Which function does not represent exponential decay? y=4 (0.5) ^x y=0.4 (0.5) ^x y=5 (4) ^x y=5 (0.4) ^x

Exponential functions are closely related to geometric sequences. A geometric sequence of numbers is one in which each successive number of the sequence is obtained by multiplying the previous number by a fixed factor m. An example is the sequence {1, 3, 9, 27, 81, ...}. If we label the numbers in the sequence as {y0, y1, y2, ...} then their values are given by the formula: yn = y0 · m n. The geometric sequence is completely described by giving its starting value y0 and the multiplication factor m. For the above example y0 = 1 and m = 3. Another example is the geometric sequence {40, 20, 10, 5, 2.5, ...} for which y0 = 40 and m = 0.5.

The exponential function is simply the generalization of the geometric sequence in which the counting integer n is replaced by the real variable x. We define an exponential function to be any function of the form: y = y0 · m x. It gets its name from the fact that the variable x is in the exponent. The " starting value " y0 may be any real constant but the base m must be a positive real constant to avoid taking roots of negative numbers.

The exponential function y = y0 · m x has these two properties: When x = 0 then y = y0.

When x is increased by 1 then y is multiplied by a factor of m. This is true for any real value of x, not just integer values of x. To prove this suppose that y has some value ya when x has some value xa. That is:Now increase x from xa to xa+1. We get:

We see that y is now m times its previous value of ya. If the multiplication factor m > 1 then we say that y grows exponentially, and if m < 1 then we say that y decays exponentially.