Section 3'1 Exponential Functions and Their Graphs

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Section 3.1 Exponential Functions and Their
Graphs
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What You Should Learn

• Recognize and evaluate exponential functions with
  base a.
• Graph exponential functions and use the One
  to-One Property.
• Recognize, evaluate, and graph exponential
  functions with base e.
• Use exponential functions to model and solve
  real-life problems.

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Which is the better deal?

• 1000 a day
• 1, 2, 4, 8

1000
1
2000
3
3000
7
10000
1023
4000
15
11000
2047
5000
31
12000
4095
6000
63
13000
7000
127
8191
14000
16383
8000
255
15000
32767
9000
511
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Will I ever leave the room?
A

• Let A be the starting point and B be the door.
• The distance from A to B is 1.
• Before I can get to the door I have to get
  halfway to the door.
• ½
• Now I am halfway to the door, and before I get to
  the door I have to get halfway to the door,
  again.
• ½ ¼ ¾
• Now I am ¾ way to the door, and before I get to
  the door I have to get halfway to the door,
  again.
• ¾ 1/8 7/8
• Now I am 7/8 of the way to the door, and before I
  get to the door I have to get halfway to the
  door, again.
• 7/8 1/16 15/16

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B
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Elementary Operation

• One of the operations of addition, subtraction,
  multiplication, division, and integer (or
  rational) root extraction.

Weisstein, Eric W. "Elementary Operation. From
MathWorld--A Wolfram Web Resource.
http//mathworld.wolfram.com/ElementaryOperation.h
tml
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Algebraic Functions

• A function which can be constructed using only a
  finite number of elementary operations together
  with the inverses of functions capable of being
  so constructed. Nonalgebraic functions are called
  transcendental functions.

Weisstein, Eric W. "Algebraic Function." From
MathWorld--A Wolfram Web Resource.
http//mathworld.wolfram.com/AlgebraicFunction.htm
l
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Definition of Exponential Functions

• The exponential function f with base a is denoted
  by
• f(x) ax
• Where a gt 0, a ? 1, and x is any real number.
• a gt 0 No negative exponents
• a ? 1 If a 1 then f(x) 1x 1
• x any real number is what makes it
  transcendental.

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Graph y 2x
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p
8.8249778
8
4
2
1
½
¼
1/8
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Graph g(x) 2-x
8
1/8
4
¼
2
½
1
1
½
2
¼
4
1/8
8
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One-to-One PropertyExample 4

• Original equation
• 9 32
• One-to-One Property
• Solve for x

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Matching
(1, 3)
(0, -1)
(-1, 3)
(0, 3)
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What is the right hand behavior of the function?
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Exploration page 222

• Use a graphing utility to graph

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Evaluate ex for the given values of x.

• x -2
• x -1
• x 0.25
• x -0.3

• 0.135353
• 0.3678794
• 1.2840254
• 0.7408182

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Formulas for Compound Interest

• Years t
• Balance in the account A

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