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8.1 exponential growth
1.
2. What are Exponential Functions?
 Exponential functions – functions that
include the expression bx where b is a
positive # other than 1.
ď‚— b is called the base.
3. What’s the Shape?
 Let’s make a table to find the general shape.
ď‚— If we use f(x) = 2x as an example:
x f(x) = 2x
-3
-2
-1
0
1
2
3
4. Asymptotes
ď‚— An asymptote is a line that a graph
approaches (but does not touch) as you
move away from the origin.
ď‚— For example:
ď‚— Our graph has a
horizontal asymptote
at y = 0.
5. Graphing y = abx
ď‚— If a > 0 and b > 1, y = abx is an
exponential growth function.
ď‚— For all y = abx , b > 1:
ď‚— Graphs pass through (0, a) (a is the y-int)
ď‚— x-axis is an asymptote
ď‚— Domain: all real #s
ď‚— Range: y > 0 if a > 0
y < 0 if a < 0
6. To graph:
ď‚— Plot 2 points: (0, a) and (1, __)
ď‚— Plug in 1 for x to fill the blank
ď‚— Connect with a smooth curve that:
ď‚— Starts left of the origin, close to the x-axis
ď‚— Moves up or down quickly to the right
9. General Exponential Functions
ď‚— General form:
ď‚— As usual:
ď‚— h is horizontal shift
ď‚— k is vertical shift
ď‚— To graph:
 Sketch the “parent graph” y = abx
ď‚— Shift using h and k
12. Exponential Growth Models
ď‚— When 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).
ď‚— (1 + r) is the growth factor.
13. Example:
ď‚— 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.
ď‚— Write a model giving the number h (in millions)
of hosts t years after 1993.
ď‚— How many hosts were there in 1996?
14. Compound Interest
ď‚— Compound interest is interest paid on the
original principal and on previously earned
interest.
ď‚— Modeled by an exponential function.
ď‚— If interest is compounded n times per
year, the amount A in the account after t
years is:
where P is the initial principal and r is the
annual interest rate.
15. Example:
ď‚— You deposit $1000 in an account that pays
8% annual interest. Find the balance after 1
year if interest is compounded:
ď‚— A. annually
ď‚— B. quarterly
ď‚— C. daily
ď‚— Which is the best investment?