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Exponential Functions and Their Graphs Exponential Functions and Their Graphs

Exponential Functions and Their Graphs - PowerPoint Presentation

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Exponential Functions and Their Graphs - PPT Presentation

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

exponential graph range function graph exponential function range units base domain translation functions horizontal shifts reserved copyright houghton special

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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