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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

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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 (12)

1 answer

Last reply by: Dr Carleen Eaton
Sun Jul 8, 2018 1:20 PM

Post by John Stedge on June 27 at 03:44:59 PM

for completing the square the number you add would just be the x/y term before you factor out the leading coefficient of the x^2/y^2 terms correct? I saw this in example 4 where the x/y terms before factoring were the same as the numbers you added to the right side of the equation in the completeing the square step.

0 answers

Post by Shiden Yemane on March 20 at 10:57:07 AM

Thanks, it was very helpful!

1 answer

Last reply by: Dr Carleen Eaton
Wed Jan 14, 2015 7:41 PM

Post by Saadman Elman on December 19, 2014

It was very helpful. Thanks.

1 answer

Last reply by: Rafael Mojica
Thu May 22, 2014 10:22 AM

Post by Rafael Mojica on May 22, 2014

Hello,

on example 4 you made a mistake on the equation that you used. Since major axis is parallel to y-axis. So, (x-h)^2/b^2-(y-k)^2/a^2=1 is our formula.

1 answer

Last reply by: Dr Carleen Eaton
Sat Sep 14, 2013 2:44 PM

Post by Anwar Alasmari on August 11, 2013

the video around 7:00, I think the point has to be (3,0), not (0,3) as well as (-3,0).

2 answers

Last reply by: Dr Carleen Eaton
Mon Jan 23, 2012 5:01 PM

Post by Yi Zhang on July 22, 2011

The left vertex on the major axis is supposed to be at V(-6,-6).

Ellipses

  • Understand the meaning and significance of the major and minor axes.
  • Understand the fundamental equation a2 = b2 + c2 and use it frequently.
  • Use symmetry to help you graph an ellipse.
  • Understand the standard formula for the equation of an ellipse.
  • Know how to put an equation in standard form by completing the square.

Ellipses

Find the equation of the ellipse by completing the square: x2 + 4y2 − 2x − 16y − 19 = 0
  • Group the x's with the x's and the y's with the y's, move the constant term to the right of the equation
  • (x2 − 2x) + (4y2 − 16y) = 19
  • Factor out a 4 from the y's
  • (x2 − 2x) + 4(y2 − 4y) = 19
  • Complete the square by adding [(b2)/4]
  • (x2 − 2x + [(b2)/4]) + 4(y2 − 4y + [(b2)/4]) = 19 + [(b2)/4] + 4( [(b2)/4] )
  • (x2 − 2x + [(( − 2)2)/4]) + 4(y2 − 4y + [(( − 4)2)/4]) = 19 + [(( − 2)2)/4] + 4( [(( − 4)2)/4] )
  • (x2 − 2x + [4/4]) + 4(y2 − 4y + [16/4]) = 19 + [4/4] + 4( [16/4] )
  • (x2 − 2x + 1) + 4(y2 − 4y + 4) = 19 + 1 + 16
  • (x − 1)2 + 4(y − 2)2 = 36
  • Divide left and right side of equation by 36. Simplify
  • [((x − 1)2)/36] + [(4(y − 2)2)/36] = [36/36]
[((x − 1)2)/36] + [((y − 2)2)/9] = 1
Graph x2 + 4y2 − 2x − 16y − 19 = 0
  • Write the ellipse in Generic Conic Form
  • Horizontal[((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
    Vertical[((y − k)2)/(a2)] + [((x − h)2)/(b2)] = 1
  • Notice how this problem is the same as 1. Use the resultfrom 1 to graph. Notice how
  • this ellipse will be a horizontal ellipse.
  • [((x − 1)2)/36] + [((y − 2)2)/9] = 1
  • Find the center (h,k)
  • Center = (h,k) = (1,2)
  • Find the horizontal major axis, and graph it
  • a2 = 36
  • a = √{36} = 6
  • Starting from the center, go 6 units to the left and 6 units to the right.
  • Find the Vertical Minor Axis
  • b2 = 9
  • b = √9 = 3
  • Starting from the center, go 3 units up and 3 units down.
  • Draw a smooth ellipse that passes through the end - points of the Horizontal Major and Vertical Minor axis.
Find the equation of the ellipse by completing the square: 4x2 + 9y2 − 72x − 126y + 189 = 0
  • Group the x's with the x's and the y's with the y's, move the constant term to the right of the equation
  • (4x2 − 72x) + (9y2 − 126y) = − 189
  • Factor out a 4 from the x's and 9 from the y's
  • 4(x2 − 18x) + 9(y2 − 14y) = − 189
  • Complete the square by adding [(b2)/4]
  • 4(x2 − 18x + [(b2)/4]) + 9(y2 − 14y + [(b2)/4]) = − 189 + 4( [(b2)/4] ) + 9( [(b2)/4] )
  • 4(x2 − 18x + [(( − 18)2)/4]) + 9(y2 − 14y + [(( − 14)2)/4]) = − 189 + 4( [(( − 18)2)/4] ) + 9( [(( − 14)2)/4] )
  • 4(x2 − 18x + [324/4]) + 9(y2 − 14y + [196/4]) = − 189 + 4( [324/4] ) + 9( [196/4] )
  • 4(x2 − 18x + 81) + 9(y2 − 14y + 49) = − 189 + 4( 81 ) + 9( 49 )
  • 4(x − 9)2 + 9(y − 7)2 = 576
  • Divide left and right side of equation by 36. Simplify
  • [(4(x − 9)2)/576] + [(9(y − 7)2)/576] = [576/576]
[((x − 9)2)/144] + [(9(y − 7)2)/64] = 1
Graph 4x2 + 9y2 − 72x − 126y + 189 = 0
  • Write the ellipse in Generic Conic Form
  • Horizontal[((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
    Vertical[((y − k)2)/(a2)] + [((x − h)2)/(b2)] = 1
  • Notice how this problem is the same as 3. Use the result from 3 to graph. Notice how
  • this ellipse will be a horizontal ellipse.
  • [((x − 9)2)/144] + [(9(y − 7)2)/64] = 1
  • Find the center (h,k)
  • Center = (h,k) = (9,7)
  • Find the horizontal major axis, and graph it
  • a2 = 144
  • a = √{144} = 12
  • Starting from the center, go 12 units to the left and 12 units to the right.
  • Find the Vertical Minor Axis
  • b2 = 64
  • b = √{64} = 8
  • Starting from the center, go 8 units up and 8 units down.
  • Draw a smooth ellipse that passes through the end - points of the Horizontal Major and Vertical Minor axis.
Find the equation of the ellipse by completing the square: 9x2 + 4y2 − 108x + 40y + 280 = 0
  • Group the x's with the x's and the y's with the y's, move the constant term to the right of the equation
  • (9x2 − 108x) + (4y2 + 40y) = − 280
  • Factor out a 9 from the x's and 4 from the y's
  • 9(x2 − 12x) + 4(y2 + 10y) = − 280
  • Complete the square by adding [(b2)/4]
  • 9(x2 − 12x + [(b2)/4]) + 4(y2 + 10y + [(b2)/4]) = − 280 + 9( [(b2)/4] ) + 4( [(b2)/4] )
  • 9(x2 − 12x + [(( − 12)2)/4]) + 4(y2 + 10y + [((10)2)/4]) = − 280 + 9( [(( − 12)2)/4] ) + 4( [((10)2)/4] )
  • 9(x2 − 12x + [144/4]) + 4(y2 + 10y + [100/4]) = − 280 + 9( [144/4] ) + 4( [100/4] )
  • 9(x2 − 12x + 36) + 4(y2 + 10y + 25) = − 280 + 9( 36 ) + 4( 25 )
  • 9(x − 6)2 + 4(y + 5)2 = 144
  • Divide left and right side of equation by 144. Simplify
  • [(9(x − 6)2)/144] + [(4(y + 5)2)/144] = [144/144]
  • [((x − 6)2)/16] + [((y + 5)2)/36] = 1
  • or
[((y + 5)2)/36] + [((x − 6)2)/16] = 1
Graph 9x2 + 4y2 − 108x + 40y + 280 = 0
  • Write the ellipse in Generic Conic Form
  • Horizontal[((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
    Vertical[((y − k)2)/(a2)] + [((x − h)2)/(b2)] = 1
  • Notice how this problem is the same as 5.Usetheresultfrom 5 to graph. Notice how
  • this ellipse will be a vertical ellipse.
  • [((y + 5)2)/36] + [((x − 6)2)/16] = 1
  • Find the center (h,k)
  • Center = (h,k) = (6, − 5)
  • Find the Vertical Major Axis , and graph it
  • a2 = 36
  • a = √{36} = 6
  • Starting from the center, go 6 units up and 12 units down
  • Find the Horizontal Minor Axis
  • b2 = 16
  • b = √{16} = 4
  • Starting from the center, go 4 units right and 4 units left
  • Draw a smooth ellipse that passes through the end - points of the Vertical Major and Horizontal Minor axis.
Find the equation of the ellipse satisfying:
Endpoints of major axis:(4,7),( − 6,7)
Foci:(3,7),( − 5,7)
  • Notice how this ellipse is going to be a Horizontal Ellipse in the form [((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
  • Find the length of the major axis
  • 2a = | − 6 − 4| = | − 10| = 10
  • a = 5
  • Find the center of the ellipse given that a = 5 and one enpoint is (4,7).
  • You may also find the center by finding the mid - poing of the major axis.
  • Center = (h,k) = (h = 4 − 5,7) = ( − 1,7)
  • So far you have
  • [((x + 1)2)/(52)] + [((y − 7)2)/(b2)] = 1
  • Find b using the foci and the relationship a2 = b2 + c2
  • The center is located at ( − 1,7) and one foci located at (3,7), therefore recall that
  • c is the distance between the center and the foci, so
  • c = | − 1 − 3| = | − 4| = 4
  • To find b, then
  • b2 = a2 − c2
  • b = √{a2 − c2} = √{52 − 42} = √{25 − 16} = √9 = 3
  • Complete the equation of the ellipse
  • [((x + 1)2)/(52)] + [((y − 7)2)/(b2)] = 1
  • [((x + 1)2)/(52)] + [((y − 7)2)/(32)] = 1
[((x + 1)2)/25] + [((y − 7)2)/9] = 1
Find the equation of the ellipse satisfying:
Endpoints of major axis:(1,8),( − 9,8)
Foci:( − 1,8),( − 7,8)
  • Notice how this ellipse is going to be a Horizontal Ellipse in the form [((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
  • Find the length of the major axis
  • 2a = | − 9 − 1| = | − 10| = 10
  • a = 5
  • Find the center of the ellipse given that a = 5 and one enpoint is (1,8).
  • You may also finbd the center by finding the mid - poing of the major axis.
  • Center = (h,k) = (h = 1 − 5,8) = ( − 4,8)
  • So far you have
  • [((x + 4)2)/(52)] + [((y − 8)2)/(b2)] = 1
  • Find b using the foci and the relationship a2 = b2 + c2
  • The center is located at ( − 4,8) and one foci located at ( − 7,8), therefore recall that
  • c is the distance between the center and the foci, so
  • c = | − 7 − ( − 4)| = | − 3| = 3
  • To find b, then
  • b2 = a2 − c2
  • b = √{a2 − c2} = √{52 − 32} = √{25 − 9} = √{16} = 4
  • Complete the equation of the ellipse
  • [((x + 4)2)/(52)] + [((y − 8)2)/(b2)] = 1
  • [((x + 4)2)/(52)] + [((y − 8)2)/(42)] = 1
[((x + 4)2)/25] + [((y − 8)2)/16] = 1
Find the equation of the ellipse satisfying:
Center = ( − 10,1)
Vertical Major axis = 12 units
Horizontal Minor axis = 8
  • Notice how this ellipse is going to be a Vertical Ellipse in the form [((y − k)2)/(a2)] + [((x − h)2)/(b2)] = 1
  • Find a using length of the vertical major axis
  • 2a = 12
  • a = 6
  • Find b using length of the horizontal minor axis
  • 2b = 8
  • b = 4
  • Fill in the formula
  • [((y − 1)2)/(62)] + [((x + 10)2)/(42)] = 1
[((y − 1)2)/36] + [((x + 10)2)/16] = 1
Find the equation of the ellipse satisfying:
Center = ( − 3, − 3)
Horizontal Major axis = 24 units
Vertical Minor axis = 10
  • Notice how this ellipse is going to be a Vertical Ellipse in the form [((x − h)2)/(a2)] + [((y − k)2)/(b2)] = 1
  • Find a using length of the horizontal major axis
  • 2a = 24
  • a = 12
  • Find b using length of the vertical minor axis
  • 2b = 10
  • b = 5
  • Fill in the formula
  • [((x + 3)2)/(122)] + [((y + 3)2)/(52)] = 1
[((x + 3)2)/144] + [((y + 3)2)/25] = 1

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

Answer

Ellipses

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

  • Intro 0:00
  • What Are Ellipses? 0:11
    • Foci
  • Properties of Ellipses 1:43
    • Major Axis, Minor Axis
    • Center
    • Length of Major Axis and Minor Axis
  • Standard Form 5:33
    • Example: Standard Form of Ellipse
  • Vertical Major Axis 9:14
    • Example: Vertical Major Axis
  • Graphing Ellipses 12:51
    • Complete the Square and Symmetry
    • Example: Graphing Ellipse
  • Equation with Center at (h, k) 19:57
    • Horizontal and Vertical
    • Difference
    • Example: Center at (h, k)
  • 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

Transcription: Ellipses

Welcome to Educator.com.0000

So far in conic sections, we have discussed parabolas and circles.0002

The next type of conic section we are going to cover is the ellipse.0006

First of all, what is an ellipse? Well, an ellipse is formally defined as the set of points in a plane0012

such that the sum of the distances from two fixed points is constant.0019

Well, what does that mean? First of all, this is the general shape of an ellipse.0024

And these two points here are the foci of the ellipse: we will call them f1 and f2.0029

And looking back at this definition, it is the set of points in the plane...0036

and if you pick any of these points (say right here) and measure the distance from this point to one focus0041

(we will call that d1), and then we measure the distance from that same point to f2,0051

to the other focus, we will call that distance d2.0059

This definition states that, if I add up these two distances, d1 + d2, they will equal a constant.0063

I could pick any point; I could then pick this point and say, "OK, here is another distance," calling that, say, d3, and this one d4.0074

And if I added up this distance and this distance, d3 + d4, I could get that same constant.0087

And I could do that with any point on the ellipse.0094

Continuing on with some properties of the ellipse: an ellipse actually has two axes of symmetry.0104

One is called the major axis, and the other is the minor axis, and these intersect at the center of the ellipse.0110

Here, we are going to look at an ellipse that is centered at the origin (it has a center at (0,0)).0117

And again, I have foci f1 and f2.0122

This is one of the vertices of the ellipse; here is a vertex; this is the second vertex of the ellipse.0127

And the major axis runs from one vertex to the other vertex.0135

And you can see that it passes through both of the foci; this is the major axis.0140

And what we are looking at here is an ellipse that has a horizontal major axis.0149

In a few minutes, we will look at ellipses that are oriented the other way--ellipses that are oriented with a vertical major axis.0154

So, that does exist; but right now we will just focus on this for the general discussion.0163

Here we have the major axis; and intersecting at the center is a second axis called the minor axis.0167

Looking more closely at the relationships between the major axis, the minor axis, the foci, and the distances relating them:0182

let's call the distance from one vertex to the center A: this distance is A.0193

Therefore, the length of the major axis is 2A.0200

And this is going to become important later on, when we are working with writing equations for ellipses,0209

or taking an equation and then trying to graph the ellipse.0214

That is my first distance: this is actually called the semimajor axis.0218

The length of the semimajor axis is A; the length of the major axis is 2A.0226

Now, looking at the minor axis: from this point to the center, this length is B.0230

Therefore, the length of the minor axis is equal to 2B.0237

Now, looking at the foci: the distance from one focus to the center is going to be C.0246

Therefore, the distance from one focus to another, or the distance between the foci, is equal to 2C.0255

There is also an equation relating these A, B, and C; and that is A2 = B2 + C2.0276

So, keep this equation in mind: again, it becomes important, because you might be given A and B, but not C;0287

or you may be given a drawing, a sketch of the ellipse, and then asked to write an equation based on it.0294

And sometimes you need to find this third component in order to write that equation.0300

And you can do that by knowing that A2 = B2 + C2.0304

Again, A is the length of the semimajor axis; B is the distance from here to the center of the minor axis0309

(2B is the length of the minor axis); and C is the distance between one focus and the center (2C gives the distance between the two foci).0319

Those are important relationships to understand when working with the ellipse.0328

Standard form: looking at what the standard form of the equation of an ellipse with a center at (0,0) and a horizontal major axis is:0333

it is x2/A2 + y2/B2 = 1.0343

This is the standard form; and again, we already discussed what A is and B is.0353

And if you figure out those from the graph, and are given those, then you can go ahead and write your equation.0359

Or, given the equation, you can graph the ellipse.0365

Let's take a look at an example: A2/9 + y2/4 = 1.0368

So again, this is just the equation for an ellipse with a horizontal major axis, so it is sketched out here that way.0379

We can make this more precise by saying, "OK, A2 = 9; therefore, A equals √9, or 3."0388

That means that the distance from here to here is going to be 3.0405

And since this is centered at the origin, this will actually be at the point (3,0).0409

So, let's write that as a coordinate pair, (0,3).0415

OK, that means that this length of A is 3; over here, the other vertex is going to be at (0,-3).0420

So, I have one vertex at (0,3) and the other at (0,-3); and I have f1 here and f2 here.0431

And the length of the major axis is actually 6: it is going to be equal to 2A.0437

2A equals 6, and that is the length of the major axis.0443

B2 = 4: since B2 = 4, B = √4, which equals 2.0452

Therefore...actually, this needs to be written the other way; this is actually (3,0), and (-3,0).0461

Now, up here, we have (0,2) and (0,-2).0474

And 2, then, is the length; that is B--that is the length of half of the minor axis.0485

2B = 4, and this is the length of the minor axis.0492

You can get a lot of information just by looking at the equation of the ellipse in standard form.0498

If I needed to, I could figure out the distance between the foci, and I could figure out what C is.0505

And remember, C is this length; because recall that A2 = B2 + C2.0512

And I know that A2 is 9; I know that B2 is 4; so I am looking for C2.0518

9 - 4 gives me 5, so C2 = 5; therefore, C = √5, and that is approximately equal to 2.2.0524

So, this distance here is roughly 2.2; and the distance between these foci would be about 4.4.0535

So, just by having this equation, I could graph the ellipse, and I could find this third component that was missing.0544

OK, so that was standard form for an ellipse that has a horizontal major axis.0555

For an ellipse that has a vertical major axis, you are going to see that the A2 is associated with the y2 term.0560

In the horizontal major axis, we had x2/A2.0567

Now, if you were given an equation, how would you know what you are dealing with--0572

if you were dealing with a vertical major axis or a horizontal major axis?0578

Well, A2 is the larger number; so let's say I was given something like y2/16 + x2/9 = 1.0581

When I look at this, I see that the larger number is associated with y2; so that tells me that I have a vertical major axis.0594

If it had been x2 associated with 16, then I would have said, "OK, that is a horizontal major axis."0602

Again, I can graph this ellipse by having this equation written in standard form.0609

I know that A2 equals 16; therefore, A equals the square root of 16; it equals 4.0614

This time, I am going to go along here for the major axis; and that makes sense,0622

because I have a focus here and another focus here, and the major axis passes through the two foci.0628

OK, so (0,4) is going to give me one vertex; (0,-4) will give me the other vertex.0638

And remember that 2A = 8, and that is going to be the length of the major axis.0647

And again, right now, we are limiting our discussion to ellipses with a center at (0,0).0658

Later on, we will expand the discussion to talk about the graphs of ellipses with centers in other areas of the coordinate plane.0664

OK, so now I have B2 = 9; therefore, √9 is going to equal B; this tells me that B equals 3.0672

So, since B equals 3, the length of half of the minor axis is going to be 3.0689

So, right here at (3,0) is going to be one point, and I am going to have the other point right here at (-3,0).0696

And the length of the minor axis...2B = 6, and that is the length of the minor axis.0704

Again, I can use the relationship A2 = B2 + C20719

to figure out what C is, and to figure out where the foci are located.0725

I know that A2 is 16; I know that B2 is 9; and I am trying to figure out C2.0730

So, I take 16 - 9; that gives me 7 = C2; therefore, C = √7.0738

And the square root of 7 is approximately equal to 2.6, so this is going to be up here at about (0,2.6).0746

And f2 is going to be down here at about (0,-2.6).0755

So again, using standard form, you can graph the ellipse, and you can find where the foci are, based on the values of A2 and B2.0758

We talked a little bit about graphing ellipses; but sometimes you are not given the equation in standard form.0771

If the equation is not in standard form, you have to put it that way.0776

And working with other conic sections, we have learned that you can put an equation into standard form for a conic section by completing the square.0780

You can also use symmetry, just as we did when graphing parabolas or circles.0788

So, let's say you are given an equation like this: 3x2 + 4y2 - 18x - 16y = -19; and you are asked to graph it.0794

You are going to put it in standard form; but it is nice, first of all, to know what kind of shape you are working with.0807

And you can tell that just by looking at this, even though it is not in standard form yet.0811

And the reason is: I see that I have an x2 term and a y2 term0815

on the same side of the equation, with the same sign, with different coefficients; that tells me that I am working with an ellipse.0820

And we are going to go into more detail in the lecture on conic sections.0827

We are going to review how to tell apart the equations for various conic sections.0831

But just briefly now: recall that a parabola would have either an x2 term or a y2 term, not both.0835

With a circle, the coefficients of x2 and y2 are the same; that is for a circle.0845

For an ellipse, the coefficients of the x2 and y2 terms are different.0861

Now, you see that these have the same sign; so with an ellipse,0876

it is important to note that the x2 and y2 terms have the same sign.0881

If they don't have the same sign, that is actually a different shape.0884

They have the same sign; but this coefficient (the x2 coefficient) is 3, and the y2 coefficient is 4.0887

That tells me I have an ellipse, not a circle.0894

My next step is to complete the square and write this in standard form,0898

so that, if I wanted to graph it, I would have all of the information that I need.0903

The first thing to do is to group the x variable terms and the y variable terms.0906

This gives me x2 - 18x + 4y2 - 16y = -19.0911

Now, when I am looking at this, I remember that, to complete the square, I want to end up with a leading coefficient of 1.0931

I am going to factor out this 3 to get 3(x2 - 6x); and I am going to need to add0937

something else over here to complete the square--a third term.0944

Here, factor out the 4 to give me (y2 - 4y) = -19.0949

Recall that, to complete the square, you are going to add b2/4 to each set of terms.0957

So, for this x group, we have b2/4 = -(6)2/4 = 36/4 = 9.0963

So, I am going to add a 9 here; and very importantly, to the right side, I am going to add 9 times 3, which is 27.0978

Add that to the right, because if I don't, the equation won't be balanced anymore.0995

Now, the y variable terms: I need to add something here to complete the square.1003

And I will work over here for this: b2/4 equals (-4)2/4 = 16/4 = 4.1013

So, I am going to add 4 here; but to the right, what I need to add is 4 times 4; so I need to add 16.1026

Now that I have this, my next step is to write this in standard form: 3...and this is x - 3, the quantity squared;1038

if I squared this, I would get this back; plus 4y - 2, the quantity squared, equals...1049

-19 plus 16 gives me, let's see, -3; 27 minus 3 gives me 24 on the right.1057

Well, recall that for standard form, I want a 1 over here on the right; I want this to equal 1.1069

So, this gives me...I need to divide both sides by 24; this is going to give me 3(x - 3)2/24 + 4(y - 2)2/24 = 24/24.1076

This finally leaves me with (x - 3)2...this 3 will cancel, and I am going to get 8 in the denominator;1101

this 4 will cancel, and I will get 6 in the denominator; equals 1.1109

Now, I have the bigger term associated with x; that tells me that I have a horizontal major axis.1115

So, the major axis is horizontal; and let me go ahead and write that up here.1126

x2/a2 + y2/b2 = 1.1144

OK, now you can see that this looks slightly different; and we have these extra terms here.1154

And what that tells us is something that we are going to discuss in a moment.1160

And what we are going to discuss is situations where the center of the ellipse is not at (0,0).1164

But we have the same information that we would have with the center being at (0,0).1174

And that is that I have A2, which equals 8; and I know I have a horizontal major axis; and I have B2 = 6.1180

Looking, now, at equations for ellipses with centers at (h,k), at somewhere other than 0:1198

if you look at the situation where we have a horizontal major axis, versus a vertical major axis,1206

you can again see that it is very similar to when the center is at (0,0).1212

The x is associated with the A2 term when the major axis is horizontal.1217

The y is associated with the A2 term when the major axis is vertical.1221

The only difference is that we now have these terms telling us where the center is.1227

And if the center was going to be 0, all that would happen is that you would have x - 0 and y - 0,1231

which then gives you back the equation we worked with before, x2/A2 + y2/B2,1237

or y2/A2 + x2/B2, all equal to 1.1244

So, let's talk about an example for this: (x - 4)2 + (y + 6)2 = 1.1252

This is all over 100, and this is all divided by 25.1264

What this tells me is that the center is at (h,k); so this is h (that is 4).1269

You need to be careful here, because what you have is a positive; but the standard form is a negative.1278

And it is perfectly fine to write this like this, but you need to keep in mind that this is really saying...1284

if you think about it, y + 6 is the same as y - -6.1291

So, when I look at it this way, I realize that, if it is in this form, k is actually going to be -6.1296

And if you are not careful about that, you can end up putting your center in the wrong place.1302

Let's let this be 2, 4, 6, -2, -4, -6; the center is at (4,-6), right here.1309

A2 = 100; my larger term is my A2term, and I have a horizontal major axis.1323

Therefore, A = √100, which is 10.1331

So, since A equals 10 and the length of the major axis is horizontal, then I am going to need to go 10 over from 4.1337

It is going to be all the way out here at 14.1351

So, this is where one vertex would be--this is going to be at 4 + 10 gives me (14,-6).1357

That is one vertex: -2, 4, 6, 8, 10, 12, 14...so -14 is going to be about here.1368

And I am going to have the other vertex here at (-14,-6).1376

I have a minor axis: B2 = 25; therefore B = 5.1384

So, what that tells me is that I come here at -6; I add 5 to that; and that is going to bring me right here at -1; that is going to be right about here.1389

And then, -8, -10, -12...-(6 + 5) is going to be down here at -11; so this ellipse is going to roughly look like this.1400

OK, and standard form allowed me to determine that the center is at (4,-6);1419

that I have a vertex; if I add 10, that is going to give me this vertex right over here1427

(let's draw this more at the vertex); that I have another vertex here; that the major axis is going to pass through here;1433

and then, I am going to have a minor axis passing through here.1441

All right, in the first example, we are asked to find the equation of the ellipse that is shown.1447

And I am going to go ahead and label some of these points.1453

This is going to be f1; and let's say we are given f1 as being at (0,3).1457

This is going to be 1, 2, 3: each mark is going to stand for one.1463

Down here, I have f2; that should actually be down here a little bit lower; so that is f2 at (0,-3).1467

Let's say we are also given a vertex at (0,5), and the other vertex at (0,-5).1480

All right, so I am asked to find the equation of the ellipse shown.1492

And the one thing I see is that there is a vertical major axis.1495

Since there is a vertical major axis, it is going to be in the general form (y - k)2/A2 + (x - h)2/B2 = 1.1503

Now, I notice that the center is at (0,0); so that tells me that h and k are (0,0).1520

So, this is actually going to become the simpler form, y/A2 + x/B2 = 1.1528

The next piece of information I have is the length of A.1537

So, this line is going to give me A; and I know that, since the center is here at (0,0), the length of this is 5; therefore, A = 5.1541

Since A equals 5, A2 is going to equal 52, or 25.1553

I don't know B; this is going to be B, but I don't know what it is.1561

However, I do know C; if I look here, this is C--from -3 to the center is 3, so C = 3.1567

Therefore, C2 = 32, or 9.1583

The other piece of information I have is the equation we have been working with,1587

that states that A2 = B2 + C2.1591

Therefore, I can find B2, which I need in order to write my equation.1602

So, I know that A2 is 25; I know that C2 is 9; and I am looking for B2.1606

25 - 9 is 16; therefore, B2 = 16, and B = 4.1620

Now, I can go ahead and write my equation; I have all the information I need, so let's put it all together right here.1628

y, divided by A2...I know that A2 is 25...(that is actually y2 right here--1633

let's not forget the squares)...plus x2 divided by B2...B2 is 16...equals 1.1647

So again, this had a center at (0,0); so remember that this is the general form of the equation.1655

When you have a center at (0,0), it becomes this form.1660

I had a vertical major axis, so I am using this form, where the A2 is associated with the y2 term.1663

And by having A2 and C2, I was able to determine what B2 is.1671

Example 2: let's find the equation of the ellipse satisfying a major axis 10 units long and parallel to the y-axis; minor axis 4 units long; center at (4,2).1678

Let's start with the center--the center is at (4,2)--the center is right here at (4,2).1690

And it says that the major axis is parallel to the y-axis.1699

What that tells me is that I have a vertical major axis.1703

Since this is a vertical major axis and I know the center (I know (h,k)), I am going to be working with the general form1711

(y - k)2/A2 + (x - h)2/B2 = 1.1719

I have h; I have k; I need to find A2 and B2.1728

The other piece of information I have is that the major axis is 10 units long; and I know that it is vertical.1732

Since it is 10 units long, that means that the major axis length is equal to 2A, and I know that that length is 10;1738

therefore, I just take 10 divided by 2, and I get that A = 5.1751

So, I am starting here; and if I take 5 + 2, I am going to get that I am going to have the vertex up here at 7.1757

5 + 2 is going to give me 7, right there; so I go to (4,7); that is where this is going to be.1775

And I got that by keeping the x where it was, and then adding 2 where the center was and adding 5 to that, which is the length of A.1793

Then, I am going to go down; again, it is going to be at 4.1803

But then, if I take 2 and subtract 5 from it, I am going to get -3.1810

So, right here at (4,-3) is the other vertex.1815

Now, I was not asked to graph this; but I am graphing it so that I have an understanding of what each of these means,1823

so that I can go ahead and write the equation.1829

What I have now is...I know what A is, which means I can figure out A2; and I know the center.1832

The last thing I know is that the minor axis length equals 2B: I am given that that is 4 units long; therefore, B = 2.1844

So, if I started here at 4, and I added 2 to that, I am going to end up with (6,2).1857

And I am going to end up with...if I start at 4 and I subtract 2, I am going to end up with (2,2).1867

And this is B; this is A.1881

I have a minor axis that has a length of 2B, which tells me that B = 2.1887

From this information, I can go ahead and write this equation.1891

I know A equals 5, so A2 equals 52, which is 25.1897

B = 2; B2, therefore, equals 22, or 4.1904

So, I have everything I need to write this: (y - k)...well, k is 2; the quantity squared, divided by A2;1911

A2 is 25; plus (x - h)...h is 4...the quantity squared, divided by B2, which is 4, equals 1.1920

So, this standard form describes the ellipse with the major and minor axis here.1936

And you could finish that out by just connecting these points and drawing the ellipse if you wanted to finish graphing it.1941

Find the equation of the ellipse satisfying endpoints of the major axis at (11,3) and (-7,3) and foci at (7,3) and (-3,3).1953

All right, endpoints of the major axis: let's do 2, 4, 6, 8, 10, 12, and -2, 4, -6...OK.1962

The endpoint is at (11,3): 11 is right here, and then we will have 3 be right here.1979

The other endpoint is at -7, which is going to be here, 3.1991

And that tells me the major axis: since the major axis is horizontal, we are going to be working with the general form1999

(x - h)2/A2 + (y - k)2/B2 = 1.2010

So, that is my first piece of information.2018

I can also figure out the length of the major axis.2020

So, since the major axis goes from -7 to 11, if I take -7 minus 11, I am going to get -18.2024

And a length is going to be an absolute value, so I am just going to take 18; the length is going to be the absolute value of this difference.2045

All right, I know that the major axis length is 18; the other thing I know is that the major axis length equals 2A, as we have discussed.2053

So, 2A = 18; therefore, A = 9; so I know that the distance from the center to this endpoint is 9.2061

Therefore, I could just say, "OK, 9 minus 11 gives me 2; and I know that the y-coordinate will be 3; so that is (2,3)."2076

Another way to solve this, without using all this graphing, would have simply been to find the midpoint.2087

I am going to go ahead and do that, as well, because the midpoint of the major axis is the center of the ellipse.2093

Let's try that, as well--the center, using the midpoint formula.2100

Recall the midpoint formula: we are going to take x1 + x2 (that is 11 + -7);2105

and we are going to take the average of that (we will divide it by 2); that is going to give me the x-coordinate.2112

For the y-coordinate, I am going to take 3 + 3, and I am going to divide that by 2.2117

And this will give you a center at...11 - 7 gives you 4; 4 divided by 2 is 2.2126

3 + 3 is 6, divided by 2 gives you 3; and that is exactly what I came up with using the graphing method.2138

So, just to show you: you could have figured this out algebraically (where the center is); or you could have figured it out using graphing.2144

This gives me (h,k), so I have (h,k), and I have A, which is 9, so I can get A2.2154

The next thing I need to do is figure out B, and they don't give me B; but what they do give me are the foci.2162

There are foci at (7,3) (7 is here--focus at (7,3)--we will call this one f1 at (7,3)); and this is the center right here.2167

There is another focus at (-3,3): f2 is going to be at (-3,3).2187

Recall that the distance from one focus to the other is 2C; the distance from one focus to the center is C.2201

Let's just work with this; this is C; 7 - 2 is 5; therefore, C = 5.2210

So, I have A = 9; I have C = 5 (again, that is the distance from the center to a focus);2221

or I could have figured out the distance from one focus to another--that is 2C--and then divided by 2;2229

So, I have A, and I have C, and I know that A2 = B2 + C2.2233

So, this gives me 92 = B2 + 52: 92 is 81, equals B2 + 25.2240

If I take 81 - 25, I am going to get 56 = B2.2254

And I don't even need to take the square root of that, because to put this into standard form, I actually need B2.2259

So, I get (x - h); remember, the center is (h,k), so that is 2; the quantity squared, divided by A2...2265

recall that A2 is 92, so it is 81; plus (y - k)2...k is 3, the quantity squared;2273

divided by B2...I determined that B2 is 56; all of this equal to 1.2284

Just by knowing the endpoints of the major axis and the location of the foci, I could figure out A22289

and B2, as well as the center, and then write this equation for the ellipse in standard form.2295

Finally, we are asked to graph an ellipse that is not given to us in standard form.2307

We have some extra work to do: we are going to actually have to complete the square in order to even graph this.2311

So, let's go ahead and start by grouping the x terms together and y terms together.2318

Also note that, since this has an x2 term and a y2 term on the same side of the equation,2329

with the same sign (they are both positive), but different coefficients, I know I have an ellipse.2335

It is not a circle, because for a circle, these coefficients would be the same.2341

Grouping the terms together gives me 14x2 - 56x + 6y2 - 24y = -38.2347

Looking at this, I see that I have a common factor of 2.2368

If I divide both sides by 2, I can simplify this equation; so the numbers I will work with will be smaller.2372

So, let's divide both sides by 2 to get 7x2 - 28x + 3y2 - 12y = -19.2378

The next thing to do is to factor out the leading coefficient, since it is not 1.2391

I am going to factor out a 7 to get x2 - 4x, plus...2397

over here, I am going to factor out a 3; that gives me y2 - 4y, all equals -19.2402

I need to complete the square, so I need to get b2/4.2411

In this case, that is going to give me 42/4 = 16/4 = 4.2415

So, over here, I am going to add a 4; very important--on the right, I have -19, and I am adding to the left 7 times 4; that is 28.2423

So, I need to add 28 to the right.2440

All right, over here, for the y terms: b2/4: again, we have a b term that is 4, so I am going to end up with the same thing, 4.2446

Now, here I am actually adding 4 times 3 (is 12), so I need to add that to the right, as well, to keep the equation balanced.2462

This is the easiest step to mess up on: you are focused here on completing the square,2470

and then you sometimes don't remember that you have to add the same thing to both sides.2474

Now, working on writing this in standard form: this is going to give me (x - 2)2 + 3(y - 2)2 = -19 + 12...2479

that is going to leave me with 28; -19 + 12 is going to be -7; 28 minus 7 equals 21.2493

To get this into standard form, I need to have a 1 on the right, so I am going to divide both sides by 21.2504

This is going to give me 7(x - 2)2/21 + 3(y - 2)2/21 = 21/21.2510

This cancels to (x - 2)2/3; this becomes (y - 2)2/7; and this just becomes 1.2528

Now, I have standard form; I can do some graphing.2547

I have a center at, let's see, (2,2); that is right here; the center is at (2,2), so that is h and k.2550

And I notice, actually, that the larger term is under the y; it is associated with the y.2577

That tells me that this has a vertical major axis; and therefore, I am going to keep that in mind--that it is going to be oriented this way.2587

The ellipse is going to end up like this, instead of like this.2600

I have my center at (2,2); therefore, A2 = 7; A = √7.2604

The square root of 7 is about 2.6, so it gets messy, as always, when you are working with radicals.2620

But it is about 2.6; so what I have to do is say, "OK, my vertex up here is going to be at x = 2, and then y is going to equal 2 + 2.6, which is 4.6."2626

So, (2,4.6): and again, I got that by saying the length of A (the length from the vertex to the center) is 2.6.2643

So, 2 + 2.6 gives me 4.6.2655

Over here, I am going to take 2, and I am going to subtract 2.6 from it; so x is still going to be 2; now y is going to be about here, which is (2, -.6).2658

Again, 2 is here; the length of A is 2.6, so it is going to be 2 - 2.6; this gives me my major axis.2674

I know, from vertex to vertex, where this ellipse is going to land.2685

Now, the minor axis, B: I know that B2 = 3; therefore, B = √3, which is approximately equal to 1.7.2690

This is not to do the same thing, but going along the x direction, the horizontal direction.2700

So again, this is A; OK, now to get B, I am going to have 2, and I am going to add 1.7 to it.2706

That is going to give me 3.7; it is going to land about here.2715

In this direction, I am going to subtract; I am going to say that I have 2 - 1.7, so that is going to land here, at about .3.2722

And in the y direction (in the vertical direction) it is still going to be up at 2.2739

So again, to get this, I said 2 + 1.7; that brought me to (3.7,2)--that is this point.2742

This point is at 2 minus 1.7, so that is (.3,2).2753

So, this gives me the general shape of this ellipse, like this; so I can get a good sketch, based on this equation.2761

I took this equation, and I recognized that it was an ellipse.2782

I completed the square to get it in standard form, and saw that it had a vertical major axis and that it had a center at (2,2).2787

I then found A to determine where the vertices would be, and then B to determine the width of the ellipse here; and then I could get a good sketch.2795

That concludes this session of Educator.com on ellipses; thanks for visiting!2806

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