Exponential Functions

An exponential function is a function where the variable is an exponent. Examples: f(x) = 3 x g(x) = 5000(1.02) x h(x) = (¾) x+2

You can evaluate exponential functions just like any other kind of function. If f(x) = 2 x, find  f(1)  f(2)  f(5)  f(-3)

You can evaluate exponential functions just like any other kind of function. If f(x) = 2 x, find  f(1)2 1 =1  f(2)2 2 =4  f(5)2 5 =32  f(-3)2 -3 = 1 / 8 or.125

If g(x) = 2  3 x, find g(4)

If g(x) = 2  3 x, find g(4) Remember the order of operations: 2  3 4 = 2  81 = 162

The graphs of exponential functions are all similar.  y = b x will always contain the points (0,1) and (1,b)  If the base is positive. the graph will always rise rapidly to the right and level off at the left

The line where it levels off (usually the x-axis) is called an asymptote.

Sketch a graph of f(x) = 5 x

Sketch a graph of f(x) = 5 x You know this will contain the points (0,1) and (1,5).

Sketch a graph of f(x) = 5 x You know this will contain the points (0,1) and (1,5). It will also contain (2,25).

Sketch a graph of f(x) = 5 x You know this will contain the points (0,1) and (1,5). It will also contain (2,25). It will level off on the left and rise on the right.

Sketch a graph of f(x) = 5 x

If the base is a fraction, the graph is reversed.  Falls from left to right  Asymptote is on the right Note this still contains (0,1) and (1,½)

The most common use for exponential functions is in problems that involve things that grow (or decay) over time. These often involve population or money.

You put $1000 in an account that earns 4% interest, compounded annually. If you leave it in there, how much will the account be worth after 30 years?

You can solve this using the function y = 1000  1.04 x

You can solve this using the function y = 1000  1.04 x In 30 years, the value would be1000  = $

Remember the problem where someone gave you 1¢ on the 1 st, 2¢ on the 2 nd, 4¢ on the 3 rd, 8¢ on the 4 th, etc. The question is how much would they give you on the 31 st of the month.

You can do this problem with the function f(x) = 2 x – 1 We need to find f(31)

You can do this problem with the function f(x) = 2 x – 1 We need to find f(31) f(31) = 2 30 = 1,073,741,824¢ or $10,737,418.24

A town has 987 people. Suppose it loses 1% of its people every year. How many people will it have 10 years from now?

A town has 987 people. Suppose it loses 1% of its people every year. How many people will it have 10 years from now? P(x) = 987 .99 x

A town has 987 people. Suppose it loses 1% of its people every year. How many people will it have 10 years from now? P(x) = 987 .99 x P(10) = 987   893