ExponentialFunctions

An exponential function is a function where the variable is an exponent.

Examples: f(x) = 3x

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) = 2x, find

f(1) f(2) f(5) f(-3)

You can evaluate exponential functions just like any other kind of function.

If f(x) = 2x, find

f(1) 21 = 1 f(2) 22 = 4 f(5) 25 = 32 f(-3) 2-3 = 1/8 or .125

If g(x) = 23x, find g(4)

If g(x) = 23x, find g(4)

Remember the order of operations:

234 = 281 = 162

The graphs of exponential functions are all similar. y = bx will always contain the

points (0,1) and (1,b) If the base is positive.

the graph will alwaysrise rapidly to theright and level offat the left

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

Sketch a graph of f(x) = 5x

Sketch a graph of f(x) = 5x

You know this will contain the points (0,1) and (1,5).

Sketch a graph of f(x) = 5x

You know this will contain the points (0,1) and (1,5).

It will also contain (2,25).

Sketch a graph of f(x) = 5x

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) = 5x

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 = 10001.04x

You can solve this using the function y = 10001.04x

In 30 years, the value would be 1000 1.0430 = $3243.40

Remember the problem where someone gave you 1¢ on the 1st, 2¢ on the 2nd, 4¢ on the 3rd, 8¢ on the 4th, etc.

The question is how much would they give you on the 31st of the month.

You can do this problem with the function f(x) = 2x – 1

We need to find f(31)

You can do this problem with the function f(x) = 2x – 1

We need to find f(31)

f(31) = 230 = 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) = 987.99x

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) = 987.99x

P(10) = 987.9910 893