | $-3$−3 | $-2$−2 | $-1$−1 | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $10$10 |
---|---|---|---|---|---|---|---|---|---|---|
$y$y | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Is $y=3^x$y=3x an increasing function or a decreasing function?
Increasing
Decreasing
Describe the rate of increase of the function.
As $x$x increases, $y$y increases at a faster and faster rate.
As $x$x increases, $y$y increases at a slower and slower rate.
As $x$x increases, $y$y increases at a constant rate.
Consider the function $y=9^{-x}$y=9−x.
The value of $9^{-x}$9−x is always greater than which number?
$0$0
$1$1
$9$9
So how many $x$x-intercepts does the graph of $y=9^{-x}$y=9−x have?