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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Slope

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Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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
VII. 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
VIII. 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
IX. 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
X. 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
XI. 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 (14)

1 answer

Last reply by: Dr Carleen Eaton
Sat Jul 12, 2014 1:33 PM

Post by Matthew Johnston on June 26, 2014

You could just switch the numerator and denominator and make it negative right?

1 answer

Last reply by: Dr Carleen Eaton
Sat Jul 12, 2014 1:37 PM

Post by abdulaziz bagabas on February 2, 2014

when asked to graph poiny passing through  (1,-3) and parallel to the graph of 2x-3y, you should have put the the point on 3 not on 4, or if i am not right can anybody correct me

0 answers

Post by abdulaziz bagabas on February 2, 2014

i really was  benefited and entertained by this useful lecture and i would like to thank educator.com team and DR.CARLEEN EATON for those  huge efforts to presenting useful lectures.

1 answer

Last reply by: Dr Carleen Eaton
Sat Apr 27, 2013 1:27 PM

Post by Iris Kim on April 6, 2013

You said that vertical lines and lines with the same slope can be parallel. What about two horizontal lines? Can't they be parallel?

1 answer

Last reply by: Dr Carleen Eaton
Fri Oct 26, 2012 11:07 PM

Post by michelle morris on October 24, 2012

Were doing that in school but were also doing standard form w/t form slopes.
also the slope intercept form of an equation , and then writing the slope intercept form given the slope and y-intercept. I dont understand that

2 answers

Last reply by: Justin Wright
Tue Sep 18, 2012 6:51 PM

Post by Sayaka Carpenter on July 14, 2012

wouldn't two horizontal lines be considered parallel as well? you only mentioned vertical and lines with the same slope.

1 answer

Last reply by: Dr Carleen Eaton
Tue Jul 3, 2012 7:15 PM

Post by Vita Kravchenya on June 25, 2012

How many feet in the vertical direction correspond to 10ft in the horizontal direction
one side of triangular 3 ft and other 12 feet

Slope

  • The slope of a straight line is a measure of how steep the line is. Lines that increase as x increases have positive slopes, those that decrease as x increases have negative slopes.
  • Slope is defined to be the change in the y-coordinates of two points divided by the change in the x-coordinates of those points.
  • The slope is the same no matter which two points you pick to compute it.
  • A vertical line has no slope, a horizontal line has a slope of 0.
  • Non-vertical parallel lines have the same slope, non-vertical perpendicular lines have slopes whose product is – 1.

Slope

Find the slope of the line passing through the points ( − 3, − 1) and (1,2)
  • Use the slope formula m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)]
  • m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)] = [(2 − ( − 1))/(1 − ( − 3))]
  • Simplify
m = [(2 − ( − 1))/(1 − ( − 3))] = [(2 + 1)/(1 + 3)] = [3/4]
Find the slope of the line passing through the points ( − 3,3) and ( − 1, − 1)
  • Use the slope formula m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)]
  • m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)] = [( − 1 − (3))/( − 1 − ( − 3))]
  • Simplify
m = [( − 1 − (3))/( − 1 − ( − 3))] = [( − 4)/( − 1 + 3)] = [( − 4)/2] = − 2
Find the slope of the line passing through the points (2,3) and (5,2)
  • Use the slope formula m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)]
  • m = [Dy/Dx] = [(y2 − y1)/(x2 − x1)] = [(2 − (3))/(5 − (2))]
  • Simplify
m = [(2 − (3))/(5 − (2))] = [( − 1)/3] = − [1/3]
Graph the line passing through ( − 3,4) with slope m = [1/2]
  • Step 1:Plot the point ( − 3,4)
  • Step 2: From ( − 3,4) use the fact that slope = [change in y /change in x] = [rise/run] = [1/2] to plot the next point
  • Go Up 1 and Right 2.
  • Step 3 : Repeat again. From previous point, use slope to go Up 1 and Right 2.
  • Step 4 : Draw the line
Graph the line passing through ( − 2, − 3) with slope m = 3
  • Step 1:Plot the point ( − 2, − 3)
  • Step 2: From ( − 2, − 3) use the fact that slope = [change in y /change in x] = [rise/run] = [3/1] to plot the next point
  • Go Up 3 and Right 1.
  • Step 3:Repeat again. From previous point, use slope to go Up 3 and Right 1.
  • Step 4:Draw the line
Graph the line passing through (4, − 3) with slope m = − 2
  • Step 1:Plot the point (4, − 3)
  • Step 2: From (4, − 3) use the fact that slope = [change in y /change in x] = [rise/run] = [( − 2)/1] to plot the next point
  • Go Down 2 and Right 1.
  • Step 3:Repeat again. From previous point, use slope to go Down 2 and Right 1.
  • Step 4:Draw the line
Graph the line passing through (3,4) parallel to the graph of 3x + 4y = 12.
  • Recall that parallel lines have the same slope. Therefore, find the slope of the line 3x + 4y = 12
  • and use the slope to plot additional points to graph the line.
  • Find x - and y - interceps to use as points to find the slope m.
  •  X-InterceptY-Intercept
     set y=0set x=0
    3x + 4y = 12  
  •  X-InterceptY-Intercept
     set y=0set x=0
    3x + 4y = 12(4,0)(0,3)
  • Use the slope formula to find m
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [(3 − 0)/(0 − 4)] = − [3/4]
  • Now that you know what the slope is, graph the next point starting at (3,4)
  • Go Down 3 ,Right 4.
  • Graph the line. Notice how it is indeed parallel to the other line.
Graph the line passing through (1,1) parallel to the graph of − 5x + 3y = 15.
  • Recall that parallel lines have the same slope. Therefore, find the slope of the line − 5x + 3y = 15
  • and use the slope to plot additional points to graph the line.
  • Find x - and y - interceps to use as points to find the slope m.
  •  X-InterceptY-Intercept
     set y=0set x=0
    -5x+3y = 15
  •  X-InterceptY-Intercept
     set y=0set x=0
    -5x+3y = 15(-3,0)(0,5)
  • Use the slope formula to find m
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [(5 − 0)/(0 − ( − 3))] = [5/3]
  • Now that you know what the slope is, graph the next point starting at (1,1)
  • Go Up 5 ,Right 3.
  • Graph the line. Notice how it is indeed parallel to the other line.
Graph the line passing through (3,4) and perpendicular to the graph of 3x + 4y = 12.
  • Recall that perpendicular lines have the property that the product of their slopes = − 1.
  • Therefore, find the slope of the line 3x + 4y = 12
  • and take the negative reciprocal to find the slope of the perpendicular line.
  • Use the slope to plot additional points to find the graph.
  • Find x - and y - interceps to use as points to find the slope m.
  •  X-InterceptY-Intercept
     set y=0set x=0
    3x + 4y = 12
  •  X-InterceptY-Intercept
     set y=0set x=0
    3x + 4y = 12(4,0)(0,3)
  • Use the slope formula to find m
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [(3 − 0)/(0 − 4)] = − [3/4]
  • Now that you know what the slope of the first line is, take the negative reciprocal to find the slope of the
  • perpendicular line.
  • m2 = − [1/(m1)] = − [1/( − [3/4])] = [4/3]
  • Use m2, the slope of the perpendicular line to graph the next point starting at (3,4)
  • Go Up 4 ,Right 3.
  • Graph the line. Notice how it is indeed perpendicular to the other line.
Graph the line passing through (1,1) and perpendicular to the graph of − 5x + 3y = 15.
  • Recall that perpendicular lines have the property that the product of their slopes = − 1.
  • Therefore, find the slope of the line − 5x + 3y = 15
  • and take the negative reciprocal to find the slope of the perpendicular line.
  • Use the slope to plot additional points to find the graph.
  • Find x - and y - interceps to use as points to find the slope m.
  •  X-InterceptY-Intercept
     set y=0set x=0
    -5x + 3y = 15
  •  X-InterceptY-Intercept
     set y=0set x=0
    -5x + 3y = 15(-3,0)(0,5)
  • Use the slope formula to find m
  • m = [(y2 − y1)/(x2 − x1)]
  • m = [(5 − 0)/(0 − ( − 3))] = [5/3]
  • Now that you know what the slope of the first line is, take the negative reciprocal to find the slope of the
  • perpendicular line.
  • m2 = − [1/(m1)] = − [1/([5/3])] = − [3/5]
  • Use m2, the slope of the perpendicular line to graph the next point starting at (1,1).
  • Go Down 3, Right 5.
  • Graph the line. Notice how it is indeed perpendicular to the other line.

*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

Slope

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
  • Definition of Slope 0:07
    • Change in Y / Change in X
    • Example: Slope of Graph
  • Interpretation of Slope 3:07
    • Horizontal Line (0 Slope)
    • Vertical Line (Undefined Slope)
    • Rises to Right (Positive Slope)
    • Falls to Right (Negative Slope)
  • Parallel Lines 7:18
    • Example: Not Vertical
    • Example: Vertical
  • Perpendicular Lines 8:31
    • Example: Perpendicular
  • 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

Transcription: Slope

Welcome to Educator.com.0000

In today's lesson, we will be discussing slope.0002

Recall the definition of slope: the slope, m, of the line passing through a point (x1,y1),0007

and another point (x2,y2), is given by:0015

the slope equals the change in y, over the change in x (the change in the y coordinates, over the change in the x coordinates).0020

Therefore, if you have any two points on a line, you can use that to find the slope.0030

For example, imagine that you are given the graph of a line that looks like this, and you are asked to find the slope.0037

Well, I have a point right here at x is 0, so my (x1,y1) point is going to be x is 0, y is 2.0053

I have a second point over here, and I am going to call this (x2,y2); and here, x is -2, y is 0.0065

So, I am using the intercepts; but I could have used any other point on this line.0076

OK, an important point is that I could have assigned either one of these as...0081

I could have said this is (x1,y1), and this is (x2,y2).0086

It doesn't matter, as long as you are consistent (I can't say (x1,y2), (x2,y1)).0090

As long as I am consistent, and I follow that consistency when I am subtracting, I am fine.0096

So, I arbitrarily assigned this as (x1,y1), and this point as (x2,y2).0103

So, I found two points on the line; now, my y2 value is 0, and I am going to subtract my y1 value (which is 2) from that.0107

The x2 value is -2; my x1 value is 0, so minus 0; so here, the slope equals -2 over -2, or 1.0120

So, I could have used any two points on this line, and found the difference between those x and y coordinates--the change in y over the change in x.0138

So, what this is telling me is that...we say m = 1, but what we really mean is m = 1/1; think about it that way, and say that.0147

For every increase in y by 1, I am going to increase x by 1; increase y by 1, increase x by 1.0156

So, I may be given a line and asked to find the slope; or I may directly be given a set of points and asked to find the slope.0167

So, if I were given two points on the line, but not given the line itself, or a graph, or anything, I could easily find the slope.0175

This is telling us the vertical change over the horizontal change.0182

OK, some cases to keep in mind when you are dealing with slope: a horizontal line has a slope of 0.0187

And let's think about why this is so: consider a horizontal line up here at, say, 4; OK, so this is 4.0198

And if I look at any point--for example, right here--here, x is 1; y is 4; and then maybe I look over here at this point; here x is 3; y is still 4.0212

Now, recall that the slope formula is the change in y over the change in x.0229

So, I have my two points; and I am going to call this (x1,y1), (x2,y2).0239

Again, I could have done it the other way around.0246

All right, so I have y2 (that is 4), minus y1;0249

and then I have my x2 (which is 3), minus x1 (which is 1).0255

It actually doesn't matter what I have down here, because, since I have a horizontal line, the y-values everywhere here are going to equal 4.0261

So, 4 - 4 is going to be 0; therefore, the slope is 0.0273

So, a horizontal line has a slope of 0, because the y-values are the same at every point, so the difference in y is going to be 0.0281

OK, vertical lines are also a special case: and they have an undefined slope.0291

For example, let's look at a vertical line right here at x = -3.0298

OK, if I look right here at (I could pick any point)...I am going to pick this point, x is -3; y is 1.0306

Then, I will pick another point right here; let's say x is -3; y is -2.0315

So, I have my (x1,y1), and my (x2,y2).0322

Again, slope is change in y over change in x; change in y is y2 (that is -2), minus 1, over change in x (-3 - -3).0330

This is going to give me -3 over...-3 minus -3 is -3 plus 3; that is over 0.0349

And this is not allowed; this is undefined.0359

We say that a vertical line has an undefined slope because, since the change in x is 0,0367

because x is the same at every point, what you end up with is a denominator that is 0, and that is undefined; that is not allowed.0372

So, for a horizontal line, the slope is 0; for a vertical line, slope is undefined.0379

A couple of other things to keep in mind regarding slope: one is that in a line that rises to the right (such as this one), m is a positive number.0393

So, it is increasing to the right; the values are increasing.0407

If you have a line that falls to the right, such as this one, here the slope is positive; in this case, the slope is negative.0413

Rises to the right--slope is positive; falls to the right--slope is negative.0428

A horizontal line has a slope of 0; a vertical line has an undefined slope.0433

Parallel lines: two lines are parallel if and only if they are vertical or have the same slope.0439

Let's first check out the case where the lines are not vertical, but they are still parallel.0446

If I have two lines that are parallel, it is going to look like this, for example.0451

These are going to have the same slope; so if this is my line 1 and line 2, m1 will equal m2--the slopes will be equal.0459

And that is because they are changing at the same rate--these two lines are changing at the same rate, so they never intersect.0468

Now, we set vertical lines aside; we put them separately; we don't say they have the same slope.0476

And remember that the reason is because a vertical line has an undefined slope.0483

So, I can't say these two have the same slope, because their slope is undefined.0492

However, we recognize that they are still parallel; so we say that two lines are parallel0496

if they are vertical (that is a separate case), or if they have the same slope; but both of these represent parallel lines.0501

Perpendicular lines: two non-vertical lines are perpendicular if and only if the product of their slopes is -1.0512

So, "perpendicular" would tell me that these two lines intersect at right angles.0522

And if this is my line 1 and line 2, and then I have a slope m1 and m2, what this is stating0535

is that the product of m1 and m2 is -1.0544

Let's look at an example: let's say I have an m1 that is equal to 4, and I say that it is perpendicular to a line L2.0552

So, this is line 1, and this is line 2--the perpendicular line.0569

I can find the slope of the perpendicular line, because I know that m1 times m20575

is -1 (since these are perpendicular lines), and I am given m1.0584

So, 4 times m2 equals -1; therefore, the perpendicular line would have a slope of -1/4.0591

So, the perpendicular line has -1/4 as the slope, and that is the negative reciprocal of m1.0606

Knowing the relationship between parallel lines and the slopes of parallel lines and the slopes of perpendicular lines0618

can be helpful in answering problems and graphing lines; and we will see that right now with the examples.0624

OK, in this first example, find the slope of the line passing through the points (-9,-7) and (-6,-3).0633

Well, the definition of slope is the change in y over the change in x.0643

And I could choose to assign this either way; but I am just going to go ahead0651

and say this is (x1,y1), and this is going to be my (x2,y2).0654

And it doesn't matter which way you assign, as long as you are consistent.0660

So, I could have called this (x2,y2), and it would have been fine; we would have gotten the same answer.0663

OK, y2 is -3, and y1 is -7, so minus -7; over x2 (which is -6), minus x1 (which is -9).0668

Simplifying this: this is -3; a negative and a negative gives me a positive; and that gives me 4/3.0686

So, the slope of a line passing through these points is 4/3, just using my slope formula.0697

OK, Example 2: Graph the line passing through the point (-2,-1) with a slope of -2/3.0705

Well, I am given the slope, and I am given a point on the line.0717

So, let's start out with the point that I am given, which is (-2,-1); that is right here.0722

Remember that the slope is the change in y over the change in x.0730

So, if the change in y is -2, then I am starting out here at -1; I am going to go to -2, -3.0735

For every 2 that y is decreased in value, x is going to increase by 3: 1, 2, 3; so, I am going to end up with this point right here.0744

You could have also looked at this as this, and I could say that for every 2 y is increased, x is decreased by 3: 1, 2, 3, right here.0756

The same thing--you could look at it either way.0771

OK, so now, I am able to graph this line because I had a starting point right here, and I have the slope.0777

So, I know, starting from here, how much the x is going to change and how much the y is going to change.0783

So then, I simply connect these lines to give my graph.0789

And notice that this line is decreasing to the right: and that fits with what I have, which is a negative slope.0800

So, simply plot the point you are given; and then use the change in y over the change in x to find additional points on the line.0806

OK, graph the line passing through (1,-3) and parallel to the graph of this equation.0820

Well, I have my point; x is 1; y is -3; so, I have my starting point.0829

I need my slope; and I am not directly given the slope; however, I am given a parallel line.0835

And recall that parallel lines have the same slope.0843

That means that, if I find the slope of this line, I am going to have the slope of the line that I am looking for.0853

In order to find the slope, I need to be able to find the change in y over the change in x; so I need two points on this line.0860

So, the slope is (y2-y1)/(x2-x1).0870

Thinking of an easy way to find a couple points on this line, I am going to use the intercept method.0876

And I could find these points and graph, or I could just find these points and plug them into my slope equation.0883

First, I am going to let x equal 0 to find the y-intercept.0890

OK, so this gives me -3y = 6, or y = -2.0901

When x is 0, y is -2; when y is 0, let's go ahead and plug that in; 2x - 3(0) = 6, so this is 2x - 0 = 6; 2x = 6.0913

Divide both sides by two; that gives me x = 3.0933

OK, so when x is 0, y is -2; when x is 3, y is 0.0937

Now, I can find the slope of the sign, because this can be my x1;0946

this can be my y1; this can be my x2; this can be my y2; so let me find the slope.0953

This is y2, which is 0, minus y1, which is -2.0959

This is x2, which is 3, minus 0, which is x1.0967

This is going to give me 0 minus -2 (0 plus 2), over 3 minus 0; so the slope is 2/3.0975

Therefore, the slope of this line is 2/3, and since parallel lines have the same slope, the slope of the line I am looking for is also 2/3.0988

Now, I was given this point, (1,-3); now I know that the vertical change is going to be an increase in y by 2;1003

and there is going to be a change in x--an increase by 3--1, 2, 3; so I have another point.1011

Now that I have this second point, I can go ahead and graph this line.1019

And this is the line I was asked to graph, so I have completed what I have been asked to do.1027

But just to look at the idea that this is indeed a parallel line, I am going to go ahead and plot this other line, as well.1032

This is the line we were asked to find; this other line--when x is 0, y is -2; when x is 3, y is 0.1038

And you can plot that out and see that, visually, this does look like a parallel line.1047

And these two are changing at the same rate: the slope of both of these is 2/3.1052

I was able to graph the line passing through this point and parallel to the graph of this line1059

by finding the slope of this line, and then realizing that my parallel line is going to have the same slope, which is 2/3.1068

Example 4: Graph the line passing through the point (-2,1) and perpendicular to the graph of this equation: 3x + 4y = 12.1077

OK, I am given a point; and let's see, let's call this -2, -4, -6, -2, -4, -6, 2, 4, 6, 8; OK.1090

-2 would be right here; and then 1 would be right about there; so that is my point.1110

But I also need the slope: recall that the slopes of perpendicular lines--the product of those slopes is equal to -1.1116

So, the product of perpendicular lines' m1 and m2 (I have two perpendicular lines;1127

the slope of one is m1, and the other is m2)--their product is -1.1133

So, I am going to call this first line line 1, and its slope is going to be m1; I am going to find m1.1137

My line I am going to call line 2; and its slope is going to be m2.1147

I need to find the slope of this line, recalling that slope is the change in y over the change in x.1153

So, in order to find this change, I need to find two points on this line.1160

I am going to go for something easy, so I am going to let x equal 0, and I am going to plug that in here.1167

I am just going to find the intercepts as my two points; but I could have found any two points on that line to get the slope.1172

This is going to give me y = 3; when x is 0, y is 3.1185

For my second point, I am going to let y equal 0; that is 3x + 4(0) = 12; so, if 3x = 12, x = 4.1189

So now, I have two points; two points means I can find the slope.1206

I am going to say that this is my (x1,y1), (x2,y2), so I can keep track of everything.1210

y2 is 0; y1 is 3; so 0 - 3, over x2 (which is 4), minus 0.1219

OK, I have found m1: I have found the slope of this line.1233

And I know that the line I am looking for, which is going to have a slope m2, is perpendicular to this line.1239

So, I am going to go right here and say that m1...I know that is equal to -3/4; and I am looking for m2.1246

m1 times m2 equals -1; so now, all I have to do is substitute this in.1258

-3/4 times m2 equals -1; I am going to multiply both sides by -4/3 to move this over to the right.1265

And that is going to give me -1, times -4/3, or m2 = 4/3.1277

OK, now I found this slope, and I need to graph the line.1285

I have my starting point here at (-2,1); and the change in y is going to be 4, so that is going to be 1, 2, 3, 4,1291

for an increase in x of 3; so that is 1, 2, 3, right there.1300

Increase in y by 4: 1, 2, 3, 4; increase x by 3: 1, 2, 3--so right about there; OK.1308

I can go ahead and plot this line out.1325

Reviewing what we did: we were given a point that the line passes through, and we needed the slope in order to graph it.1329

We were told that this line is perpendicular to the line described by this equation.1335

Therefore, we found the slope of this line; and we did that by finding two points in the line (the x- and y-intercepts) and plugging that into the slope formula.1340

Once I found the slope of this perpendicular line, I wanted to find the slope of my line.1351

And I did that by recalling that the product of the slopes of perpendicular lines equals -1.1356

So, -3/4 times the slope of the perpendicular line is -1.1363

I figured out that the line I am looking for has a slope of 4/3, and then I just took my point that I was given and increased y by 4, and then x by 3.1368

That concludes this lecture about slope on Educator.com.1382

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