Math Revision 15.5 - Exponential Graphs

In these revision notes for Exponential Graphs, we cover the following key points:

• What are exponential functions?
• What are the names of the elements in the formula of an exponential function?
• What is the simplest form of an exponential function? How do we call it?
• How to plot the graph of an exponential function?
• What is the asymptote of an exponential function graph?
• How does the sign of coefficients in an exponential function affect the orientation and size of an exponential curve?
• How to find the formula of an exponential function from its graph?
• How to deal with exponential functions with a negative base?
• How to deal with exponential functions that have Euler's number as a base?
• What are some applications in the practice of exponential functions?

Exponential Graphs Revision Notes

Exponential functions are those functions that have the independent variable written as an index (exponent). The simplest form of exponential functions is

where a is a known number called the base.

The general form of an exponential function is

where the first function y(x) = ax is known as the parent function of the latter one.

It is clear that in the simplest form of exponential functions y = ax we have k = 1, m = 1 and t = 0.

There are some limitations in the possible values an exponential function (i.e. y-values) can take. Exponential functions always take the sign of the base a for every value of the independent variable.

The Y-axis acts as a horizontal asymptote for the parent function y(x) = ax. In the general form y = k · amx + t however, this asymptote is at y = t. Exponential functions have no vertical asymptotes, as there are no limitations in the values of x in such functions.

We can plot the graph of an exponential function by giving some values to the variable x and finding the corresponding y-values. In this way, we obtain a number of points on the graph, which when connected smoothly give the graph's shape. In exponential functions, it is better to take an odd number of values that have a certain symmetry in distribution. We said odd because we also take into consideration x = 0.

The following rules apply in exponential graphs:

1. For a > 1, when the base a increases, the graph becomes 'narrower'.
2. For a < 1, the graph flips horizontally in respect to the corresponding exponential graph with a reciprocal base.

As for the other coefficients, we have the following rules:

The greater the coefficient k, the faster the increase in the values of y. This brings a narrower graph compared to the corresponding parent graph. On the other hand, every decrease in the value of the coefficient k results in a wider graph compared to the parent one.

The same thing is true for the other coefficient m as well. Thus, when m increases, the graph becomes narrower as the y-values increase faster than before and when m decreases the graph becomes wider, as the y-values increase slower than before.

We can find the formula of an exponential function from its graph. For this, we have to consider some distinct points, as we must find four different values that correspond to the two coefficients k and m, the base a and the constant t.

Any negative sign preceding the base inverts an exponential graph down, as occurs with all the other types of graphs as well.

The most famous exponential functions perhaps are those that have Euler's Number as a base. In such functions, we write the letter 'e' as a base instead of numbers such as 2, 3, 4, etc. The exponential graphs produced by such functions are similar to those explained at the beginning of this tutorial, which have a base greater than 1.

The parent function of exponential functions that include Euler's Number is

and the most frequent exponential function is

The most typical situations involving Euler's number as the base of exponential functions are those that show the number of elements in a sample as a function of time. For example, the number of bacteria left in a sample of a living organism as a function of the time elapsed since the start of the medication, the number of the non-decayed nuclei as a function of time elapsed in a radioactive sample, etc.