Transcript 3-1

Chapter 3
Exponential and
Logarithmic Functions
3.1 Exponential Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1
Objectives:
•
•
•
•
Evaluate exponential functions.
Graph exponential functions.
Evaluate functions with base e.
Use compound interest formulas.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
2
What is an exponential function?
What does an exponential function look like?
Obviously, it must have something to do with an
exponent!
Dependent
Variable
Just some
number
that’s not 0
Base
Exponent
and
Independent
Variable
Why not 0?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
3
The Basis of Bases
The base carries the meaning of the
function.
1) determines exponential growth
or decay.
2)base is a positive number;
however, it cannot be 1.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
4
Let’s examine exponential functions. They are
different than any of the other types of functions we’ve
studied because the independent variable is in the
exponent.
x
3
2
1
0
-1
-2
-3
2x
8
4
2
1
1/2
1/4
1/8
f x   2
Let’s look at the graph of
this function by plotting
x some points.
8
7
6
5
4
3
2
1
BASE
Recall what a
negative exponent
means:
1
1
f  1  2 
2
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-2
-3
-4
-5
-6
-7
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
5
Using graphing calculator plot (graph) :
y2
y3
x
f x   4 x
x
y4
x
f x   3x
Compare graphs in your groups:
1) Similarities
f x   2 x
2) Differences
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
6
Similarity of Graphs of
Exponential Function f x   a x
where a > 1
f x   4 x
1. Domain is all real numbers
f x   3x
2. Range is positive real numbers
3. There are no x intercepts because
there is no x value that you can put
in the function to make it = 0
4. The y intercept is always (0,1)
because a 0 = 1
5. The graph is always increasing
f x   2 x
Are these
What
the
What
isisisthe
What
Can
you
seexrange
the
exponential
What
is
the
What
is
the
y
domain
of an
of
an exponential
different?
horizontal
functions
intercept
of
intercept
ofthese
these
exponential
function?
asymptote
for
increasing
or
exponential
exponential
Steepness
function?
these
functions?
decreasing?
functions?
functions?
6. A horizontal asymptote y = 0 (x-axis)
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
7
Graphing Exponential functions by
Transformations
What do you remember about Transformations of any
functions:
---- shifts, reflections, stretching, compressing
Coefficients/constants “a”, “h” and “k”
On your graphing calculators :
y2
x
y  2 3
x
y  2x  4
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
8
Transformations
Shifts the graph up if
k>0
Shifts the graph down
if k < 0
Vertical translation
f(x) = bx + k
y  2x
y  2x  3
y  2x  4
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
9
On your graphing calculators :
y2
y  2x
x
y  2 x 3
y  2 x4
y  2( x 3)
Shifts to the left if h > 0
Shifts to the right if h < 0
y  2( x 4)
Horizontal translation: f(x)=bx-h
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
10
On your graphing calculators :
y2
y  2x
x
y  1  2 x
y  21 x
y  2 x
y  2 x
Reflecting
f ( x)  1 b xreflects over the x-axis.
1 x
f ( x)  b reflects over the y-axis.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
11
On your graphing calculators :
y  2x
y  2x
y  42
1 x
y  2
4
x
y  4(2 x )
Stretches the graph if a > 1
Shrinks the graph if 0 < a < 1
y
1 x
(2 )
4
Vertical stretching or shrinking, f(x)=abx:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
12
Foldable: Graphing Exponential Functions
13
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
13
Let’s take a second look at the base of
an exponential function.
(It can be helpful to think about the base as the object
that is being multiplied by itself repeatedly.)
Why can’t the base be negative?
Why can’t the base be zero?
Why can’t the base be one?
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
14
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
15
Example: Evaluating an Exponential Function
The exponential function f ( x)  42.2(1.56) x models the
average amount spent, f(x), in dollars, at a shopping
mall after x hours. What is the average amount spent, to
the nearest dollar, after three hours at a shopping mall?
We substitute 3 for x and evaluate the function.
f ( x)  42.2(1.56) x
f (3)  42.2(1.56)3  160.20876  160
After 3 hours at a shopping mall, the average amount
spent is $160.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
16
Example: Graphing an Exponential Function
x
f
(
x
)

3
Graph:
x
–2
–1
One of the ways to graph it:
“T” chart - plot points
f ( x )  3x
We set up a table of coordinates,
1
2
then plot these points, connecting
f (2)  3 
9 them with a smooth, continuous
1 curve.
1
f (1)  3 
3
0
f (0)  30  1
1
f (1)  31  3
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
17
Characteristics of Exponential Functions of the Form
f ( x)  b x
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
18
Example: Transformations Involving Exponential
Functions
x 1
x
g
(
x
)

3
f
(
x
)

3
Use the graph of
to obtain the graph of
x
f
(
x
)

3
Begin with
 1, 1 


 3
We’ve identified three
points and the asymptote.
(1,3)
(0,1)
Horizontal
asymptote
y=0
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
19
Example: Transformations Involving Exponential
Functions (continued)
x 1
x
g
(
x
)

3
f
(
x
)

3
Use the graph of
to obtain the graph of
(1,3)
The graph will shift
1 unit to the right.
Add 1 to each
x-coordinate.
(0,1)
 1, 1 


 3
 0, 1 


 3
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
(2,3)
(1,1)
Horizontal
asymptote
y=0
20
The Natural Base e
The number e is defined as the value that 1  1 


 n
n
approaches as n gets larger and larger. As n  
the approximate value of e to nine decimal places is
e  2.718281827
The irrational number, e, approximately 2.72, is called
the natural base. The function f ( x)  e x is called the
natural exponential function.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
21
Example: Evaluating Functions with Base e
The exponential function f ( x)  1066e0.042 x models the
gray wolf population of the Western Great Lakes, f(x), x
years after 1978. Project the gray wolf’s population in
the recovery area in 2012.
Because 2012 is 34 years after 1978, we substitute 34
for x in the given function.
f ( x)  1066e0.042 x
f (34)  1066e0.042(34)  4446
This indicates that the gray wolf population in the
Western Great Lakes in the year 2012 is projected to
be approximately 4446.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
22
Formulas for Compound Interest
After t years, the balance, A, in an account with
principal P and annual interest rate r (in decimal form)
is given by the following formulas:
1. For n compounding periods per year:
nt
r

A  P 1  
 n
2. For continuous compounding:
A  Pert
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
23
Example: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%.
Find the balance in the account after 5 years subject to
quarterly compounding.
We will use the formula for n compounding periods per
year, with n = 4.
nt
4 5
r
0.08 

A  P  1  
A  10,000 1 
 14,859.47

 n
4 

The balance of the account after 5 years subject to
quarterly compounding will be $14,859.47.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
24
Example: Using Compound Interest Formulas
A sum of $10,000 is invested at an annual rate of 8%.
Find the balance in the account after 5 years subject to
continuous compounding.
We will use the formula for continuous compounding.
A  Pe
rt
A  10,000e
0.08(5)
 14,918.25
The balance in the account after 5 years subject to
continuous compounding will be $14,918.25.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
25