Exponential functions are functions of the form f(x) = a^x or y = a^x, where the base number "a" is constant (a number) and the exponent is the variable x.

A REAL EXAMPLE

Some types of bacteria reproduce by "mitosis", cell division whereby one cell divides into two after a short space of time, in some cases every 15 minutes. How many bacteria would be produced in this way from one cell after one day?

x represents the 15 minute intervals:...2⁴ = 16 after one hour, 2⁸ = 256 after two hours,... 2^24·4 = 2⁹⁶ = 7,9·10²⁸. after a day! This example gives us a clear idea of what exponential growth is! This term is used to describe something which grows very quickly.

2.- Look at how the value of "y" changes when we change the value of x. (Change the values in the corresponding box at the bottom of the window).

3.- Now alter the "a" values. What happens to the blue graph?

4.- What do you notice about the graph when a = 1, a >1, a <1 but a positive value?

5.-What happens when the value is negative?

These observations allow us to come to some preliminary conclusions about exponential functions: 

6.- Note that a > 0 if we want the function to exist and to be able to draw its graph. Do you know why this is the case? Think about what would happen if a = -2. What would the value of (-2)^1/2 be? A similar thing would happen for other values of x as the function wouldn't exist. Note that when a = 0 then we are dealing with the function 0, which is of little interest.

7.- Note that when a > 1 the graph of the function is very different from when a < 1, and that when a = 1 the graph is a straight line.

From now on let us suppose that a > 0 and that a # 1 .In the following window we are going to focus on the properties or characteristics of exponential functions.

 The 1^st property:

We can say that the function always exists or that the DOMAIN of the function is all of R (real numbers).

2.- Note that the graph of the function always goes through the fixed point (0,1) (just make x = 0). Therefore:

The 2^nd property: IT CUTS THE Y-AXIS at the point (0,1).

3.- Note that the y values are always positive (try as many x values as you want)

therefore:

The 3^rd property: THE FUNCTION ALWAYS HAS A POSITIVE VALUE for any value of x.

4.- Note that the graph always increases or decreases (for any value of x), depending on the values of the base number "a".

The 4^th property: the graph of the function increases when a>1 and decreases when 0<a<1

5.-Note that the graph approaches the X-axis without actually cutting the axis at any point. It moves towards the right when a<1 and towards the left when a>1.

The 5th property:  THE X-AXIS IS A HORIZONTAL ASYMPTOTE (Going towards the left when a>1 and towards the right when a<1)

Finally, the following window shows the graphs of the two types of exponential function, as well as the graph of the constant function when the base number is 1; for  the values a =2, a = 1 and a = 1/2.

In addition, the red graph shows the exponential function whose base number is 3, which can be altered to whichever value you wish.