INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Exponential Functions

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (22)

2 answers

Last reply by: Jerry Jin
Wed Jul 24, 2019 11:06 AM

Post by Matthew Johnston on August 8, 2016

In example 3 shouldn't 8/38=4/19 instead of 2/19?

0 answers

Post by Krishna Vempati on April 6, 2015

I watched this lecture and I tried solving a problem in math class but I had trouble here is the problem.....    
Your parents offer to pay you exponentially to study for your Algebra test. They say that if you study for one hour you'll get $6, two hours gets you a total $7, three hours $9, four hours $13, etc. What equation are they using to come up with those values?
Could you please help me solve this?

1 answer

Last reply by: Dr Carleen Eaton
Sun Mar 15, 2015 11:20 PM

Post by Daija Jenkins on March 9, 2015

Directions: Write an exponential function for the graph that passes through the given points.
How do I solve: (0,-5)and (-3, -135)

1 answer

Last reply by: Dr Carleen Eaton
Sat Sep 14, 2013 2:58 PM

Post by Tami Cummins on August 27, 2013

Isn't negative 3 raised to a negative 2 power still a positive 1/9.  When you square the negative 3 doesn't it become positive?

1 answer

Last reply by: Dr Carleen Eaton
Tue Jul 3, 2012 7:24 PM

Post by Laura Gilchrist on June 27, 2012

If there is no variable in the exponent, will it just be a power function instead? Does it have to have variable for it to be exponential? Thanks!!

1 answer

Last reply by: Dr Carleen Eaton
Mon Apr 16, 2012 10:18 PM

Post by Ed Grommet on April 13, 2012

FOr some reason it will not play. Question is IF i have a expo equations of y=-5^x same as y=(-1)(5^x) ? Also is it decay or growth since it is not above the x axis?

1 answer

Last reply by: Dr Carleen Eaton
Mon Mar 19, 2012 3:51 PM

Post by Ding Ye on March 19, 2012

This is a really nice video. Thanks a lot!

1 answer

Last reply by: Dr Carleen Eaton
Thu Jan 26, 2012 7:53 PM

Post by Jose Gonzalez-Gigato on January 24, 2012

In the slide labeled 'Properties', at about 12:50, you mention f(x) is 'one-to'one' and give the reason that it passes the vertical line test. For a function to be 'one-to-one' it must pass the horizontal line test.

1 answer

Last reply by: Dr Carleen Eaton
Wed Jan 11, 2012 12:38 AM

Post by Arlene Francis on January 9, 2012

Are there extra examples of problems.

1 answer

Last reply by: Dr Carleen Eaton
Wed Dec 28, 2011 9:10 PM

Post by Jonathan Taylor on December 27, 2011

Dr carleen must the base be the same in all exponential equation are is this only when your working with certain exponential fuction

0 answers

Post by Guillermo Marin on August 8, 2010

Dr. Eaton is really OUTSTANDING!

0 answers

Post by Dr Carleen Eaton on May 18, 2010

Correction to Example III: The solution, x = 8/38 reduces to 4/19, not 2/19

Exponential Functions

  • Know the graph of the exponential function and its properties.
  • If the base is greater than 1, the function is exponential growth. If it is between 0 and 1, it is exponential decay.
  • Solve exponential equations with the same base by equating the exponents.
  • Solve exponential inequalities with the same base by applying the same inequality to the exponents.

Exponential Functions

Graph f(x) = 3*( [1/2] )x. On the graph correctly identify the y - intercept.
  • Create a Table of values in order to graph the Exponential Function.
  • xf(x) = 3*( [1/2] )x
    -23*( [1/2] ) − 2 = 3*( [2/1] )2 = 12
    -13*( [1/2] ) − 1 = 3*( [2/1] )1 = 6
    03*( [1/2] )0 = 3*1 = 3
    13*( [1/2] )1 = [3/2]
    23*( [1/2] )2 = [3/4]
    33*( [1/2] )3 = [3/8]
  • Draw a smooth curve. Your graph should never touch the x - axis.
Graph f(x) = 4*2x. On the graph correctly identify the y - intercept.
  • Create a Table of values in order to graph the Exponential Function.
  • xf(x) = 4*2x
    -64*( 2 ) − 6 = 4*( [1/2] )6 = [4/64] = [1/16]
    -44*( 2 ) − 4 = 4*( [1/2] )4 = [4/16] = [1/4]
    -24*( 2 ) − 2 = 4*( [1/2] )2 = [4/4] = 1
    04*( 2 )0 = 4*1 = 4
    24*( 2 )2 = 4*4 = 16
    44*( 2 )4 = 4*16 = 64
  • Draw a smooth curve. Your graph should never touch the x - axis.
Does this function represent growth or decay?
f(x) = 7*( [1/4] ) − x
  • Rewrite the equation in standard form :f(x) = a*bx
  • Remember that a − n = [1/(an)]
  • Rewritten in standard form, the function becomes f(x) = 7*4x
Since a > 0 and b > 1, this function represents Exponential Growth
Does this function represent growth or decay?
f(x) = 3*(6) − x
  • Rewrite the equation in standard form :f(x) = a*bx
  • Remember that a − n = [1/(an)]
  • Rewritten in standard form, the function becomes f(x) = 3*( [1/6] )x
Since a > 0 and b < 1, this function represents Exponential Decay
Solve ( [1/5] )m − 3 = 252m + 3
  • In order to solve for m, we need to have the same base. Rewrite problem in base 5.
  • Rewrite [1/5] as base 5 =
  • Rewrite 25 as base 5 =
  • That would give you [1/5] = (5) − 1 and 25 = 52
  • Rewrite the problem.
  • (5 − 1)m − 3 = (52)2m + 3
  • Distribute the exponets.
  • 5 − m + 3 = 54m + 6
  • Now that you have the problem in the same base, eliminate the base and solve for m
  • − m + 3 = 4m + 6
m = − [3/5]
Solve 343 − x = ( [1/7] )3x
  • In order to solve for m, we need to have the same base. Rewrite problem in base 7.
  • Rewrite [1/7] as base 7 =
  • Rewrite 343 as base 7 =
  • That would give you [1/7] = (7) − 1 and 343 = 73
  • Rewrite the problem.
  • (73) − x = (7 − 1)3x
  • Distribute the exponets.
  • 7 − 3x = 7 − 3x
  • Now that you have the problem in the same base, eliminate the base and solve for x
  • − 3x = − 3x
Whenever this situation happens, the solution is all real numbers.
Solve 36 − x = 216 − 2x − 1
  • In order to solve for x, we need to have the same base.
  • The only base that seems to work will be base 6.
  • Rewrite 36 as base 6 =
  • Rewrite 216 as base 6 =
  • That would give you 36 = (6)2 and 216 = (6)3
  • Rewrite the problem.
  • (62) − x = (63) − 2x − 1
  • Distribute the exponets.
  • 6 − 2x = 6 − 6x − 3
  • Now that you have the problem in the same base, eliminate the base and solve for x
  • − 2x = − 6x − 3
x = − [3/4]
Solve 625 − 3x = 253x − 3
  • In order to solve for x, we need to have the same base.
  • The only base that seems to work will be base 5.
  • Rewrite 625 as base 5 =
  • Rewrite 25 as base 5 =
  • That would give you 625 = (5)4 and 25 = (5)2
  • Rewrite the problem.
  • (54) − 3x = (52)3x − 3
  • Distribute the exponets.
  • 5 − 12x = 56x − 6
  • Now that you have the problem in the same base, eliminate the base and solve for x
  • − 12x = 6x − 6
x = [1/3]
Solve 64 − 3x< 16
  • In order to solve for x, we need to have the same base.
  • The only base that seems to work will be base 4.
  • Rewrite 64 as base 4 =
  • Rewrite 16 as base 4 =
  • That would give you 64 = (4)3 and 16 = (4)2
  • Rewrite the problem.
  • (43) − 3x< 42
  • Distribute the exponets.
  • 6 − 9x< 42
  • Now that you have the problem in the same base, eliminate the base and solve for x
  • − 9x < 2
  • Divide both sides by − 9. Remember that whenever you divide by a negative, the inequality must be switched.
x >− [2/9]
Solve 16 − 3x + 2> 4 − 2x
  • In order to solve for x, we need to have the same base.
  • The only base that seems to work will be base 4.
  • Rewrite 16 as base 4 =
  • That would give you 16 = (42)
  • Rewrite the problem.
  • (42) − 3x + 2> 4 − 2x
  • Distribute the exponets.
  • 4 − 6x + 4> 4 − 2x
  • Now that you have the problem in the same base, eliminate the base and solve for x.
  • − 6x + 4 >− 2x
  • − 4x + 4 > 0
  • − 4x >− 4
  • Divide both sides by − 4. Remember to switch the inequality whenever you divide by a negative.
x < 1

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Exponential Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • What is an Exponential Function? 0:12
    • Restriction on b
    • Base
    • Example: Exponents as Bases
    • Variables as Exponents
    • Example: Exponential Function
  • Graphing Exponential Functions 2:33
    • Example: Using Table
  • Properties 11:52
    • Continuous and One to One
    • Domain is All Real Numbers
    • X-Axis Asymptote
    • Y-Intercept
    • Reflection Across Y-Axis
  • Growth and Decay 15:06
    • Exponential Growth
    • Real Life Examples
    • Example: Growth
    • Example: Decay
    • Real Life Examples
  • Equations 17:32
    • Bases are Same
    • Examples: Variables as Exponents
  • Inequalities 21:29
    • Property
    • Example: Inequality
  • Example 1: Graph Exponential Function 24:05
  • Example 2: Growth or Decay 27:50
  • Example 3: Exponential Equation 29:31
  • Example 4: Exponential Inequality 32:54

Transcription: Exponential Functions

Welcome to Educator.com.0000

Today begins the first in a series of lectures on exponential and logarithmic relations, starting out with exponential functions,0003

beginning with the definition: What is an exponential function?0012

Well, an exponential function is a function of the form f(x) = a times bx, where a is not 0,0016

because if a were to equal 0, this would all just drop out, and you wouldn't have a function.0027

b is greater than 0: we are restricting this definition to values of b that are positive; and b does not equal 1.0032

If b were to equal 1, no matter what you made x, that would still remain 1.0039

And then, you wouldn't really have a very interesting function.0044

So, the base of the function here is b.0047

Now, recall earlier on, when we worked with functions and equations, we have seen things like this: x2 + 2x - 1...f(x) equals this.0050

Here, the base was a variable, and here the exponent was a constant.0062

Now, we are going the other way around: in this case, with exponential functions, we are going to be working with situations0070

where the base is a number (a constant) and the exponent contains a variable.0084

And that is what makes these functions fundamentally different from some of the other functions that we have seen so far.0101

Examples would be something like f(x) = 6x: here the base is 6, the coefficient is just 1 (a = 1), and the exponent is x.0110

Or you could have something a little bit more complicated: f(x) = (3 times 1/2)4x - 2.0128

I have an algebraic expression, not just a single variable, as the coefficient.0138

So here, I have a base equal to 1/2; the coefficient is 3; and then the exponent is 4x - 2.0142

Starting out by looking at the graphs of exponential functions: as we have done with other types of functions,0153

we can use a table of values to graph an exponential function.0160

And we are going to look at a few different permutations of these functions and see what we end up with.0163

I am going to start out with letting f(x) equal 3x; let's find some values for x and y.0170

I am going to just rewrite f(x) as y.0180

If x is -3, then what is y? Well, this is giving me 3 to the -3 power.0190

Recall that a-n equals 1/an; so what this is really saying is 1/33, which equals 1/27.0197

At this point, if you are not really comfortable with exponents and the rules and properties governing working with exponents,0215

you should go back and review the earlier lecture on that, because we are going on to solve equations using these.0220

So, you need to have the rules learned, as we will be applying them frequently.0227

When x is -2, this is going to give me 3-2, which is 1/32, or 1/9.0231

And we continue on: when x is -1, that is 3-1 = 1/31, or 1/3.0239

Then, getting back to some more familiar territory: when x is 0, this gives me 30.0248

A number or variable, or anything, to the 0 power, is going to give me 1; when x is 1, f(x), or y, is 3.0257

When x is 2, that is 32 is 9; when x is 3, that is 33, to give me 27.0266

Let's go ahead and plot these values out; let's do that so that this is -2, -4, -6, -8, so we can look at more values right on this graph.0276

2, 4, 6, 8, 2, 4, 6, 8, 10; -2, 4, 6, 8, 10; so -10 is down here.0288

When x is -3, the graph is just slightly above 1; it is 1/27; when x is -2, we get a little farther away from 1--it becomes 1/9.0299

When x is -1, the graph rises up even a little more and becomes -1/3.0312

When x is 0, y is 1; when x is 1, y is 3; when x is 2, y is right up here at 9; and then, when x is 3, y gets very large.0321

This gives me a sense of the shape of the graph--that as x becomes positive, y rapidly becomes a very large number;0336

as x is negative, what I can see happening is that f(x) is approaching 0, but it never quite gets there.0345

Therefore, what I have is that, for this graph, the x-axis is a horizontal asymptote.0358

So, this is the graph of f(x); let's look at a different case--let's look at the graph of...this is in the form f(x) = abx;0377

let's look at the graph f(x) = a(1/b)x.0392

Well, a is just 1, so I am going to look at the situation g(x) = 1 over 3 to the x power.0398

I am making a table of values, again, for x and y.0410

Again, when x is -3, let's look at what y is; it is going to be 1/3 to the -3.0419

Using this property, I am then going to get 33, which is 27.0426

For -2, I am going to get (1/3)-2, or 32, which is 9.0433

-1 is going to give me (1/3)-1 = 3; 0...anything to the 0 power is simply going to be 1.0440

(1/3)1 is 1/3; (1/3)2 is 1/9; and (1/3)3 is 127.0453

Let's see what happens with this graph: here, when x is -3, y is very large--it is some value way up there, 27 (that is off my graph).0469

Let's look at -2: when x is -2, y is 9 (that is right here).0479

And I know that, when I get out to -3, this is increasing; so I know the shape up here.0484

When x is -1, y is 3; when x is 0, y is 1; they have the same y-intercept, right here at (0,1).0489

When x is 1, y is 1/3; when x is 2, we see that the graph is approaching the x-axis, because now, we are here at 1/9.0502

And when x is 3, the value of the function is 127.0515

So again, I see this x-axis, again, being the horizontal asymptote for both graphs.0519

I know that y is getting very large as x is becoming negative.0532

And I know that, as x is positive, the graph is approaching, but not reaching, the x-axis.0538

Here, the y-axis also forms an axis of symmetry; so these two graphs are mirror images of each other.0547

f(x) and g(x) are mirror images reflected across the y-axis.0553

Let's look at one other case: let's look at the case...let's call it h(x) = 3x, but let's take the opposite of that.0558

Therefore, this will be simpler to figure out the values: let's just leave the x-values the same as they were for f(x),0569

and let's just extend this graph out: these were my values for f(x), and now let's figure out what h(x) is going to give me.0581

Well, if x is -3, and I put a -3 in here, again, I am going to get 1/27, but I want the opposite of that; so I am going to get -1/27.0592

If x is -2, again, I am going to get 3-2, which is 1/9, but I am going to take the opposite.0602

So, all I have to do is change the signs on these to get my h(x) values.0608

And then, we can see what this graph looks like.0615

This is f(x); I have g(x) here; for h(x), when x is -3, h(x) is right here; it is very close to the x-axis, but not quite reaching it.0620

And at -2, it is a little bit farther away at -1/9; at -1, we are down here at -1/3; at 0, here we have the y-intercept at (0,-1).0633

When x is 1, y is -3, right about here; when x is 2, y is down here at -9; and when x is 3, we are going to be way down here at -27.0649

So, for h(x), what I am going to have (I'll clean this up just a bit)...0664

This is the graph of h(x), and again, I am seeing the x-axis here, acting as a horizontal asymptote.0686

And I am seeing that, as x becomes positive, y becomes very large, but this time it is in the negative direction.0699

It is giving me a mirror image here with f(x); but now, my values are going down in the negative direction.0706

So, these are several graphs of exponential functions; let's go ahead and sum up the properties of these functions.0712

What we saw is that f(x) is continuous and one-to-one.0721

I am just very briefly sketching these three situations: I let f(x) equal 3x, g(x) equal (1/3)x, and h(x) equal (-3)x.0724

So, for the graph of f(x), what I got is this...actually, rising faster; so let's make that rise much faster.0746

And I saw that it is continuous; there weren't any gaps or discontinuities.0759

When we worked with graphing some rational functions, we saw that there were actually discontinuities; there aren't any here.0765

It is also one-to-one; and I could use the vertical line test.0771

Recall that, if you draw a vertical line across the curve of a graph, if it only crosses the graph once,0774

at one point, everywhere you possibly could try, you have a one-to-one relationship; you have a function.0782

So, no matter where I drew a vertical line, I would only cross this curve one time.0790

The domain of f(x) is all real numbers: you see that I could make x a negative value; I could make it a positive value.0795

However, the range is either all positive real numbers or all negative real numbers.0802

Here, the range is positive real numbers; for h(x), I graphed that, and that turned out like this.0807

Here, the domain was all real numbers, but the range was the negative real numbers.0820

We saw that the x-axis acts as an asymptote that the graphs of these functions approach, but never reach.0835

The y-intercept is at (0,a); so for both g(x) and f(x), here a = 1, so the coefficient is 1; so the y-intercept is at (0,1).0842

However, for h(x), a = -1; the coefficient is -1, so I have a y-intercept here at (0,-1).0861

Finally, we saw that the graphs of f(x) = abx and f(x) = a/(1/bx) are reflections across the y-axis.0872

So, that was this graph, which shares the same y-intercept, but is reflected here across the y-axis.0880

So, looking at these three different situations and their graphs can tell us a lot about what is going on with exponential functions.0893

Introducing the concept of growth and decay: if you have a function in this form, f(x) = abx,0907

and a is greater than 0 (it is a positive number), and b is greater than 1, then this represents exponential growth.0916

And the concepts of exponential growth and decay are very frequently used in real-world applications.0925

So, we are going to delve into this topic in greater depth in a separate lecture.0931

But thinking about exponential growth: that could be something like working in finance, or thinking about your own savings and compound interest.0937

That works in such a way that the growth is exponential.0947

An example of exponential growth would be something like f(x) = 1/4(2x).0952

And that is because here, b is greater than 1; so this is growth.0966

If I had another function, f(x) = 4((1/8)x), here this is representing exponential decay,0972

because b is greater than 0, but it is less than 1: it is a fraction between 0 and 1.0983

And we sometimes talk about things such as radioactive decay and half-life in terms of an exponential function.0990

Another way to think of this for a minute is: recall that a-n = 1/an.0999

So, I can actually rewrite this function as 4(8-x).1006

If I had a base here that was greater than 1, but the exponent was negative, then I also know that I have decay.1014

If you write it in the standard form where the exponent is positive, then all you need to do is look and see the value of b.1022

If it is greater than 1, you have growth; if it is between 0 and 1, you have decay.1029

If it is a number greater than 1, and it is not in standard form, and you see I have a negative exponent,1033

that also gives you a clue that you are looking at decay.1039

But you can always put them in standard form, like this, and then just go ahead and look at the value of the base.1043

Working with exponential equations: in exponential equations, variables occur as exponents.1053

That is what I mentioned in the beginning, when I was talking about exponential expressions and exponential functions.1059

But now, we are talking about equations: again, you are used to working with things where the base may be a variable, but the exponent is a number.1064

And now, we are going to change that around and actually have situations where variables are exponents.1072

There are some properties of this that we can use to solve these equations.1079

If the bases are the same (if bx equals by), then x must equal y.1083

If the bases are the same, in order for the left half of the equation to equal the right half, the exponents have to be the same; they have to be equivalent.1090

Look at this in a very simple case: 63x - 5 = 67.1100

6 is the same base as 6; so for this left half to equal the right half, if the bases are equal, these two must be equal.1108

So, in order to solve these, I am just going to take 3x - 5 and put that equal to 7, which leaves me with a simple linear equation to solve.1116

I am going to add 5 to both sides, which is going to give me 3x = 12.1125

Then, I am going to divide both sides by 3 to get x = 4.1129

Things get a little more complicated if the bases are not the same.1137

If the bases are not the same, the first thing to do is try to make them the same.1141

If that is not done reasonably simply, then we can use another technique that we are going to learn about in a later lecture.1145

But for right now, we are going to stick to situations where either the bases are the same, or you can pretty easily make them the same.1152

For example, I could be given an exponential equation 2x - 3 = 4.1159

These bases are not the same, so I can't use this technique.1168

However, you can pretty easily see that you can make them the same base1171

by saying, "OK, 2 squared is 4; so instead of writing this as 4, I am going to write it this way."1175

Now, I am back to the situation where the bases are the same; I am going to set the exponents1185

equal to each other (because they must be equal to each other) and solve for x: x = 5.1191

You could have a little bit more complicated situation, where 3x - 1 (a separate example) = 1/9.1199

And I can see that I want these to be the same, and I know that 1/3 squared is 1/9.1208

So, I am pretty close; but I need this to be 3.1214

Well, recall that I could rewrite this as 3-2: 3-2 is the same as 1/3 squared.1217

This also equals 1/9; and it is fine that this is a negative exponent--I can go ahead and use it up here, rewriting 1/9 as 3-2.1227

Now, -x - 1 = -2, so x = -1.1236

OK, so basically, when you are working with exponential equations,1252

if they are the same base, you simply set the exponents equal to each other,1259

because this property tells us that that must be the case.1268

If the bases are not the same, try to make them the same: that is going to be your first approach.1272

And then, once you have written them as the same base, then you go ahead and solve by setting the exponents equal to each other.1278

Now, let's look at exponential inequalities: exponential inequalities involve exponential functions.1289

We just talked about exponential equations, and this is a similar situation, except we are working with an inequality, not an equation1296

(greater than, less than, greater than or equal to, less than or equal to).1303

And there are some properties that we can use to help us solve these.1307

If b > 1, then bx > by if and only if x > y.1311

And this makes sense: if I have the same base (these are the same),1320

the only way that this left half is going to be greater than the right half1325

is if these exponents hold the same relationship, where x is greater than y.1330

And this could be greater than or equal to, or less than, or less than or equal to.1335

So, bx < by if and only if x < y--the same idea.1339

If the bases are the same, and this on the left is less than the one on the right,1346

then that relationship, x < y, must be holding up.1350

We are going to use this property to solve inequalities.1355

For example, 4x + 3 > 42: I know that the bases are equal, so that x + 3 must be greater than 2.1357

So, I just solve, subtracting 3 from both sides to get x > -1.1371

Again, if you are trying to work with a situation where you are solving an exponential equation or inequality,1378

and the bases aren't the same, see if you can make them the same.1384

56x < 115 well, I know that 52 is 25; if I multiply 25 times 5, I am going to get 125.1389

Therefore, 52 times 5...that is 53...equals 125.1404

I am going to rewrite this as 53; now that the bases are the same, I can say, "OK, 6x is less than 3, so x is less than 1/2."1410

Again, the idea is to get these in the form where the base is the same,1426

and then use the property that the relationship between the exponents has to be maintained,1431

according to the inequality, if the bases are the same.1439

Looking at examples: let's go back to talking about graphing.1446

We are asked to graph this function, f(x) = 3(2x)--graphing an exponential function.1449

x...and we need to find y, so that we can do some graphing.1457

When x is -3, this is going to be 3 times 2 to the -3 power.1467

Again, this is equal to 3 times 1 over 23; this is going to be equal to 3 times...2 times 2 is 4, times 2 is 8; or 3/8.1474

When x is -2, I get 3 times 2-2; 3 times 1/2 squared...2 times 2 is 4, so this gives me 3 times 1/4, or 3/4.1492

-1: 3 times 2-1 equals 3 times 1/21...remember that this -1 tells us that we need to take 1/the first power.1509

So, that is 3 times 1/2, which is 3/2.1530

0 gives me 3 times 20, or 3 times 1, which equals 3.1537

Using 1 as the x-value gives me 3 times 21, or 3 times 2, which equals 6.1546

Using 2 gives me 3 times 2 squared, which is 3 times 4, or 12.1555

And then, one more: 3 times 2 cubed equals 3 times 8, or 24.1562

Plotting these values: when x is -3, y is 3/8 (just a little bit greater than 0); when x is -2, y gets a little bit bigger; it is 3/4.1570

When x is -1, then y becomes 3/2, or 1 and 1/2; 3/2 is going to be about here.1596

When x is 0, y is 3; when x is 1, y is up here at 6, rising rapidly; when x is 2, y will be all the way up here at 12.1607

I have enough points to form my curve; and I see the typical properties of exponential functions.1620

Remember that the y-intercept is going to be at (0,a): in this case, a = 3, so my y-intercept is going to be at (0,3), and that is exactly what I see.1629

And I can see that over here in the table of values.1646

I also see that the x-axis is an asymptote, and that the graph is approaching, but never reaching, the x-axis.1648

And I also can see that, as x becomes large in the positive direction, the value of y rapidly increases.1659

This is a graph of a typical exponential function.1667

Example 2: Does this function represent exponential growth or decay?1671

Recall that, if a is greater than 0, then we can look at a function in the form f(x) = abx and evaluate b.1677

If the base is greater than 1, then we have exponential growth.1691

If b is greater than 0, but less than 1, we have exponential decay.1698

But the caveat is that it has to be in the standard form; and I have an equation here that is not in standard form.1704

But I can use my rule that a-n equals 1/an.1711

Therefore, I am going to rewrite this as f(x) = 4((1/5)x).1716

So, I just took the reciprocal of the base and rewrote it.1727

And this is talking about a different 'a' than I am talking about here, with the coefficient: here, the coefficient is a.1735

All right, now that I have it in this form, I can evaluate; and I see that my coefficient is greater than 0, and that b is actually between 0 and 1.1742

Therefore, this is exponential decay.1751

I also could have looked back at the original and recalled that, if b is greater than 1,1756

and then you have a negative exponent, that would also give me a clue that this is exponential decay.1760

But one way to go about it is just to put it in the standard form, and then evaluate the value of the base.1765

All right, in Example 3, I have an exponential equation; and recall that, if the bases are the same,1773

then the exponents must be equal in order for the equation to be true.1779

The problem is that the bases aren't the same; so I need to get them to be the same.1787

Looking at these, they are both even; so I am going to try 2.1793

And we know that 2 cubed is 8; so let's start from there: 8 times 2 is 16; 16 times 2 is 32, so this gives me 16, 32...1796

times 2 is 64; times 2 is 128; so 2 to the third, fourth, fifth, sixth, seventh, equals 128.1817

If 27 is 128, then 2-7 = 1/128.1831

So, I have this written as a power of 2; let's look at the right.1839

I know that 27 is 128; 128 times 2 is 256; 27 times 21...1844

I add the exponents, so this is actually 28 = 256.1856

Now, I can write this equation using the base of 2.1863

On the left, I am going to have 2-7 raised to the 2x power, equals...1867

and then, on the left, 28 times 3x - 1.1875

So, this is going to give me 2 to the -7 times 2x (is going to be -14x), equals 2 to the 8, times 3x - 1.1883

Now, I have it in this form, where the bases are the same, and I have an x and a y.1897

Therefore, -14x = 8(3x - 1); and then I am just left with a simple linear equation.1901

-14x = 24x - 8; I am going to add 14x to both sides to get 38x (I added 14 to both sides).1911

At the same time, I am going to add 8 to both sides to get it over here.1926

I could have done it the other way; then I would have had a negative and a negative, and then divided or multiplied by -1.1930

Anyway, I am going to go ahead and divide both sides by 38; and this is going to give me x = 8/38.1936

I can do a little simplification, because I have a common factor of 2.1949

I can cancel that out to get 2/19.1952

The hardest part for solving this was simply getting them into the same base.1956

And I was able to do that because 2-7 is 1/128, and 28 is 256.1961

Once I did that, it was simply a matter of putting the exponents equal and solving a linear equation.1967

Here we have an exponential inequality: again, the first step is to get the bases the same,1976

because I know that, if I have a base raised to a certain power, and it is less than that same base1981

raised to another power, that the relationship between these two exponents has to hold up.1992

Well, this time (last time I had some even numbers; I tried 2, hoping I could find a base there2001

that I could easily make them into a common base, and I did), since I see a 27, I am going to work with 3's.2008

So, I know that 3 cubed is 27; therefore, 3-3 = 1/27.2015

243...let's look at that: if 33 is 27, I multiply that times 3; that is going to give me 81.2028

If I multiply that by 3, fortunately, I am going to get 243.2037

So, this is 33 times 3; that is 34, times 3, so this gives me 35 = 243.2044

Let's go ahead and work on this, then: rewrite this as, instead of 243...I am going to write it right here as 35,2052

raised to 4x - 3, is less than...instead of 1/27, I am going to write this as 3-3, all to 6x - 4.2063

So, this is going to give me 3 to the 5, times 4x - 3, is less than 3 to the -3 times 6x - 4.2074

Now, I have the same bases: I can just look at the relationship between the exponents:2087

that 5(4x - 3) is less than -3(6x - 4), and solve this linear equation.2092

This gives me 20x - 15 < -18x + 12; that is 38x - 15 (I added 18 to both sides) < 12.2101

So, 38x (I am going to add 15 to both sides to get this) < 27.2116

And then, I am just going to divide both sides by 38 to get x < 27/38.2123

Again, the most difficult step was just getting these to be written as the same base.2129

And I was able to do that by using 3 as the base.2135

And then, I had equivalent bases; I did a little simplifying, set them equal, and solved this linear inequality.2140

That concludes this lesson of Educator.com on exponential equations and inequalities; thanks for visiting!2150

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