4.01 Exponential functions

A base form of an exponential function is $f\left(x\right)=a^x$f(x)=ax, where $a$a is a positive number and the variable is in the exponent. Unlike linear functions which increase or decrease by a constant, exponential functions increase or decrease by a constant multiplier. Let's first look at cases for $a>1$a>1, where we have exponential growth and identify key characteristics of such functions.

 

Graphs of $y=a^x$y=ax for $a>1$a>1

Let's create a table for the function $y=2^x$y=2x:

$x$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

We can see our familiar powers of two and as $x$x increases by one, the $y$y values are increasing by a constant multiplier - here they are doubling. This causes the differences between successive $y$y values to grow and hence, $y$y is increasing at an increasing rate. Let's look at what this function looks like when we graph it.

Key features:

• As the $x$x-values increase, the $y$y-values increase at an increasing rate.
• The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=2^0=1$y=20=1.
• As $x$x becomes a larger and larger negative number, $y$y becomes a smaller and smaller fraction. In the graph we see as $x$x becomes a larger negative number the graph approaches but does not reach the line $y=0$y=0. Hence, $y=0$y=0 is a horizontal asymptote. We can write this asymptotic behaviour mathematically as follows: As $x\ \rightarrow-\infty,y\ \rightarrow0^+$x →−∞,y →0+ (As $x$x approaches negative infinity, $y$y approaches zero from above).
• Domain: $x$x is any real number.
• Range: $y>0$y>0

How does this compare to other values of $a$a? Let's graph $y=2^x$y=2x, $y=3^x$y=3x and $y=5^x$y=5x on the same graph. You can create a table for each to confirm the values sketched in the graph below:

We can see all of the key features mentioned above were not unique to the graph of $y=2^x$y=2x.

Key features:

• As the $x$x-values increase, the $y$y-values increase at an increasing rate.
• The $y$y-intercept is $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=a^0=1$y=a0=1, for any positive value $a$a.
• $y=0$y=0 is a horizontal asymptote for each graph. As $x\ \rightarrow-\infty,y\ \rightarrow0^+$x →−∞,y →0+ 
• Domain: $x$x is any real number.
• Range: $y>0$y>0

The difference is that for $x>0$x>0 the higher the $a$a value the faster the graph increases. Each graph goes through the point $\left(1,a\right)$(1,a) and we can see the larger the $a$a value the higher this point will be.

For $x<0$x<0 the higher the $a$a value the quicker the graph approaches the horizontal asymptote. 

 

Practice questions

question 1

question 2

 

Graphs of $y=a^x$y=ax for $00<a<1

Let's create a table for the function $y=\left(\frac{1}{2}\right)^x$y=(12​)x:

$x$x	$-4$−4	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2	$3$3	$4$4
$y$y	$16$16	$8$8	$4$4	$2$2	$1$1	$\frac{1}{2}$12​	$\frac{1}{4}$14​	$\frac{1}{8}$18​	$\frac{1}{16}$116​

Again we can see our familiar powers of two but this time as $x$x increases by one the $y$y values are decreasing by a constant multiplier - here they are halving. The differences between successive $y$y values is shrinking and hence, $y$y is decreasing at a decreasing rate. Let's look at what this this function looks like when we graph it.

Key features:

• As the $x$x-values increase, the $y$y-values decrease at decreasing rate.
• The $y$y-intercept is still $\left(0,1\right)$(0,1), since when $x=0$x=0, $y=\left(\frac{1}{2}\right)^0=1$y=(12​)0=1.
• As $x$x becomes a larger and larger positive number, $y$y becomes a smaller and smaller fraction. So again we have $y=0$y=0 as a horizontal asymptote, however this time the graph approaches this line as $x$x gets larger. We can write this asymptotic behaviour mathematically as follows: As $x\ \rightarrow\infty,y\ \rightarrow0^+$x →∞,y →0+ (As $x$x approaches infinity, $y$y approaches zero from above).

And as before:

• Domain: $x$x is any real number.
• Range: $y>0$y>0

All graphs of the form $y=a^x$y=ax where $00<a<1 will have these similar key features. They will all be exponential decreasing (decaying) functions, since our multiplier is a fraction.

How did the graph and table of $y=\left(\frac{1}{2}\right)^x$y=(12​)x compare that of $y=2^x$y=2x? Can you see they are a reflection of each other in the $y$y-axis? The values in the tables were reversed and the $y$y-value for $y=\left(\frac{1}{2}\right)^x$y=(12​)x at $x=k$x=k was the same as $y=2^x$y=2x at $x=-k$x=−k. We can see why this is the case by using our index laws to rewrite $y=\left(\frac{1}{2}\right)^x$y=(12​)x as follows:

Let $g(x)=\left(\frac{1}{2}\right)^x$g(x)=(12​)x and $f(x)=2^x$f(x)=2x. 

$g(x)$g(x)	$=$=	$\left(\frac{1}{2}\right)^x$(12​)x
	$=$=	$\left(2^{-1}\right)^x$(2−1)x
	$=$=	$\left(2\right)^{-x}$(2)−x
	$=$=	$f\left(-x\right)$f(−x)

In general, for $a>0$a>0 the graph of $g\left(x\right)=\left(\frac{1}{a}\right)^x$g(x)=(1a​)x is equivalent to $g\left(x\right)=a^{-x}$g(x)=a−x, which is a decreasing exponential function and a reflection of the graph $f(x)=a^x$f(x)=ax in the $y$y-axis. 

 

Practice questions

Question 3

Question 4

Question 5