Download SECTION 10.3 LECTURE NOTES

Lecture Notes, Detailed Comments, and Additional Explorations
253
Lecture Notes, Detailed Comments, and Additional Explorations
253
SECTION 10.3 LECTURE NOTES
Math098 Lecture (10.3) SECTION 10.3 LECTURE NOTES
Objectives
1. Sketch the graph of an exponential function.
Objectives
1.
the graphical
graph of an
exponential
2. Sketch
Know the
significance
offunction.
a and b for a function of the form f (x) = abx .
x
2.
significance
of athe
and
b for a function
of the form
f (x) and
= ab
3. Know
Know the
the graphical
base multiplier
property,
increasing
or decreasing
property,
the. reflection property.
3. Know
base multiplier
property,
the increasing
or decreasing property, and the reflection property. 253
Lecture
Notes,the
Detailed
Comments,
and Additional
Explorations
Main point: Graph exponential functions.
SECTION 10.3 LECTURE NOTES
Main point: Graph exponential functions.
OBJECTIVE
1
x
OBJECTIVE
1
1. Create a table
Objectives of solutions of f (x) = 2 and graph f by hand.
✓ ◆x
1. Create
a table
solutions
f (x) = 2xfunction.
and
1 graph f by hand.
1. Sketch
theof
graph
of an of
exponential
2. Create
a table
of
solutions
of g(x) = 8✓
◆x and graph g by hand.
12
2. Know
the graphical
significance
b forgraph
a function
of the form f (x) = abx .
2. Create
a table
of solutions
of g(x) =of
8 a and and
g by hand.
Use a graphing calculator to verify your graphs2in Problems 1 and 2.
3. Know the base multiplier property, the increasing or decreasing property, and the✓reflection
property.
◆x
Use a graphing calculator to verify your graphs in Problems 1 and 2.
1
OBJECTIVES 2 and 3 Refer to your tables of solutions of f (x) = 2x and g(x) = 8✓ ◆ to motivate the
x
12
x
base Main
multiplier
property.
OBJECTIVES
2
and
3
Refer
to
your
tables
of
solutions
of
f
(x)
=
2
and
g(x)
=
8
to motivate the
point: Graph exponential functions.
2
base multiplier property.
OBJECTIVE 1
Base Multiplier Property
Multiplier
1. Create a table of solutions of f (x) =Base
2x and
graph f Property
by hand.
For an exponential function of the form y = abx , if the value of the independent variable increases by 1, the
✓
◆
x
value of the dependent variable variable is multiplied
b.
1 x theby
For
exponential
function
of the
form=
y=
value gofby
thehand.
independent variable increases by 1, the
2. an
Create
a table of
solutions
of g(x)
8 ab , ifand
graph
2
value of the dependent variable variable is multiplied
by b.
Describe the base multiplier property as applied to the given function.
Use a graphing calculator to verify your graphs in Problems 1 and 2.
3.
given
function.
✓ 2 ◆x
Describe
base
x multiplier property as applied to the
f (x) =the
5(4)
✓ ◆x
4. g(x) = 7✓ ◆
3= x2x and g(x) = 8 1 to motivate the
x 2 and 3 Refer to your tables of solutions of f (x)2
OBJECTIVES
3. f (x) = 5(4)
2
4. g(x) = 7
3
base
multiplier
Use
the base
OBJECTIVE 2property.
multiplier
and 3 property to explain the following connections:
Use
the base
property
to explain
• f (x)
= 2xmultiplier
is an increasing
function
and the
b =following
2 > 1 (seeconnections:
Problem 1).
Base Multiplier Property
✓ ◆x
x
• f (x) = 2 is
1 an increasing function and b = 2 > 1 (see Problem 1).
• g(x)
= 8✓ ◆ is
a decreasing function and xb = <value
1 (see
2).
For
an exponential
of Problem
the independent
variable increases by 1, the
2 x function of the form y = ab , if the
2
1
1
value
of
the
dependent
variable
variable
is
multiplied
by
b.
• g(x) = 8
is a decreasing function and b = < 1 (see Problem 2).
2
2
Increasing or Decreasing Property
x
Describe the base multiplier property as applied
to the given function.
1
fIncreasing
( x) = 6 ⎛⎜ ⎞⎟ or Decreasing Property
x
✓ ◆x
= ab , xwhere
a > 0. Then
⎝ 2 ⎠ Let
3. ff (x)
(x)
Consider 2
= 5(4) the function, 4. g(x) = 7
x
Let f• (x)
3
, where
> 0. Then
If b=>ab
1, then
theafunction
f is increasing. We say the function grows
exponentially.
By the base multiplier property, as the value of x increases by 1, the value of y is multiplied by one half. Values are ••in If
>
then
function
f to
is increasing.
say
function
growsdecays
exponentially.
If
<t1,
b<
1,bthe
then
the
function
f is decreasing.
Wethe
say
the function
exponentially.
Use
the
base
multiplier
property
explain
the We
following
connections:
shown tb0he able elow. If 0 =
< 2bx<is1,anthen
the function
f isand
decreasing.
sayProblem
the function
• •f (x)
increasing
function
b = 2 > We
1 (see
1). decays exponentially.
Draw graphs to illustrate this property.
✓ ◆x
1
1
Draw
graphs
this property.
• g(x)
= 8 to illustrate
is a decreasing
function and b = < 1 (see Problem 2).
2
2
c 2015
Copyright
Pearson Education,
Increasing
or Decreasing
PropertyInc.
c 2015 Pearson Education,
Inc.
Let
f (x)
= ab
x , where
a >
0.
Copyright
Then
• If b > 1, then the function f is increasing. We say the function grows exponentially.
• If 0 < b < 1, then the function f is decreasing. We say the function decays exponentially.
Draw graphs to illustrate this property.
BasexMultiplier Property
For
anan
exponential
function
ofof
thethe
form
y y==abab
,xif, if
thethe
value
ofof
thethe
independent
variable
increases
For
exponential
function
form
value
independent
variable
increasesbyby1,1,thethe
value
of
the
dependent
variable
variable
is
multiplied
by
b.
value
the dependent
variable
variable
multiplied
b. of the independent variable increases by 1, the
For anofexponential
function
of the
form yis=
abx , if thebyvalue
value of the dependent variable variable is multiplied by b.
Describe
thethe
base
multiplier
property
asas
applied
toto
thethe
given
function.
Describe
base
multiplier
property
applied
given
function.
✓✓◆x◆
Describe the xbase multiplier property as applied to the given function.
22 x
x
3. 3.f (x)
=
5(4)
f (x) = 5(4)
4.4.g(x)
g(x)==7 7 ✓
3 32 ◆x
3. f (x) = 5(4)x
4. g(x) = 7
3
Use
the
base
multiplier
property
to
explain
the
following
connections:
Use the base multiplier property to explain the following connections:
the
base
property
to explain
the followingProblem
connections:
x multiplier
• •Use
f (x)
==
2x2is
anan
increasing
function
and
b=
1).1).
f (x)
is
increasing
function
and
b =2 2>>1 (see
1 (see Problem
✓
◆
x x
• f (x) = 2✓
x increasing function and b = 2 >
1 1is◆an
1 11 (see Problem 1).
• •g(x)
=
8
is
a decreasing
function
and
b=
< 1 (see Problem 2).
◆
g(x) = 8 ✓
is
a
decreasing
function
and
b
=
x
2 21
2 21 < 1 (see Problem 2).
• g(x) = 8
is a decreasing function and b = < 1 (see Problem 2).
2
2
Increasing
oror
Decreasing
Property
Increasing
Decreasing
Property
Increasing
or
Decreasing
Property
x x
Let
f (x)
==
abab
, where
a a>>0.0.Then
Let
f (x)
, where
Then
x
Let
=1,abthen
, where
a > 0. fThen
• •fIf(x)
b>
thethe
function
increasing.
say
thethe
function
grows
If
b>
1, then
function fis is
increasing.We
We
say
function
growsexponentially.
exponentially.
•
If
b
>
1,
then
the
function
f
is
increasing.
We
say
the
function
grows
exponentially.
• •If If
0<
b<
1, 1,
then
thethe
function
f fis is
decreasing.
say
thethe
function
0<
b<
then
function
decreasing.We
We
say
functiondecays
decaysexponentially.
exponentially.
• If 0 < b < 1, then the function f is decreasing. We say the function decays exponentially.
Draw
graphs
toto
illustrate
this
property.
Draw
graphs
illustrate
this
property.
254
CHAPTER
LectureNotes,
Notes,Detailed
DetailedComments,
Comments,and
andAdditional
AdditionalExplorations
Explorations 254
CHAPTER
2 2 Lecture
Draw graphs to illustrate this property.
254
CHAPTER 2 Lecture
x Notes, Detailed Comments, and Additional Explorations
x Notes,
Substitute0 0for
forxxininthe
theCHAPTER
generalequation
equation
show
thefollowing
following
property.
254Substitute
2 Lecture
Detailed
Comments,
and Additional Explorations
general
y y==abab
totoshow
the
property.
c 2015
Copyright
Pearson
Education, Inc.
Copyright
2015
Pearson
Inc.
Substitute 0 for x in the general
equationcy of
=
abxExponential
to showEducation,
the following
y-Intercept
an
Function property.
x Exponential Function
y-Intercept
of
an
Substitute 0 for x in the general Copyright
equation y c= 2015
ab toPearson
show the
followingInc.
property.
Education,
Forananexponential
exponentialfunction
functionofofthe
theform
form
y-Intercept
of an Exponential Function
For
y-Intercept of an
Exponential
Function
x x,
y y==abab
,
Fory-intercept
an exponential
function
of the form
the
(0,
a).
the
y-intercept
isis(0,
a).
For
an exponential
function
of the form
y = abx ,
y = abx ,
the the
y-intercept
is (0,
Find
they-intercept
y-intercept
ofa).
thegraph
graphofofthe
thefunction.
function.
Find
of
the
the y-intercept is (0, a).
✓✓ 2◆x◆x
x
x
y==8(5)
8(5)
2
the
y-intercept of the graph of the function.
5.5.yFind
6.6.y y== 4 4 7
Find the y-intercept of the graph of the function.
7✓ ◆x
5. y = 8(5)x
✓
◆2xx
6.
y
=
4
Warning:
x
2 xb is (0, 1), not (0, b).
They-intercept
y-interceptofofthe
thegraph
graphofofa afunction
functionofofthe
the formy y==
5.Warning:
y = 8(5)
The
6. yform
= 4 b 7 is (0, 1), not (0, b).
Warning: The y-intercept of the graph of a function of the form y = b7x is (0, 1), not (0, b).
xbase multiplier property to
Find
they-intercept
y-intercept
thegraph
graph
ofthe
thegiven
function.
Then
use
thebbase
graphthe
the
Warning:
The y-intercept
of
theofgraph
ofgiven
a function
of Then
the
form
ythe
=
is (0,
1), not property
(0, b). to graph
Find
the
ofofthe
function.
use
multiplier
x
Warning:
The y-intercept of the graph of a function of the form y = b is (0, 1), not (0, b).
function
hand.
function
bybyhand.
Find the y-intercept of the graph of the given function. Then use
◆base multiplier property to graph the
✓✓the
xx
1◆base
xx
the=
y-intercept
of the graph of the given function. Then use the
multiplier property to graph the
7.f f(x)
(x)
2(3)
1
function
by=
hand.
7.Find
2(3)
g(x)==1212
8.8.g(x)
function by hand.
3✓3 ◆x
7. f (x) = 2(3)x
✓ ◆1x
8. g(x) = 12
x
x
x
x1x
7.
f
(x)
=
2(3)
Sketchand
andcompare
comparethe
thegraphs
graphsofoff f(x)
(x)== 4(2)
4(2) and
andg(x)
g(x)=
9.9.Sketch
8.
g(x)
==4(2)
124(2). 3.
3
x
9. Sketch and compare the graphs of f (x) = 4(2) and g(x) = 4(2)x .
9. Sketch and compare the graphs of f (x)Reflection
=
4(2)x Property
and
g(x) = 4(2)x .
Reflection
Property
x
x x and g(x) = ab
Thegraphs
graphsofoff f(x)
(x)== abab
arereflections
reflections
eachother
otheracross
acrossthe
thex-axis.
x-axis.
Reflection
Property
The
and g(x) = abx are
ofofeach
Reflection Property
The graphs
of f (x)
=
abx and g(x) = abx are reflections of each other across the x-axis.
Graph
thefunction
function
hand.
Graph
the
byby
hand.
The graphs of f (x) = abx and g(x) = abx are reflections of each other
across the x-axis.
✓✓ ◆x◆x
xx
1
1
10.
f
(x)
=
3(4)
10. fGraph
(x) =the3(4)
function by hand.
11.g(x)
g(x)== 1616
11.
Graph the function by hand.
4✓4 ◆x
x
10. f (x) = 3(4)
✓ ◆1x
11. g(x)
16
x x where a > 0 and do the
x and
Discuss
domain
andrange
rangeofofananexponential
exponentialfunction
function
the=
form
(x)
Discuss
ofofthe
form
f1f(x)
a > 0 and do the
10.
f (x) the
=thedomain
3(4)
4 ==abab where
11.
g(x)
=
16
samewhere
wherea a<<0.0.
same
4
Discuss the domain and range of an exponential function of the form f (x) = abx where a > 0 and do the
same
Find
Discuss
Find
where
the
domain
the
domain
a domain
< 0. and
and
range
range
range
of
of
the
the
function.
an
exponential
a > 0 and do the
function.
and
of
function of the form f (x) = abx where
the
same where a < 0.
✓✓ ◆x◆x
xx
11
12.fFind
f(x)
(x)=
=3(2)
3(2)
12.
the
domain
and range of the function.
13.g(x)
g(x)== 1010
13.
22
Find the domain and range of the function.
For an exponential function of the form
9. Sketch and compare the graphs of f (x) =
the y-intercept is (0, a).
3
y = abx ,
4(2)x and g(x) = 4(2)x .
Reflection Property
Find the y-intercept of the graph of the function.
✓ ◆x
2 across the x-axis.
abx and g(x) = abx are reflections of each other
6. y = 4
7
Graph the function by hand.
Warning: The y-intercept of the graph of a function of the form y = bx is (0, 1), not (0, b).
✓ ◆x
1
10. f (x) = 3(4)x
11.
g(x)
=
16
Find the y-intercept of the graph of the given function. Then use the
4 base multiplier property to graph the
function by hand.
x
For all eDiscuss
xponential functions, the xof
-­‐axis is a horizontal asymptote. the domain
and range
an exponential
function
of the form ✓
f (x)
=
ab
where
a
>
0
and
do
the
◆x
x
1
f (x)where
= 2(3)
a < 0.
7. same
8. g(x) = 12
3
Find the domain and range of the function.
x
9. Sketch and compare the graphs of f (x) = 4(2)x and g(x) = 4(2)
✓ ◆x.
x
1
12. f (x) = 3(2)
13. g(x) = 10
2
Reflection Property
x of f (x) =
graphs
5. y The
= 8(5)
SHORTofHW
5, 11, 15, x21, 27, 29, 33, 37, x41, 45, 47, 63, 71, 77, 85
The
graphs
f (x) = ab and g(x) = ab are reflections of each other across the
x-axis.
MEDIUM HW 1, 5, 11, 15, 17, 21, 23, 27, 29, 33, 37, 41, 45, 47, 61, 63, 71, 75, 77, 85, 87, 99, 101, 103
Graph the function by hand.
✓ ◆x
x
1
10. f (x) =
3(4)
The domain of any exponential function f(x) = abx is t11.
he sg(x)
et of =
real 16
numbers. 4
The range of an exponential function f(x) = abx is the set of all positive real numbers if a > 0, and the range is the set of all negative real and
numbers f an exponential function of the form f (x) = abx where a > 0 and do the
Discuss
the domain
range iof
a < 0where
. same
a < 0.
Find the domain and range of the function.
Copyright c 2015 Pearson Education, Inc.
✓ ◆x
1
12. f (x) = 3(2)x
13. g(x) = 10
2
SHORT
HW 5, 11, 15, 21, 27, 29, 33, 37, 41, 45, 47, 63, 71, 77, 85
MEDIUM
HW 1, 5, 11, 15, 17, 21, 23, 27, 29, 33, 37, 41, 45, 47, 61, 63, 71, 75, 77, 85, 87, 99, 101, 103
The graph of an exponential function f is shown below. Copyright c 2015 Pearson Education, Inc.
1. Find f(2). Ans: f(2) = 8 2. Find x when f(x) = 2. Ans: x = –2 when f(x) = 2. 3. Find x when f(x) = 0. Ans: there is no value of x where f(x) = 0. Because the graph of an exponential functions gets close to, but never reaches, the x-­‐axis. <Homework 10.3> 𝒇 𝒙 = 𝟐𝒙 + 𝟑𝒙 #51. Find 𝑓(2) #52. Find 𝑓(0) #53. Find 𝑓(−2) #54. Find 𝑓(−1) 𝒇 𝒙 = 𝟑𝒙 #55. Find x when 𝑓 𝑥 = 3. #56. Find x when 𝑓 𝑥 = 9. #57. Find x when 𝑓 𝑥 = 1. 1
#58. Find x when 𝑓 𝑥 = 3. #75. Use an equation of the form 𝑓 𝑥 = 𝑎𝑏 ! , where a and b are constants you specify. What equation works?