INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Analyzing the Graphs of Quadratic 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 (11)

0 answers

Post by William Zhang on May 5, 2020

Where is the x squared in 25:11?

0 answers

Post by William Zhang on May 4, 2020

At 13:04 how is 8+9=15??

1 answer

Last reply by: John Stedge
Mon Jun 4, 2018 3:36 PM

Post by John Stedge on June 4, 2018

what if a=0

1 answer

Last reply by: Dr Carleen Eaton
Wed May 1, 2013 8:00 PM

Post by Norman Cervantes on April 30, 2013

Great job explaining the lessons Dr. Eaton. you make learning extremely easy. Thank you.

1 answer

Last reply by: Dr Carleen Eaton
Thu May 10, 2012 10:02 PM

Post by ALI SAAD on May 8, 2012

how is 8+9 = 15 ???

0 answers

Post by julius mogyorossy on March 27, 2012

I guess my formula does not work for fractions, I wondered about that, I forgot that Grant showed us you can have a perfect square with fractions.

1 answer

Last reply by: Dr Carleen Eaton
Sun Jan 29, 2012 4:42 PM

Post by Edmund Mercado on January 28, 2012

Dr. Eaton:
Wonderful, fast-paced delivery in only 30 minutes. Thanks!

Analyzing the Graphs of Quadratic Functions

  • Know and understand the vertex form of a quadratic function.
  • The coefficient of the quadratic term determines the direction that the graph opens and the shape of the parabola.
  • The values of h and k in the vertex form determine how the graph is translated vertically and horizontally.
  • Complete the square to put a quadratic function in vertex form. Factor the coefficient of the quadratic term first if it is not 1.

Analyzing the Graphs of Quadratic Functions

Write in vertex form: y = 3x2 + 6x − 9
  • Isolate the x - variable on the right side
  • y + 9 = 3x2 + 6x
  • Since the right side is not in standard form, factor out a 3
  • y + 9 = 3(x2 + 2x)
  • Complete the squre by adding [(b2)/4] to both sides. Be careful, when adding to the left,
  • it must be multiplied by 3, the number factored earlier.
  • y + 9 + 3( [(b2)/4] ) = 3(x2 + 2x + [(b2)/4])
  • y + 9 + 3( [(22)/4] ) = 3(x2 + 2x + [(22)/4])
  • y + 9 + 3( [4/4] ) = 3(x2 + 2x + [4/4])
  • y + 9 + 3 = 3(x2 + 2x + 1)
  • y + 12 = 3(x2 + 2x + 1)
  • Factor the right side using the formula x2 + bx + [(b2)/4] = (x + [b/2])2
  • y + 12 = 3(x + [2/2])2 = 3(x + 1)2
  • Subtract 12 from both sides
y = 3(x + 1)2 − 12
Write in vertex form: y = 3x2 − 9x − 2
  • Isolate the x - variable on the right side
  • y + 2 = 3x2 − 9x
  • Since the right side is not in standard form, factor out a 3
  • y + 2 = 3(x2 − 3x)
  • Complete the squre by adding [(b2)/4] to both sides. Be careful, when adding to the left,
  • it must be multiplied by 3, the number factored earlier.
  • y + 2 + 3( [(b2)/4] ) = 3(x2 − 3x + [(b2)/4])
  • y + 2 + 3( [(( − 3)2)/4] ) = 3(x2 − 3x + [(( − 3)2)/4])
  • y + 2 + 3( [9/4] ) = 3(x2 − 3x + [9/4])
  • y + 2 + [27/4] = 3(x2 − 3x + [9/4])
  • y + [35/4] = 3(x2 − 3x + [9/4])
  • Factor the right side using the formula x2 + bx + [(b2)/4] = (x + [b/2])2
  • y + [35/4] = 3(x + [( − 3)/2])2 = 3(x − [3/2])2
  • Subtract [35/4] from both sides
y = 3(x − [3/2])2 − [35/4]
Write in vertex form: y = x2 − 4x + 9
  • Isolate the x - variable on the right side
  • y − 9 = x2 − 4x
  • Complete the squre by adding [(b2)/4] to both sides.
  • y − 9 + ( [(b2)/4] ) = x2 − 4x + [(b2)/4]
  • y − 9 + ( [(( − 4)2)/4] ) = x2 − 4x + [(( − 4)2)/4]
  • y − 9 + ( [16/4] ) = x2 − 4x + [16/4]
  • y − 9 + 4 = x2 − 4x + 4
  • y − 5 = x2 − 4x + 4
  • Factor the right side using the formula x2 + bx + [(b2)/4] = (x + [b/2])2
  • y − 5 = (x + [( − 4)/2])2 = (x − 2)2
  • Add 5 to both sides
y = (x − 2)2 + 5
Write in vertex form: y = x2 + 2x − 6
  • Isolate the x - variable on the right side
  • y + 6 = x2 + 2x
  • Complete the squre by adding [(b2)/4] to both sides.
  • y + 6 + ( [(b2)/4] ) = x2 + 2x + [(b2)/4]
  • y + 6 + ( [((2)2)/4] ) = x2 + 2x + [((2)2)/4]
  • y + 6 + ( [4/4] ) = x2 + 2x + [4/4]
  • y + 6 + 1 = x2 + 2x + 1
  • y + 7 = x2 + 2x + 1
  • Factor the right side using the formula x2 + bx + [(b2)/4] = (x + [b/2])2
  • y + 7 = (x + [2/2])2 = (x + 1)2
  • Subtract 7 from sides
y = (x + 1)2 − 7
Write in vertex form: y = x2 − 6x − 8
  • Isolate the x - variable on the right side
  • y + 8 = x2 − 6x
  • Complete the squre by adding [(b2)/4] to both sides.
  • y + 8 + ( [(b2)/4] ) = x2 − 6x + [(b2)/4]
  • y + 8 + ( [(( − 6)2)/4] ) = x2 − 6x + [(( − 6)2)/4]
  • y + 8 + ( [36/4] ) = x2 − 6x + [36/4]
  • y + 8 + 9 = x2 − 6x + 9
  • y + 17 = x2 − 6x + 9
  • Factor the right side using the formula x2 + bx + [(b2)/4] = (x + [b/2])2
  • y + 17 = (x + [( − 6)/2])2 = (x − 3)2
  • Subtract 17 from sides
y = (x − 3)2 − 17
Find an equation in Vertex Form y = a(x − h)2 + k for the parabola with vertex at ( − 4, − 2) and passing through ( − 2,0)
  • Use the Vertex Form equation to find a, h and k.
  • Vertex = (h,k) = ( − 4, − 2)
  • h =
  • k =
  • h = − 4
  • k = − 2
  • Substitute h and k into the vertex form.
  • y = a(x − h)2 = a(x − ( − 4))2 − 2 = a(x + 4)2 − 2
  • y = a(x + 4)2 − 2
  • Use the point ( − 2,0) to find a
  • 0 = a( − 2 + 4)2 − 2
  • 0 = a(2)2 − 2
  • 0 = 4a − 2
  • Solve for a
  • a = [1/2]
  • Write equation in vertex form
y = [1/2](x + 4)2 − 2
Find an equation in Vertex Form y = a(x − h)2 + k for the parabola with vertex at ( − 4,3) and passing through (1,8)
  • Use the Vertex Form equation to find a, h and k.
  • Vertex = (h,k) = ( − 4,3)
  • h =
  • k =
  • h = − 4
  • k = 3
  • Substitute h and k into the vertex form.
  • y = a(x − h)2 = a(x − ( − 4))2 + 3 = a(x + 4)2 + 3
  • y = a(x + 4)2 + 3
  • Use the point (1,8) to find a
  • 8 = a(1 + 4)2 + 3
  • 8 = a(5)2 + 3
  • 5 = 25a
  • Solve for a
  • a = [1/5]
  • Write equation in vertex form
y = [1/5](x + 4)2 + 3
Find an equation in Vertex Form y = a(x − h)2 + k for the parabola with vertex at (2,5) and passing through (3,11)
  • Use the Vertex Form equation to find a, h and k.
  • Vertex = (h,k) = (2,5)
  • h =
  • k =
  • h = 2
  • k = 5
  • Substitute h and k into the vertex form.
  • y = a(x − h)2 = a(x − (2))2 + 5 = a(x − 2)2 + 5
  • y = a(x − 2)2 + 5
  • Use the point (3,11) to find a
  • 11 = a(3 − 2)2 + 5
  • 11 = a(1)2 + 5
  • 6 = a
  • Write equation in vertex form
y = 6(x − 2)2 + 5
Find an equation in Vertex Form y = a(x − h)2 + k for the parabola with vertex at (5, − 1) and passing through (2, − 10)
  • Use the Vertex Form equation to find a, h and k.
  • Vertex = (h,k) = (5, − 1)
  • h =
  • k =
  • h = 5
  • k = − 1
  • Substitute h and k into the vertex form.
  • y = a(x − h)2 = a(x − (5))2 − 1 = a(x − 5)2 + − 1
  • y = a(x − 5)2 − 1
  • Use the point (2, − 10) to find a
  • − 10 = a(2 − 5)2 − 1
  • − 10 = 9a − 1
  • − 9 = 9a
  • − 1 = a
  • Write equation in vertex form
y = − (x − 5)2 − 1
Find an equation in Vertex Form y = a(x − h)2 + k for the parabola with vertex at (6,3) and passing through (12, − 9)
  • Use the Vertex Form equation to find a, h and k.
  • Vertex = (h,k) = (6,3)
  • h =
  • k =
  • h = 6
  • k = 3
  • Substitute h and k into the vertex form.
  • y = a(x − h)2 + 3 = a(x − (6))2 + 3 = a(x − 6)2 + 3
  • y = a(x − 6)2 + 3
  • Use the point (12, − 9) to find a
  • − 9 = a(12 − 6)2 + 3
  • − 10 = 36a − 1
  • − 9 = 36a
  • Solve for a
  • − [1/3] = a
  • Write equation in vertex form
y = − [1/3](x − 6)2 + 3

*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

Analyzing the Graphs of Quadratic 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
  • Vertex Form 0:12
    • H and K
    • Axis of Symmetry
    • Vertex
    • Example: Origin
    • Example: k = 2
    • Example: h = 1
  • Significance of Coefficient a 7:13
    • Example: |a| > 1
    • Example: |a| < 1
    • Example: |a| > 0
    • Example: |a| < 0
  • Writing Quadratic Equations in Vertex Form 10:22
    • Standard Form to Vertex Form
    • Example: Standard Form
    • Example: a Term Not 1
  • Example 1: Vertex Form 19:47
  • Example 2: Vertex Form 22:09
  • Example 3: Vertex Form 24:32
  • Example 4: Vertex Form 28:23

Transcription: Analyzing the Graphs of Quadratic Functions

Welcome to Educator.com.0000

In today's lesson, we will be working more on quadratic equations and functions.0002

And we are going to start out by analyzing the graphs of quadratic functions.0007

Recall: in previous lessons, we discussed quadratic functions in standard form.0012

Today, I am going to introduce a new form, which is the vertex form of a quadratic function.0019

And this is a very useful form of the function, and it is given as y = a, times (x - h) squared, plus k.0024

h and k are the vertex of the parabola; and x = h is the line that is the axis of symmetry.0033

And as you will recall, the vertex is the maximum point for a downward-opening parabola;0041

and it is the minimum point for an upward-opening parabola.0049

So, by having a function in this form, we already have information about what the graph is going to look like.0053

So, let's start out with just a very simple function, y = x2.0059

OK, so looking at it in this form, we have just an a that is 1, and h and k are 0.0068

So, the vertex is going to be right here; so h equals 0 and k equals 0, so the vertex is (0,0).0076

Now, drawing some points, x and y: when x is -1, y is 1; when x is 1, y is 1; when x is 2, y is 4; and when x is -2, y is also 4.0087

So, it is a pretty simple graph.0106

Now, by starting out with this graph, we can look at what happens--the significance of h and k on the shape of the graph.0114

So, we are starting out with just this graph y = x2.0122

Now, let's look at a situation where y = x2 + 2.0131

All right, so now, I still have an h that equals 0; but here, k equals 2; therefore, the vertex is at (0,2).0142

So, my vertex is right here; now let's look at some points, x and y.0155

When x is -1, then x2 will be 1, plus 2--that gives me 3.0166

When x is 1, again, y is 3; when x is 2, y is 6; and when x is -2, that is -2 squared is 4, plus 2 is 6.0175

So, look at what happens with this graph: when x is 1, y is 3; when x is -1, y is 3; when x is 2, y is going to be up right about here at 6, and right here.0188

OK, so as you can see, the graph here is shifted starting right up 2; so, the vertex is going to be right here;0209

and when x is 1, y is 3; when x is 2...there we go.0224

So, the vertex is right here; OK, so what this is going to give me is a graph that is shifted up by 2.0231

So, you see what k does to the graph: it is the same basic graph, but it is shifted up by 2.0244

If k were to be -2, then the graph would be shifted down by 2.0253

Now, h also has an effect on the graph; so let's look at what h could do to the graph.0257

Let's take the example y = (x - 1)2.0267

Here, I have h = 1, k = 0; so the vertex is at (1,0); now let's find some points.0278

The vertex is over here at (1,0): some points x and y for this function: when x is -1, then that is going to give me -1 and -1 is -2, squared is 4.0293

When x is 0, 0 minus 1 is -1; squared is going to give me 1.0314

I already have the vertex at (1,0), so let's pick 2: 2 - 1 is 1, squared is 1.0320

One more point: 3 - 1 is 2, squared is 4; that is going to give me (-1,4), (0,1), (1,0), which is the vertex, (2,1), and then finally (3,4).0328

So, I have this graph right here; this graph is y = (x - 1)2.0352

You can see, what happened is: the graph here is shifted to the right by 1.0368

h shifts to the right or left; if this had been -1, the graph would have shifted over to the left.0378

So again, k is going to shift the graph either up or down; h is going to shift to the left or right.0386

And we will talk about a in a minute.0394

So again, vertex form is very useful, because it gives you a lot of information.0396

It gives you the vertex, which is (h,k), and it also gives you the axis of symmetry.0400

And recall that the axis of symmetry is the line at x = h, which bisects this parabola into two symmetrical halves.0407

For example, if I had a parabola here, a downward-facing one, and the vertex is right here, the axis of symmetry would be right here0417

at the line x = h, where the vertex is some point (h,k).0428

Now, let's talk more about the coefficient a; what the coefficient a tells you is how steep or flat the parabola is.0434

For example, if I had one parabola, and let's say it had a = 1, and it looked like this, for example:0443

and then I had a very similar parabola, but this time with a = 2; well, as it says here,0455

if the absolute value of a is greater than 1, the parabola is narrower.0463

I have a = 1; and if I have a = 2, it is going to be a narrower parabola, meaning that, for every change in x, y changes more dramatically.0479

So, it goes up more steeply--each section of the parabola.0493

Now, if you have an absolute value of a that is less than 1 (that is a fraction), the parabola is going to be wider.0497

So, a = 1/2 might look something like this; for every change in x, the change in y is less dramatic.0507

OK, so the first thing that the absolute value of a tells us is how narrow or wide the parabola is.0517

A larger number is going to be narrower; a smaller number, especially a fraction, is going to have a much flatter, wider parabola.0524

The second thing that a tells us is which direction the parabola opens.0531

All of these parabolas that I showed you opened upward, and that means that a is positive.0535

If a is greater than 0, the parabola opens upward; if a is less than 0, the parabola opens downward.0541

And here, I have drawn a pretty wide parabola, so it would have a fairly small value of a; and it would have a negative value.0553

The absolute value of a would be small, and the value of a itself would actually be negative.0561

Here, all of these have an absolute value of a that is...these two are larger; this one is smaller; but all of them have a value of a that is positive.0569

So they open upward; so if a is positive, the parabola opens upward; if a is negative, it opens downward.0585

If the absolute value is larger, you have a narrower parabola; if the absolute value of a is smaller, you have a wider, flatter parabola.0592

Between what we just learned about h and k and the vertex,0601

and what we learned here about a, you can get a lot of information0606

just by looking at the equation about the shape of the graph of the parabola0609

before you have even found points beyond the vertex.0615

OK, now since this is an important form of the equation, you need to know how to write it in vertex form.0620

And you may actually be given the quadratic equation in standard form and be asked to write it in vertex form.0627

And the way you go about that, if you are given a quadratic equation in standard form, is to complete the square.0635

And that will allow you to write the equation in vertex form.0643

Now, it says here that, if a does not equal 1, then you need to factor out a before you go on to complete the square.0647

So, we will deal with that in a minute; let's start out with a simpler case where y equals x2 + 6x - 8.0655

So, I have standard form; and I am going to go ahead and write it in vertex form.0679

Let's just recall vertex form: y - a(x - h)2 + k.0687

So, I am starting out with standard form; and when I complete the square, the first thing I need to do is isolate the variable terms on the right.0699

Here, I have my x variable terms, but I also have a constant here.0715

So, what I am going to do is add 8 to both sides to get this.0720

Now, I have my x variable terms isolated on the right.0724

Isolate the x; I should specify x variable terms on the right; the y stays on the left.0729

OK, next complete the square; and recall that, in order to complete the square, you need to add b2/4 to both sides.0738

So, here let's figure out what this would give me...y + 8, b is 6, so that is going to be plus 62/4 = x2 6x + 62/4.0763

Doing some simplification: 62 is 36, divided by 4 is simply 9.0779

OK, I have completed the square on the right; I am going to do some simplification here, because this is actually y + 15.0787

And this is not quite in vertex form, because if I look up here, vertex form would have y isolated on the left.0794

So, I am going to actually change this, and I am going to subtract 15 from both sides.0800

And this is going to give me, let's see, -8.0808

Actually, one more step: before we do that, I have completed the square; so I want to go ahead0819

and write this out closer to vertex form before I even go on.0827

And this is x2 + 6x + 9, and that is a perfect square, so I am going to write it in this form.0831

Then, I am going to go ahead and move my 15 over.0838

OK, so I have written this in vertex form; and I accomplished that by first isolating the x variable terms on the right,0842

then adding b2/4 to both sides; I did that, and I ended up with this right here.0853

And I see, on the right, I have a perfect square; so I can rewrite that as (x + 3)2.0861

And then, finally, this is vertex form.0867

And although vertex form has a negative here, it is fine to write it like this.0869

If I wanted to, I could have written it as x - -3, but this is actually acceptable, as well.0874

Now, it gets a little bit more complicated if the a term is not equal to 1; so let's go ahead and look at that situation.0881

If I had something like y = 2x2 - 8x + 5,0895

I am first going to isolate the x variable terms on the right (that is going to give me y - 5 = 2x2 - 8x),0904

and then, as it says here, I need to factor a out, since a is not 1, before I complete the square.0913

So, this is going to give me 2x2 - 4x; now, I complete the square by adding b2/4 to both sides.0922

But you have to be careful on the left; let's see what happens.0932

This is going to give me x2 - 4x, and then b2/4 is actually going to be (-4)2/4.0936

Now, I can't just add that to the right, because in reality, what I am adding is not just (-4)2/4.0945

But it is actually (-4)2/4, times 2; that is what I need to add to the left.0952

Otherwise, the equation won't be balanced.0958

So, let me go ahead and simplify this (-4)2/4; that equals 16/4, which equals 4.0960

Now, what I am really adding to both sides is actually 4 times 2; 4 times 2 equals 8.0967

This is 2 times 4; so that is what I need to add to this side, as well.0977

So, this is y - 5 + 8; it is going to give me y + 3 = 2 times x2 - 4x + 4.0982

Now, I am almost there, because I have a perfect square, which is (x - 2)2.0995

So then, I can rewrite this in vertex form.1001

And as you see, it was more complicated, because I started out with a leading coefficient that is other than 1.1008

In that case, I have to first go ahead and isolate my x variables, and then factor that out, so that this becomes 1.1016

Once I have done that, I can complete the square; but I have to be very careful, on the left,1025

that I am not just adding b2/4, but that I am adding b2/4,1029

times what I factored out, to keep the equation balanced.1034

And now, you see, I have vertex form; and here a equals 2; h equals 2; and k equals -3.1038

OK, finally, if you are given the vertex and another point on the graph, you can write the equation of the graph in vertex form.1046

For example, if I am given that the vertex equals (2,11), and I am also told that a point on the graph1055

is at (1,6), then let's look at what I have.1066

Well, I have y, because I have a point on the graph; so this is h; this is k; this is x; this is y.1071

So, I have y; I have x; I have h; and I have k.1078

The only thing that I am missing is a, but since I have everything else, I can solve for a.1083

So, I am going to go ahead and do that: y is 6; 6 equals a times...x is 1, minus h squared, plus k.1088

OK, so this gives me 6 = a, and this is -1 squared plus 11, so 6 equals a times 1, plus 11; 6 = 11 + a; therefore, a equals -5.1101

Now that I have solved for a, I can go ahead and write this in vertex form.1136

I have y = -5(x -...I know that h is 2) + k, which is 11.1141

So again, if I am given vertex form and a point on the graph, then I can write this, substituting in everything I have, and then solving for a.1154

And here, I found that a equals -5; so now, I have h, k, and a, and all I need to write an equation in vertex form is h, k, and a.1165

So, if you are either given the quadratic equation in standard form, or you are given1174

the vertex and a point on the graph, either of those is enough to rewrite the equation in vertex form.1180

OK, practicing this with some examples: here I am given a quadratic equation in standard form, and I am asked to write it in vertex form.1188

Remember that the first step is to isolate the x variable; I am going to do that by subtracting 7 from both sides.1195

OK, so I have y - 7 = x2 -4x; next, I am going to complete the square.1209

And remember, you do that by adding b2/4 to both sides.1216

And here, b is -4; so this is going to be (-4)2/4 = x2 - 4x + (-4)2/4.1223

Simplifying: this gives me 16/4; simplifying further, I get simply 4.1239

OK, so I have -7 + 4; that is going to give me y - 3 = x2 - 4x + 4.1254

And what I have on the right is a perfect square, so I am going to rewrite that as y - 3 = (x - 2)2.1261

One more step to getting this in vertex form is to move the constant over to the right.1279

And recall that vertex form is y = a(x - h)2 + k.1285

So, here I have a = 1; h is 2; and k equals 3; and the vertex is at (2,3).1292

And I accomplished that by isolating the x variables on the right, completing the square1308

by making sure I added b2/4 to both sides, and then once I had that completed,1314

I could rewrite this in this form, the (x - h)2 form, and then moving the constant over to the right.1322

OK, here: find an equation in vertex form for the parabola with vertex at (-4,3), and passing through the point (-2,15).1329

So, first I am looking at what I have and at what I need.1339

Vertex form: y = a(x - h)2 + k.1344

So, to write in vertex form, I need a, h, and k.1348

What do I have? Well, I have the vertex; this is h and k; and I have a point--this is x and y.1354

What I am missing is a; but since I have everything else, I can solve for a.1361

I am going to substitute in here for y; that is 15; I don't know a; I know that x is -2; I know that h is -4; and k is 3.1367

So, solving: a negative and a negative is actually a positive, so this gives me 15 = a...1386

-2 plus 4 is 2, squared; so then, this is 15 =...this is actually...1396

I am down to here; I am going to subtract 3 from both sides to get 4a = 12.1416

Isolate a by dividing both sides by 4 to give me a = 3.1421

So, now that I have a, I can write in vertex form: I have y = 3 times x, minus h (is -4).1428

And I can either write this as x + 4 or x - -4, plus k, which is 3.1437

This is vertex form; and I accomplished that by recognizing that I had everything I needed except for a,1443

substituting in, solving for a, and then going back and writing this in vertex form.1450

And here, a equals 3; h equals -4; and k equals 3 (vertex form).1460

OK, again, I am asked to write the quadratic equation in vertex form.1472

But notice this time: this is with the standard form; a does not equal 1.1478

Recall that I am going to have to do an extra step before completing the square in this situation.1483

So, the first step is to isolate the x variables on the right.1489

Now, the extra step I have to take is to factor out a.1494

Now that I have done that, I can complete the square; and that is accomplished by adding b2/4 to both sides.1504

So, I need to add b2/4 to both sides.1520

And here, we have b2 is actually -3, but I need to be careful,1530

because I am not just adding b2/4; I actually have factored out the 4;1548

and if I don't add that back in on the left, my equation won't be balanced.1556

So, this is actually 4 times b2/4.1562

x - 3x + (-3)2/4: now, I already have 4 times b2/4 here, so I needed to balance it by having that on the left side, as well.1566

This is going to give me y - 20 + 4, times 9/4, equals 4 times x, minus 3x, plus 9/4.1585

OK, this is going to give me y - 20; and the 4's cancel here, which is kind of convenient;1600

so that is going to give me...these will cancel; I will just get 9.1606

And -20 + 9 is -11, so this is y - 11 equals 4 times...and this actually is a perfect square, and you can check it out for yourself.1615

This equals (x - 3/2)2; this is a perfect square.1625

Now, I am almost in vertex form; I just have to move this 11 over to the right by adding 11 to both sides.1632

And this is vertex form; so again, vertex form is y = a(x - h)2 + k.1640

Here, I have a = 4, h = 3/2, and k = 11; so the vertex is going to be at (3/2,11).1650

This was a little more complicated, because I did have a coefficient here, an a, that did not equal 0.1660

So, right after I isolated my x variable terms, I factored that out.1667

And then, when I complete the square, I need to make sure that I add 4 times b2/4 to the left, since that is what I am adding to the right.1672

And then, I just went on and did that; I got a perfect square on the right.1680

And this is going to end up...this is actually all x2's, and this is going to end up giving me (x - 3/2)2 as a perfect square.1685

And then, I just move the 11 over to the right.1699

OK, now find an equation in vertex form of the parabola with vertex at (2,-3) and passing through the point (4,-19).1702

OK, vertex form: y = a(x - h)2 + k.1713

Now, I look at what I am given; and I am given the vertex, so that is h and k, and a point (that is x and y).1724

Since I have all of these, I can find a; I don't know what a is--that is my unknown.1733

Well, I have y = -19; I don't know a; x is 4; h is 2; and k is -3.1739

4 minus 2 gives me 2, squared, minus 3; so that is -19, equals a, times 4, minus 3;1757

-19 equals 4a - 3; so I am going to add 3 to both sides to give me -16 = 4a; therefore, a equals -4.1766

Now that I have found a, I just go back in here and write this as y = -4x, and then I have h as 2, and k is -3.1778

So, here I have vertex form; I know the vertex is (2,-3).1791

And now I know that a equals -4; so this is going to be a parabola that opens downward.1795

OK, this concludes this session of Educator.com.1802

And I will see you again next lesson.1806

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