EXPONENTIAL FUNCTIONS_2
Section: Calculus
 
1. SPECIAL CASES
1. 1. EXPONENTIAL FUNCTIONS WHEN a >1 AND WHEN a=e

In the following window the graphs of the functions 2x and 3x are shown in blue and the graph of the function y = ex  in green. You may already know the number "e". For those of you who don't, it is an irrational number, which therefore has an infinite number of non-periodic decimal places, and its value is 2.718281.... up to the first six decimal places.

The exponential function whose base number is e is of special interest. You will go into more detail about this when you focus on limits of functions and logarithms.

It can be written as y = ex and y=exp(x), as it is the most commonly used exponential function.

1.- Look at the properties of the function with the help of this window:

- the graph of the function always increases.

- the X-axis is an asymptote going towards the left, whilst the function tends to infinity towards the right.
1.2. THE EXPONENTIAL FUNCTION WHEN a <1 

These kinds of exponential functions are of little interest and you will not see many examples of them in the future.

The following window first shows the exponential function whose base number is 1/2 = 0.5.
1.- Look at the properties of the function with the help of this window:

- All of them always decrease (remember that a>0)

- The X-axis is a horizontal asymptote towards the right, whilst the function tends to infinity when x is very small.

2.- Give "a" other positive values which are less than 1.

1.3. THE EXPONENTIAL FUNCTION WITH DIFFERENT EXPONENTS

In some exponential functions the exponent is not x but -x, 2x, x+2, x-1, etc.
1.- Look at the graph of e-x (exp(-x)) . Do you recognise it? Of course you do! If we are aware that e-x is the same as 1/ex=(1/e)x, and that 1/e is less than 1 then we have an exponential function whose base number is less than 1, as we saw earlier. Obviously, the same thing would happen with other base numbers which aren't "e". 

2.- Now look at the graphs of the exponential functions whose exponent is of the form x+1, x+2, x-1, etc.

1.4. THE EXPONENTIAL FUNCTION ex

The following window shows ex (in red), and compares it with ex+1 .
1.- Note which of the numbered properties listed earlier are still true and which change. Write them down in your exercise book.

2.- Change the function ex+1, written in blue, for whichever one you want. Delete the present function and write the new one into the box on the left. You can change the exponent to x+2, x-1, 2x, etc and watch how the graph changes, by always comparing it with ex.

2.  EXERCISES

Vary the function exp(x) in the following window and use it to help you answer the following questions.

1.- What is the difference between the exponential functions whose exponent is "x" and those whose exponent is "-x" ?
To return to the initial diagram reload the whole page with the browser.

2.- How does the exponential function change when the exponent "x" changes to "x+1", "x+2", "x-1", "x-3" and in general "x ± c"? What happens when it changes to 2x, 3x, etc ?

3.- Which function obtained from the exponential functions above would not have the X-axis as a horizontal asymptote?

4.- Which functions obtained from the exponential function above would cut the Y-axis at the point 0, 1, 2, -1, -2  etc?

     
         
  Leoncio Santos Cuervo
 
Spanish Ministry of Education, Social Afairs and Sport. Year 2001
 
 

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