8.1 exponential growth
TRANSCRIPT
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
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
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.