Chapter 3: Exponential, Logistic, and Logarithmic Functions

3.1a &bHomework: p. 286-287 1-39 odd

Overview of Chapter 3So far in this course, we have mostly studied algebraicfunctions, such as polys, rationals, and power functionsw/ rat’l exponents…

The three types of functions in this chapter (exponential,logistic, and logarithmic) are called transcendentalfunctions, because they “go beyond” the basic algebraoperations involved in the aforementioned functions…

Consider these problems:

Evaluate the expression without using a calculator.

1. 3 216 33 6

2. 3125

8

3

3

125

8

5

2

6

3.2 327 2 333 23 9

4. 5 24 5 222 52 32

We begin with an introduction to exponential functions:

2f x xFirst, consider:

This is a familiar monomial,and a power function…

one of the “twelve basics?” 2xg x

Now, what happenswhen we switchthe base and the

exponent ???

This is an example ofan exponential function

Definition: Exponential Functions

xf x a b

Let a and b be real number constants. An exponential functionin x is a function that can be written in the form

where a is nonzero, b is positive, and b = 1. The constant a isthe initial value of f (the value at x = 0), and b is the base.

Note: Exponential functions are defined and continuous for allreal numbers!!!

Identifying Exponential Functions

3xf x

Which of the following are exponential functions? For those thatare exponential functions, state the initial value and the base.For those that are not, explain why not.

1.

Initial Value = 1, Base = 3

46g x x2.

Nope! g is a power func.!

2 1.5xh x 3.

Initial Value = –2, Base = 1.5

7 2 xk x 4.

Initial Value = 7, Base = 1/2

5 6q x 5.

Nope! q is a const. func.!

More Practice with Exponents 2xf x Given

4f1.42 16

, find an exact value for:

0f2.02 1

3f 3.32

3

1 10.125

2 8

1

2f

4.1 22 2 1.414

3

2f

5.3 22 3 2

1

2

1

8 2

4

Determine the formula forthe exp. func. g:

Finding an Exponential Function from its Table of Values

x g(x)

–2 4/9

–1 4/3

0 4

1 12

2 36

xg x a b

The Pattern?

x 3

x 3

x 3

x 3

General Form:

0 4 4g a Initial Value:

11 4 12g b Solve for b:

3b

4 3xg x Final Answer:

Determine the formula forthe exp. func. h:

Finding an Exponential Function from its Table of Values

x h(x)

–2 128

–1 32

0 8

1 2

2 1/2

xh x a b

The Pattern?

x 1/4

x 1/4

x 1/4

x 1/4

General Form:

0 8 8h a Initial Value:

11 8 2h b Solve for b:

1 4b

184

x

h x

Final Answer:

How an Exponential Function Changes (a recursive formula)

xf x a b For any exponential function and any realnumber x,

If a > 0 and b > 1, the function f is increasing and is anexponential growth function. The base b is its growth factor.

1f x b f x

If a > 0 and b < 1, f is decreasing and is an exponential decayfunction. The base b is its decay factor.

Does this formula make sense with our previous examples?Does this formula make sense with our previous examples?

Graphs of Exponential

Functions

We start with an “Exploration”We start with an “Exploration”Graph the four given functions in the same viewingwindow: [–2, 2] by [–1, 6]. What point is common toall four graphs?

1 2xy 2 3xy 3 4xy 4 5xy

Graph the four given functions in the same viewingwindow: [–2, 2] by [–1, 6]. What point is common toall four graphs?

1

1

2

x

y

2

1

3

x

y

3

1

4

x

y

4

1

5

x

y

We start with an “Exploration”We start with an “Exploration”

Now, can we Now, can we analyze analyze these graphs???these graphs???

xf x b1b

0,1

1,b

xf x b

0 1b

0,1 1,b

Exponential Functions f(x) = b x

Domain: , Range: 0,

Continuity: Continuous

Symmetry: None

Boundedness: Below by y = 0 Extrema: None

H.A.: y = 0 V.A.: None

If b > 1, then also

• f is an increasing func.,

• lim 0x

f x

limx

f x

If 0 < b < 1, then also

• f is a decreasing func.,

• limx

f x

lim 0x

f x

In Sec. 1.3, we first saw the “The Exponential Function”:

xf x e(we now know that it is an exponential growth function why?)

But what exactly is this number “e”???

Definition: The Natural Base e

1lim 1

x

xe

x

Natural

Analysis of the Natural Exponential FunctionAnalysis of the Natural Exponential Function

xf x e

The graph:

0,Domain: All reals

Range:

Continuous

Increasing for all x

No symmetry

Bounded below by y = 0

No local extrema

H.A.: y = 0 V.A.: None

lim 0x

xe

End behavior: lim x

xe

Guided PracticeGuided Practice

2xf x Describe how to transform the graph of f into the graph of g.

12xg x 1. Trans. right 1

2xf x 2 xg x 2. Reflect across y-axis

xf x e 2xg x e3. Horizon. shrink by 1/2

3xf x 23 xg x 4. Reflect across both axes,Trans. right 2

xf x e 5 2xg x e 5. Reflect across y-axis,Vert. stretch by 5,Trans. up 2

Guided PracticeGuided PracticeDetermine a formula for the exponential function whose graphis shown.

y g x

0,2

21,e

xg x a b 1b 00 2g a b

2a 11 2 2g b e

1b e

12 2

xxg x e

e

Whiteboard…Whiteboard…State whether the given function is exp. growth or exp. decay,and describe its end behavior using limits.

1x

f xe

lim 0x

f x

limx

f x

Exponential Decay

0.75 xk x

limxk x

lim 0x

k x

Exponential Growth

4

3

x

Whiteboard…Whiteboard…Solve the given inequality graphically.

xx > 0 > 0

6 8x x The graph?

xx > 0 > 0

1 1

3 2

x x

The graph?