Section 31 The exponential function f with base a is defined by f x a x where a gt 0 a 1 and x is any real number For instance f x 3 ID: 593760
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Slide1
Exponential Functions and Their Graphs
Section 3-1Slide2
The
exponential function f with base a is defined by f(x) = axwhere a > 0, a 1, and x is any real number.
For instance, f(x) = 3x and g(x) = 0.5xare exponential functions.
Definition of Exponential FunctionSlide3
The value of
f(x) = 3x when x = 2 is f(2) = 32
=The value of g(x) = 0.5x when x = 4 is
g
(
4
) = 0.5
4 =
The value of f(x) = 3x when x = –2 is
9
f
(–2) = 3–2 =
0.0625
Example: Exponential FunctionSlide4
The graph of
f(x) = ax, a > 1
y
x
(0, 1)
Domain: (–
, )
Range: (0,
)
Horizontal Asymptote
y
= 0
Graph of Exponential Function (
a
> 1)
4
4
Exponential Growth FunctionSlide5
The graph of
f(x) = ax, 0 < a < 1
y
x
(0, 1)
Domain: (–
, )
Range: (0,
)
Horizontal Asymptote
y
= 0
Graph of Exponential Function (0 <
a
< 1)
4
4
Exponential Decay FunctionSlide6
Exponential Function
3 Key Parts1. Pivot Point (Common Point)2. Horizontal Asymptote3. Growth or DecaySlide7
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Manual GraphingLets graph the following together:f(x) = 2xSlide8
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Sketch the graph of f(x) = 2x.
x
x
f
(
x
)
(
x
,
f
(
x
))
-2
¼
(-2, ¼)
-1
½
(-1, ½)
0
1
(0, 1)
1
2
(1, 2)
2
4
(2, 4)
y
2
–2
2
4
Example: Graph
f
(
x
) = 2
xSlide9
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Definition of the Exponential FunctionHere are some examples of exponential functions.f (x) = 2
x g(x) = 10x h(x) = 3x
Base is 2.
Base is 10.
Base is 3.
The exponential function
f
with base
b
is defined by
f
(x) =
b
x
or
y
=
b
x
Where
b
is a positive constant other than and
x
is any real number.Slide10
Calculator Comparison
Graph the following on your calculator at the same time and note the trendy1 = 2xy2= 5xy3 = 10xSlide11
When base is a fraction
Graph the following on your calculator at the same time and note the trendy1 = (1/2)xy2= (3/4)xy3 = (7/8)xSlide12
Transformations Involving Exponential Functions
Shifts the graph of f (x) = bx upward c units if c > 0.
Shifts the graph of f (x) = bx downward c units if c < 0.g(x) = bx+ c
Vertical translation
Reflects the graph of
f
(
x) = bx about the x-axis.
Reflects the graph of f (x) = b
x about the y-axis.
g(x) = -bxg(x) = b-x
Reflecting
Multiplying
y
-coordintates of
f
(
x
) =
b
x
by
c
,
Stretches the graph of
f
(
x
) =
b
x if c > 1. Shrinks the graph of f
(x) = bx if 0 < c < 1.
g(x) =
cbx
Vertical stretching or shrinking
Shifts the graph of f (x) = bx to the left c units if
c > 0. Shifts the graph of f (x) = bx to the right c units if
c < 0.
g(x) = bx+c
Horizontal translation
Description
Equation
TransformationSlide13
Example: Sketch the graph of
g(x) = 2x – 1. State the domain and range.
x
y
The graph of this function is a vertical translation of the graph of
f
(
x
) = 2
x
down
one unit .
f
(
x
) = 2
x
y
= –1
Domain: (–
, )
Range: (–1,
)
2
4
Example: Translation of GraphSlide14
Example: Sketch the graph of
g(x) = 2-x. State the domain and range.
x
y
The graph of this function is a reflection the graph of
f
(
x
) = 2
x
in the
y
-axis.
f
(
x
) = 2
x
Domain: (–
, )
Range: (0,
)
2
–2
4
Example: Reflection of GraphSlide15
Discuss these transformations
y = 2(x+1)Left 1 unity = 2x + 2Up 2 unitsy = 2-x – 2Ry, then down 2 unitsSlide16
Special Symbols
Math uses special symbols at times to represent special numbers used in calculations.The symbol (pi) represents 3.14…..The symbol “i” represents Slide17
(The Euler #) e is an irrational #, where
e 2.718281828… is used in applications involving growth and decay.The number eSlide18
The graph of
f(x) = ex
y
x
2
–2
2
4
6
x
f
(
x
)
-2
0.14
-1
0.38
0
1
1
2.72
2
7.39
Graph of Natural Exponential Function
f
(
x
) =
e
x
Natural Exponential FunctionSlide19
Homework
WS 6-1