Exponential function

Proportionality of exponential graphs

• For , the gradient of the graph at any point is directly proportional to the value of at that point.

Graph of

• The graph of passes through the point .
• The graph of increases faster than any other exponential graph where .
• The graph of decreases slower than any other exponential graph where .
• The derivative of is equal to itself:

Any exponential function can be expressed in terms of :

Asymptotes

For the function :

• There is a horizontal asymptote at (the x-axis).

Approaching infinity

• For :
  • As , .
  • As , .
• The rate of increase of becomes faster as increases.
• The rate of decrease of becomes slower as decreases.
• The function never actually reaches ; it only approaches it as an asymptote.

Graph of an inverse function

• The inverse of a graph is the reflection of the graph in the line .