8.1 exponential growth

What are Exponential Functions?

Exponential functions – functions that

include the expression bx where b is a

positive # other than 1.

b is called the base.

What’s the Shape?Let’s make a table to find the general shape.

If we use f(x) = 2x as an example:x f(x) = 2x

-3

-2

-1

0

1

2

3

AsymptotesAn asymptote is a line that a graph

approaches (but does not touch) as you

move away from the origin.

For example:

Our graph has a

horizontal asymptote

at y = 0.

Graphing y = abx

If a > 0 and b > 1, y = abx is an

exponential growth function.

For all y = abx , b > 1:

Graphs pass through (0, a) (a is the y-int)

x-axis is an asymptote

Domain: all real #s

Range: y > 0 if a > 0

y < 0 if a < 0

To graph:Plot 2 points: (0, a) and (1, __)

Plug in 1 for x to fill the blank

Connect with a smooth curve that:

Starts left of the origin, close to the x-axis

Moves up or down quickly to the right

ExamplesGraph:

Your Turn!Graph

General Exponential Functions

General form:

As usual:

h is horizontal shift

k is vertical shift

To graph:

Sketch the “parent graph” y = abx

Shift using h and k

ExamplesGraph and state the domain and range:

Your Turn!Graph and state the domain and range:

Exponential Growth ModelsWhen a real-life quantity increases by a

fixed % each year, the amount of the

quantity after t years can be modeled by:

y = a(1 + r)t

where a is the initial amount and r is the %

increase (as a decimal).

(1 + r) is the growth factor.

Example: In January, 1993, there were about 1,313,000

Internet hosts. During the next five years, the

number of hosts increased by about 100% per

year.

Write a model giving the number h (in millions)

of hosts t years after 1993.

How many hosts were there in 1996?

Compound InterestCompound interest is interest paid on the

original principal and on previously earned interest.

Modeled by an exponential function.

If interest is compounded n times per year, the amount A in the account after tyears is:

where P is the initial principal and r is the annual interest rate.

Example:You deposit $1000 in an account that pays

8% annual interest. Find the balance after 1

year if interest is compounded:

A. annually

B. quarterly

C. daily

Which is the best investment?