5.01 Graphs and characteristics of exponential functions

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:

• On day 0, we start with $1$1 cell.
• On day 1, this cell splits into two and forms $2$2 cells.
• On day 2, each of the two cells splits and forms $4$4 cells, and so on.

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.

• The exponential equation $y=a^x$y=ax could be used to represent quantities that increase over time (exponential growth).
• The exponential equation $y=a^{-x}$y=a−x could be used to represent quantities that decrease over time (exponential decay).

 

Graph of $y=a^x$y=ax

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

• When $x=0$x=0, we have $y=2^0=1$y=20=1, so the $y$y-intercept of the graph is at $\left(0,1\right)$(0,1).
• As $x$x increases (into positive values), the function values get very large. For example, when $x=10$x=10 we have $y=2^{10}=1024$y=210=1024.
• As $x$x decreases (into negative values), the function values get very small and close to zero. For example, when $x=-10$x=−10 we have $y=2^{-10}=\frac{1}{2^{10}}=\frac{1}{1024}$y=2−10=1210​=11024​.

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 graph passes through the $y$y-axis at the point $\left(0,1\right)$(0,1).
• The graph doesn't cross the $x$x-axis, but approaches it for negative values of $x$x.
• The graph represents an increasing function.

The horizontal line (the $x$x-axis) that the curve approaches but never crosses is called an asymptote.

• The asymptote of this curve is the line with equation $y=0$y=0.

 

Graph of $y=a^{-x}$y=a−x

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:

• The graph still crosses the $y$y-axis at the point $\left(0,1\right)$(0,1), and still doesn't cross the $x$x-axis.
• The graph now gets very large for negative values of $x$x, and approaches zero for positive values of $x$x.
• The asymptote of this curve is the line with equation $y=0$y=0.
• The graph represents a decreasing function.

 

 

Practice questions

Question 1

Question 2