When something increases or decreases by the same percentage in each period, you have exponential (percentage) growth. The exponential growth may be negative, meaning the graph decreases to the right rather than ascending upwards as it otherwise would.
In exponential functions, you can either use an arbitrary number as the base or as the base.
Theory
Exponential functions can have as the base or an arbitrary number as the base. In both cases, is a constant. They look like this:
Note! These functions are reformulations of each other, so they have identical graphs .
Note that the variable is now in the exponent! The symbols , and are numbers.
When the value of in the function is positive, the graph looks like one of the two graphs below.
Rule
is the value when , is the growth factor.
blue graph, red graph.
Note! You are expected to be able to convert from and from .
Rule
The following is an overview of how the function behaves for different values of and .
and :
The graph runs along the -axis and rises sharply to the right.
and :
The graph decreases sharply to the right and flattens along the -axis.
and :
The graph is a horizontal line through .
and :
The graph runs along the -axis and decreases sharply below the -axis.
and :
The graph rises sharply and flattens along the -axis.
In general, gives you a fixed percentage increase, gives you a fixed percentage reduction, and gives you no change.
The number acts as a growth factor. The value of affects the sign of the functional values.
Example 1
You have the function . This intersects the -axis at and grows exponentially. This form of growth is very powerful. References to this type of graph are also often used in everyday speech, when someone says something is experiencing “exponential” growth.
Example 2
You have the function . This intersects the -axis at and decreases exponentially. This form of reduction is also very powerful. It’s like the downward version of the previous example.
Example 3
In a particular chemical reaction, the concentration of a substance is given by
where is the time measured in seconds, and is measured in mmol/L.
You are given the following tasks:
Now you need to find out how long it takes before the concentration is mmol/L. Here you have to solve the equation . You set it up and get
You subtract on both sides, and get
Then you divide by , and the equation becomes
You now have an equation in the form . You apply on both sides, and you get
Finally, you divide by and find the solution
This means that it takes 134 seconds before the concentration is mmol/L
To find out which value the concentration is approaching if the reaction lasts for a very long time, you can either look at the graph and see that it goes towards , or insert a properly large and get . This means that if the reaction takes a very long time, the concentration approaches mmol/L Another way you can say this is to say that is the horizontal asymptote.
Using this rule, you get
Thus the function of the reaction rate is
You want to find the reaction rate when it is mmol/L. As you saw in Exercise 2, that it happens for , so you insert 134 and get