Trigonometric Exponential Functions
Written by tutor Danielle R.
Exponential functions are typically used to model natural phenomena that increase or decrease at an exponential rate. For instance bacteria and many other populations can grow at an exponential rate. The amount of radioactive substances will normally decrease at an exponential rate. The way we calculate interest is an exponential function. Frequently the value varies over time so you will see exponential functions written with the independent variable as t rather than x.
At its most basic an exponential function has the form f(x)=abx. For the time being let’s assume b must be greater than one. This means an exponential function looks like this, if a is positive.
For this particular example the function is f(x)=2x. It is important to notice that this graph includes the point (0,1). In fact all exponential functions of this basic form will include the point (0,a). For any value of b≠0 it is true that a=a·b0. f(x)=2x can be written as f(x)=1·2x, so it includes the point (0,1). Because the independent value is often going to be t in an exponential function; this value, a, corresponds to the value where t=0. So we call it the initial value. Another easy to locate point is the value (1,a·b). This is because a·b=a·b1. The value b indicates the rate at which the value is growing. Because b is greater than 1 the function above represents exponential growth. If b is allowed to vary between 0 and 1 the function represents exponential decay. Exponential decay looks like this:
This function is f(x) = (1/2)x. You might notice that this is also the function of f(x) = 2-x. This means you can generate this function as a graph transformation of the original most "basic" function f(x) = 2x. Where you consider f(-x) = 2-x as a reflection about the y-axis.
Similarly, you might look at the graph transformation of f(x) = 2x into -f(x) = -2x, which is a reflection about the x-axis.
And then last there would be the transformation of f(x) = 2x into -f(-x) = -2-x. Here the graph transformation is a reflection across the y-axis and then a reflection about the x-axis. This would look like the following:
The advantage to thinking about them as transformations is that you only need to remember the graph of the most basic exponential function and you can find the others as graph transformations rather than memorizing them all.
The typical questions that would be asked about exponential functions are usually about finding the function itself. For instance: If I know a bacteria population has a hundred members to start and an hour later has 400, what is the exponential function that would model its growth?
For this problem you would start with the basic function:
f(t) = abt
Now to find b I also know that the function includes the point (1,400) so:
400 = 100b1
400/100 = b
4 = b
So the final function to model the growth of the bacteria is:
f(t) = 100(400t)
This means I can predict what the population will be in, say, three hours.
f(3) = 100(4003) = 6400000000 bacteria.
And another slightly more difficult problem:
Suppose I know the population of rabbits on an island is 50 to start with and 2 years later is 300.
f(t) = abt
I know the initial value is 50, so a=50.
f(t) = 50bt
And I also know that the point (2,300) is on the graph.
b = ±√6
Exponential functions don’t have negative base values so: