4.01 Exponential functions

A base form of an exponential function is $$f(x)=ax, where $$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, where we have exponential growth and identify key characteristics of such functions.

 

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

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

$$x	$$−4	$$−3	$$−2	$$−1	$$0	$$1	$$2	$$3	$$4
$$y	$$116​	$$18​	$$14​	$$12​	$$1	$$2	$$4	$$8	$$16

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

Key features:

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

How does this compare to other values of $$a? Let's graph $$y=2x, $$y=3x and $$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=2x.

Key features:

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

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

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

 

Practice questions

question 1

question 2

 

Graphs of $$y=ax for $$0<a<1

Let's create a table for the function $$y=(12​)x:

$$x	$$−4	$$−3	$$−2	$$−1	$$0	$$1	$$2	$$3	$$4
$$y	$$16	$$8	$$4	$$2	$$1	$$12​	$$14​	$$18​	$$116​

Again we can see our familiar powers of two but this time as $$x increases by one the $$y values are decreasing by a constant multiplier - here they are halving. The differences between successive $$y values is shrinking and hence, $$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 $$x-values increase, the $$y-values decrease at decreasing rate.
• The $$y-intercept is still $$(0,1), since when $$x=0, $$y=(12​)0=1.
• As $$x becomes a larger and larger positive number, $$y becomes a smaller and smaller fraction. So again we have $$y=0 as a horizontal asymptote, however this time the graph approaches this line as $$x gets larger. We can write this asymptotic behaviour mathematically as follows: As $$x →∞,y →0+ (As $$x approaches infinity, $$y approaches zero from above).

And as before:

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

All graphs of the form $$y=ax where $$0<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 $$y=(12​)x compare that of $$y=2x? Can you see they are a reflection of each other in the $$y-axis? The values in the tables were reversed and the $$y-value for $$y=(12​)x at $$x=k was the same as $$y=2x at $$x=−k. We can see why this is the case by using our index laws to rewrite $$y=(12​)x as follows:

Let $$g(x)=(12​)x and $$f(x)=2x. 

$$g(x)	$$=	$$(12​)x
	$$=	$$(2−1)x
	$$=	$$(2)−x
	$$=	$$f(−x)

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

 

Practice questions

Question 3

Question 4

Question 5