SECTION 1.5: Exponential Functions

If the percentage increase or decrease in a quantity is a constant, then the function is defined as an exponential function. Population growth in certain areas is an example often used for demonstrating exponential growth.

Exponential functions are defined as P = P0 (a^t) where
P represents the vertical axis of a graph
P0 represents the initial quantity (when t=0), and
a represents the the factor by which the function increases or decreases. It is called a base.
If a>1, there is said to be exponential growth, if a<1, there is said to be exponential decay. Larger values of a imply faster growth. and values of a near 0 im
ply faster decay.
It can also be written as a = 1+r, where r is the decimal representation of the percent rate of change. It is positive for exponential growth and negative for exponential decay.

For an exponential decay to occur, as t increases the functions values get closer and closer to zero. Therefore the t-axis represents the horizontal asymptote of the function.

It is also important to note that graphs of exponential functions are always concave up.

To differentiate between an exponential function and a linear function, it is important to understand that the former refers to a constant relative rate of change while the later refers to an absolute value for the rate of change.

In everyday situations the most common base number is defined as e= 2.71828... This is often called the natural base.

When reading this section, the question that arose in my mind was whether or not the base a could be a negative number, and if not then why?

This material was relevant to what had been studied in the earlier section, especially in discussing why certain data when extrapolated beyond certain intervals could still be considered functions and yet not change at a constant absolute rate. The section helped clearly identify the differences between linear and exponential functions.