Download Exponential Functions Objectives Exponential Function

Objectives
Exponential Functions
Exponential Function
– Evaluate exponential functions.
– Graph exponential functions.
– Evaluate functions with base e.
– Use compound interest formulas.
Definition of Exponential Function
• The exponential function is very important
in math because it is used to model many
real life situations.
• For example: population growth and
decay, compound interest, economics,
and much more.
Graphing Exponential Functions
• To graph an exponential function, follow the
• How is this different from functions that we
worked with previously?
• Some DID have exponents, but NOW, the
variable is found in the exponent.
– (Example:
exponential function)
is NOT an
Graph of an Exponential Function
• Graph
steps listed:
1. Compute some function values and list
the results in a table.
2. Plot the points and connect them with a
smooth curve. Be sure to plot enough
points to determine how steeply the
curve rises.
1
Graph of
Graph ____________
• Subtract 3 from x-values
(move 3 units left)
• Subtract 4 from y-values
(move 4 units down)
Note: Point (0,1) has now
been moved to (-3,-3)
Example
Graph the exponential function y = f(x) = 3x.
x
y = f(x) = 3x
(x, y)
0
1
(0, 1)
1
3
(1, 3)
2
9
(2, 9)
3
27
(3, 27)
−1
1/3
(−1,
1/3)
(−2,
1/9)
(−3,
1/27)
−2
1/9
−3
1/27
Example
Graph the exponential function
x
0
−1
−2
−3
1
2
3
1
3
9
27
1/3
1/9
1/27
(x, y)
(0, 1)
(−1, 3)
(−2, 9)
(−3, 27)
(1, 1/3)
(2, 1/9)
(3, 1/27)
Graph y = 3x + 2.
Example
The graph is the graph of y = 3x shifted _____ 2 units.
x
y
−3
1/3
−2
1
−1
3
0
9
1
27
2
81
3
243
Characteristics of _________
• The domain consists of all real numbers: (- ∞, ∞)
• The range consists of all positive real numbers: (0, ∞)
• The y-intercept is the point (0,1) (a non-zero base
raised to a zero exponent = 1).
• If the base b is greater than 1, the graph extends up
as you go right of zero, and gets very close to zero
as you go left. (Is an increasing function)
• If the base b lies between 0 & 1, the graph extends
up as you go left of zero, and gets very close to zero
as you go right. (Is a decreasing function)
2
The Number e
Other Characteristics of _________
• This function is one-to-one and has an inverse
that is a function.
• The graph approaches, but does not touch, the xaxis. The x-axis (y=0) is a horizontal asymptote.
• Transformations of the exponential function are
treated as transformation of polynomials. (Follow
order of operations, x’s do the opposite of what
you think.)
• e is known as the natural base
(Most important base for exponential
functions.)
• e is an irrational number
(can’t write its exact value)
• We approximate e
Natural Exponential Function
Graphs of Exponential Functions, Base
Graph f(x) =
ex
x
f(x)
−2
0.135
Remember
−1
0.368
• e is a number
0
1
• e lies between 2 and 3
1
2.718
2
7.389
Graph f(x) =
e x + 2.
x
f(x)
−4
0.135
−3
0.368
−2
1
−1
Example
Graph
e
Example
f (x) = 2 − e −3x.
x
f(x)
−2
−401.43
−1
−18.09
2.718
0
1
0
7.389
1
1.95
1
20.086
2
1.99
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Applications of Exponential Functions
• Exponential growth (compound interest!)
Example
• A father sets up a savings account for his daughter.
He puts $1000 in an account that is compounded
quarterly at an annual interest rate of 8%.
How much money will be in the account at the end
of 10 years? (Assume no other deposits were made
after the original one.)
• Exponential decay (decomposition of radioactive
substances)
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