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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Hyperbolas

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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 (5)

1 answer

Last reply by: Dr Carleen Eaton
Sun Apr 8, 2018 2:39 PM

Post by Shiden Yemane on March 21 at 12:52:14 PM

In example 3, when you were completing the square for the x terms, shouldn't the the (b^2) over 4 be (-2)^2 over 4, instead of 2^2 over 4? Let me know if I'm wrong.

2 answers

Last reply by: julius mogyorossy
Fri Aug 15, 2014 2:23 PM

Post by Kenneth Montfort on March 6, 2013

So, you said in the lecture on ellipses, that a^2 was the larger term, it seems here that b^2 is the larger term...is that another clue about how to tell which formula you are using?

Hyperbolas

  • Understand the concepts of vertices, transverse axis, and conjugate axis.
  • Understand the role of the asymptotes in graphing a hyperbola. Know their equations.
  • Understand the fundamental equation c2 = a2 + b2.
  • Use symmetry to help you graph a hyperbola.
  • Understand the standard formula for the equation of a hyperbola.
  • Know how to put an equation in standard form by completing the square.

Hyperbolas

Find the equation of the hyperbola by completing the square: 4x2 − y2 − 40x − 16y + 20 = 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 − 40x) + ( − y2 − 16y) = − 20
  • Factor out a 4 from the x's and negative from the y's
  • 4(x2 − 10x) − (y2 + 16y) = − 20
  • Complete the square by adding [(b2)/4]
  • 4(x2 − 10x + [(b2)/4]) − (y2 + 16y + [(b2)/4]) = − 20 + 4( [(b2)/4] ) − ( [(b2)/4] )
  • 4(x2 − 10x + [(( − 10)2)/4]) − (y2 + 16y + [(162)/4]) = − 20 + 4( [(( − 10)2)/4] ) − ( [(162)/4] )
  • 4(x2 − 10x + [100/4]) − (y2 + 16y + [256/4]) = − 20 + 4( [100/4] ) − ( [256/4] )
  • 4(x2 − 10x + 25) − (y2 + 16y + 64) = − 20 + 4( 25 ) − ( 64 )
  • 4(x − 5)2 − (y + 8)2 = 16
  • Divide left and right side of equation by 16. Simplify
  • [(4(x − 5)2)/16] − [((y + 8)2)/16] = [16/16]
[((x − 5)2)/4] − [((y + 8)2)/16] = 1
Graph: 4x2 − y2 − 40x − 16y + 20 = 0
  • Write in Standard Form equation of the hyperbola
  • Notice how this problem is the same as problem # 1 Proceed with the the solution [((x − 5)2)/4] − [((y + 8)2)/16] = 1
  • Find the center of the hyperbola and graph it.
  • Center = (h,k) = (5, − 8)
  • Find the lenght of a and b to write the rectangle in order to draw the asymptotes
  • a2 = 4
  • a = √4 = 2
  • b2 = 16
  • b = √{16} = 4
  • Starting from the center, move 2 units to the right, label it A, then starting from the center move 2 units to the left, label it B
  • Starting from the center, move 4 units up, label it C; starting from the center, move 4 units down, label it D.
  • Draw a rectangle that passes through the points A, C, B, D.
  • Draw your asymptotes
  • Draw your asymptotes, these must pass throgh the corners of the rectangle formed by points A, B, C, D.
  • Draw the hyperbola
  • Points B and A are your vertex. Sketch the hyperbola. The hyperbola opens to the left and to the right
Find the equation of the hyperbola by completing the square: 9x2 − 25y2 + 108x + 200y − 301 = 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) + ( − 25y2 + 200y) = 301
  • Factor out a 9 from the x's and − 25 from the y's
  • 9(x2 + 12x) − 25(y2 − 8y) = 301
  • Complete the square by adding [(b2)/4]
  • 9(x2 + 12x + [(b2)/4]) − 25(y2 − 8y + [(b2)/4]) = 301 + 9( [(b2)/4] ) − 25( [(b2)/4] )
  • 9(x2 + 12x + [(122)/4]) − 25(y2 − 8y + [( − 82)/4]) = 301 + 9( [(122)/4] ) − 25( [( − 82)/4] )
  • 9(x2 + 12x + [144/4]) − 25(y2 − 8y + [64/4]) = 301 + 9( [144/4] ) − 25( [64/4] )
  • 9(x2 + 12x + 36) − 25(y2 − 8y + 16) = 301 + 9( 36 ) − 25( 16 )
  • 9(x + 6)2 − 25(y − 4)2 = 225
  • Divide left and right side of equation by 225. Simplify
  • [(9(x + 6)2)/225] − [(25(y − 4)2)/225] = [225/255]
[((x + 6)2)/25] − [((y − 4)2)/9] = 1
Graph: 9x2 − 25y2 + 108x + 200y − 301 = 0
  • Write in Standard Form equation of the hyperbola
  • Notice how this problem is the same as problem # 3 Proceed with the the solution [((x + 6)2)/25] − [((y − 4)2)/9] = 1
  • Find the center of the hyperbola and graph it.
  • Center = (h,k) = ( − 6,4)
  • Find the lenght of a and b to write the rectangle in order to draw the asymptotes
  • a2 = 25
  • a = √{25} = 5
  • b2 = 9
  • b = √9 = 3
  • Starting from the center, move 5 units to the right, label it A, then starting from the center move 5 units to the left, label it B.
  • These are your vertices.
  • Starting from the center, move 3 units up, label it C; starting from the center, move 3 units down, label it D.
  • Draw a rectangle that passes through the points A, C, B, D.
  • Draw your asymptotes
  • Draw your asymptotes, these must pass throgh the corners of the rectangle formed by points A, B, C, D.
  • Draw the hyperbola
  • Points B and A are your vertex. Sketch the hyperbola. The hyperbola opens to the left and to the right
Find the equation of the hyperbola satisfying:
Vertices at (11,0) and ( − 11,0)
Conjugate Axis has lenght − 16
  • Identify type of hyperbola
  • Looking at the information provided, you can see that the hyperbola will be in the format [(x2)/(a2)] − [(y2)/(b2)] = 1
  • which means it will open to the left and to the right
  • Find a and b
  • Since a is the length from the center (0,0) to one of the vertices, a = 11
  • Since the lenght of the conjugate axis equals 2b
  • 2b = 16
  • b = 8
  • Write the formula given a and b
  • [(x2)/(a2)] − [(y2)/(b2)] = 1
  • [(x2)/(112)] − [(y2)/(82)] = 1
[(x2)/121] − [(y2)/64] = 1
Find the equation of the hyperbola satisfying:
Vertices at (6,0) and ( − 6,0)
Conjugate Axis has lenght of 18
  • Identify type of hyperbola
  • Looking at the information provided, you can see that the hyperbola will be in the format [(x2)/(a2)] − [(y2)/(b2)] = 1
  • which means it will open to the left and to the right
  • Find a and b
  • Since a is the length from the center (0,0) to one of the vertices, a = 6
  • Since the lenght of the conjugate axis equals 2b
  • 2b = 18
  • b = 9
  • Write the formula given a and b
  • [(x2)/(a2)] − [(y2)/(b2)] = 1
  • [(x2)/(62)] − [(y2)/(92)] = 1
[(x2)/36] − [(y2)/81] = 1
Find the equation of the hyperbola satisfying:
Vertices at (14,0) and ( − 14,0)
Conjugate Axis has lenght of 22
  • Identify type of hyperbola
  • Looking at the information provided, you can see that the hyperbola will be in the format [(x2)/(a2)] − [(y2)/(b2)] = 1
  • which means it will open to the left and to the right
  • Find a and b
  • Since a is the length from the center (0,0) to one of the vertices, a = 14
  • Since the lenght of the conjugate axis equals 2b
  • 2b = 22
  • b = 11
  • Write the formula given a and b
  • [(x2)/(a2)] − [(y2)/(b2)] = 1
  • [(x2)/(142)] − [(y2)/(112)] = 1
[(x2)/196] − [(y2)/121] = 1
Find the equation of the parabola with
Center(2,4)
Vertical Transverse axis of length 16
Conjugate axis of length 26
  • Given that the transverse axis is vertical, the equation of the hyperbola will be in the format [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • Find a and b
  • 2a = transverse axis
  • 2a = 16
  • a = 8
  • 2b = conjugate axis
  • 2b = 26
  • b = 13
  • Write equation with the ceenter and a and b
  • [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • [((y − 4)2)/(82)] − [((x − 2)2)/(132)] = 1
[((y − 4)2)/64] − [((x − 2)2)/169] = 1
Find the equation of the parabola with
Center(10,6)
Vertical Transverse axis of length 20
Conjugate axis of length 8
  • Given that the transverse axis is vertical, the equation of the hyperbola will be in the format [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • Find a and b
  • 2a = transverse axis
  • 2a = 20
  • a = 10
  • 2b = conjugate axis
  • 2b = 8
  • b = 4
  • Write equation with the ceenter and a and b
  • [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • [((y − 6)2)/(102)] − [((x − 10)2)/(42)] = 1
[((y − 6)2)/100] − [((x − 10)2)/16] = 1
Find the equation of the parabola with
Center( − 7,10)
Vertical Transverse axis of length 2
Conjugate axis of length 6
  • Given that the transverse axis is vertical, the equation of the hyperbola will be in the format [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • Find a and b
  • 2a = transverse axis
  • 2a = 2
  • a = 1
  • 2b = conjugate axis
  • 2b = 6
  • b = 3
  • Write equation with the ceenter and a and b
  • [((y − k)2)/(a2)] − [((x − h)2)/(b2)] = 1
  • [((y − 10)2)/(12)] − [((x + 7)2)/(32)] = 1
[((y − 10)2)/1] − [((x + 7)2)/9] = 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

Hyperbolas

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 Hyperbolas? 0:12
    • Two Branches
    • Foci
  • Properties 2:00
    • Transverse Axis and Conjugate Axis
    • Vertices
    • Length of Transverse Axis
    • Distance Between Foci
    • Length of Conjugate Axis
  • Standard Form 5:45
    • Vertex Location
    • Known Points
  • Vertical Transverse Axis 7:26
    • Vertex Location
  • Asymptotes 8:36
    • Vertex Location
    • Rectangle
    • Diagonals
  • Graphing Hyperbolas 12:58
    • Example: Hyperbola
  • Equation with Center at (h, k) 16:32
    • Example: Center at (h, k)
  • 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

Transcription: Hyperbolas

Welcome to Educator.com.0000

Today, we are going to be discussing the last type of conic section, which is hyperbolas.0002

So far, we have covered parabolas, circles, and ellipses.0006

As you can see, hyperbolas are a bit different in shape than the other conic sections we have worked with.0012

And one thing that makes them unique is that there are two sections referred to as branches; there are two branches in this hyperbola.0018

The formal definition is that a hyperbola is a set of points in the plane,0025

such that the absolute value of the differences of the distances from two fixed points is constant.0029

What does that mean? First, let's look at the foci.0037

These two fixed points are the foci; and here is a focus, f1, and here is the other, f2.0041

If I take a point on the hyperbola, and I measure the distance to f1,0048

and then the distance to f2, that is going to give me d1 and d2.0056

Recall that, with ellipses, we said that the distance from a point on the ellipse--if you measured the distance to one focus,0067

and then the other focus, and then added those, that the sum would be a constant.0074

Here, we are talking about the difference: the absolute value of this distance, d1, minus (the difference) d2, equals a constant.0078

That is the formal definition of a hyperbola.0092

Again, I could take some other point: I could take a point up here on this other branch.0094

And I could find a distance, say, d3, and then the distance to f1 could be something--say d4.0100

Again, the absolute value of those differences would be equal to that same constant.0107

All right, properties of hyperbolas: A hyperbola, like an ellipse, has two axes of symmetry.0117

But these have different names: here you have a transverse axis and a conjugate axis, and they intersect at the center.0124

We are looking here at a hyperbola with the center at (0,0).0133

One thing to note is that you can also have a hyperbola that is oriented as such.0139

But right now, we are looking at this one, with a more horizontal orientation.0147

But just to note: this does exist, and we will be covering it.0153

All right, first discussing the transverse axis: the transverse axis is going to go right through here--it is going to pass through the center.0156

And this is the vertex, and this is the vertex of the other branch, and this is the transverse axis.0166

The distance from one vertex to the center along this transverse axis is going to be A.0179

Again, a lot of this is going to be similar from when we worked with ellipses; but there are some important differences, as well.0186

So, the length of the transverse axis equals 2A; from here to here would be 2A.0195

The foci: if you look at foci, say f1, f2...let's look at f2...0212

it would be the same over on f1: if I looked at the distance from one focus to the center, that is going to be C.0218

The distance between the foci is therefore 2C; if I measured from here to here, that length is going to be 2C.0229

There is a second axis called the conjugate axis; and the two axes intersect here at the center.0246

The length of half...if you take half the length of this conjugate axis, it is going to be equal to B.0260

The transverse axis lies along here; the conjugate axis, in this case, is actually along the y-axis (and the transverse axis is along the x-axis).0273

The length of the conjugate axis is 2B.0288

As with the ellipse, there is an equation that relates A, B, and C; but it is a slightly different equation.0297

Here, A, B, and C are related by C2 = A2 + B2.0303

This relationship will help us to look at the equation for a hyperbola and graph the hyperbola,0311

or look at the graph, and then go back and write the equation.0317

So again, there are two axes: transverse, which goes from vertex to vertex; and conjugate, which intersects the transverse axis0321

at the center of the hyperbola and has a length of 2B (the transverse axis has a length of 2A).0331

The distance from focus to focus (between the two foci) is 2C.0338

The standard form of the hyperbola is also going to look somewhat familiar, because it is similar to an ellipse, but with a very important difference.0346

Here we are talking about a difference instead of a sum.0352

So, if you have a hyperbola with a center at (0,0) and a horizontal transverse axis, the equation is x2/A2 - y2/B2.0355

And here, we again have that the center is at the origin, (0,0).0368

Although the center certainly does not have to be at the origin, right now we are going to start out0373

working with hyperbolas with a center at the origin, just to keep things simple.0377

And again, by being given an equation in standard form, you can look at it and get a lot of information about what the hyperbola looks like.0384

Therefore, the vertex is going to be at (A,0); the other vertex will be at (-A,0).0396

You are going to have a point up here that is going to be (B,0); and this length, B, gives the length of half of the conjugate axis.0408

And then, you are going to have another point...actually, that is (0,B), because it is along the y-axis...another point, (0,-B).0420

This distance is B, from this point to the center; this distance is A.0429

And then here, I have f1 and f2; and the distance from one of those to the center is C.0436

As I mentioned, you can have a hyperbola that is oriented vertically.0447

So, if the transverse axis is vertical, and the center is at (0,0), the standard form is such that y2 is associated with the A2 term.0451

And here, it is positive: so you are taking y2/A2 - x2/B2 = 1.0463

In this case, what you are going to have is a vertex right here at (0,A), the other vertex is here at (0,-A); here is the transverse axis.0470

And then, you are going to have the conjugate axis; the length of half of that is going to be B; the length of the entire thing is 2B.0482

So, this is going to be some point, (B,0); and B2 is given here, so you could easily find B by taking the square root.0495

And then, over here is (-B,0).0502

So again, there are two different standard forms, depending on if you are working with a hyperbola0505

that has a horizontal transverse axis or one that has a vertical transverse axis.0508

Something new that we didn't talk about with ellipses is asymptotes.0516

Recall that an asymptote is a line that a curve on a graph approaches, but it never actually reaches.0520

And asymptotes are very useful when you are trying to graph a hyperbola.0528

The equations are given here: let's go ahead and draw these first.0534

Now, recall that this vertex is at a point (A,0); this vertex is at (-A,0).0536

If I measure the length...this is the transverse axis, and it is horizontal...let's say that it turns out that B is right up here, (0,B).0545

And B is going to be the length from this point to the center; 2B will be the length of the conjugate axis.0555

Then, I am going to have another point down here, (0,-B).0564

What I can do is form a box, a rectangle; and the rectangle is going to have vertices...I am going to go straight up here and across here.0568

Therefore, this is going to be given by (-A,B); that is going to be one vertex.0578

I can go over here and do the same thing; I am going to go straight up from this vertex, and straight across from this point.0586

And that is going to give me the point (A,B).0592

I'll do the same thing here: I go down directly and draw a line across here; this point is going to be (-A,-B).0597

One final vertex is right here: and this is given by (A,-B).0608

OK, now you draw a box using these A and B points; and then you take that rectangle and draw the diagonals.0618

If you continue those diagonals out, you will have the two asymptotes for the hyperbola.0630

OK, so each of these lines is an asymptote.0650

And notice that the hyperbola is going to approach this, but it is not actually going to reach it.0654

So, it is going to continue on and approach, but not reach, it; it is going to approach like that.0663

All right, now this is one way to just graph out the asymptotes.0671

You can also find the equation; and for right here, we are working with a hyperbola with a horizontal transverse axis.0676

So, we are going to look at this equation.0686

If I was working with a vertical one, I would look at this equation.0688

Now, what does this mean? Well, y equals ±(B/A)x.0691

What this actually is: this B/A gives the slope of the asymptote.0697

Recall that y = mx + b; since the center is at (0,0), the y-intercept is 0; so here, b = 0, so I am going to have y = mx.0703

The slope, m, is B/A; B/A for this line is increasing to the right (m = B/A);0714

and the slope here equals -B/A, where the line is decreasing as we go towards the right.0724

So again, there are two ways to figure out these asymptotes.0734

You can just sketch it out by drawing this rectangle with vertices at (-A,B), (A,B), (-A,-B)...that is actually (A,B), positive (A,B)...or (-A,-B).0737

Draw that rectangle and extend the diagonals.0754

Or you can use the formula, which will give you the slope for these two asymptotes.0758

If you just started out knowing the A's and B's and drew these, then you could easily sketch the hyperbola,0765

because you know that it is going to approach these asymptotes.0772

OK, so we have talked a lot about graphing.0778

And just to bring it all together: you are going to begin by writing the equation in standard form.0780

And then, for hyperbolas, you are going to graph the two asymptotes, as I just showed.0786

So, let's start out with an example: let's make this x2/9 - y2/4 = 1.0790

Since I have this in the form x2/A2 (this x2 term is positive here),0801

divided by y2/B2 = 1, what I have is a horizontal transverse axis.0809

So, this tells me that there is a horizontal transverse axis.0819

So, that is how this is just roughly sketched out already, showing the transverse axis along here.0830

Since it is in this form, I know that A2 = 9; therefore, A = 3.0841

This has a center right here at (0,0).0853

And this point here is going to be A, which is 3, 0.0860

Right here, I am going to have -A, or -3, 0.0867

So, my goal is to make that rectangle extend out the diagonals.0872

And then, I would be able to graph this correctly.0876

OK, B2 = 4; therefore, B = 2; so right up here at (I'll put that right there) (0,2)...that is going to be B.0882

And then, right down here at (0,-2)...0903

Now, all I have to do is extend the line up here and here; and these are going to meet at (2,3).0907

Extend a line out here; I am going to have a vertex right here at (-3,2).0916

I am going to have another vertex here at (-3,-2), and then finally, one over here at (3,-2).0924

Now, this was already sketched on here for me; but assuming it was not there, I would have started out by drawing this box,0936

and then, drawing these lines extending out--the asymptotes.0944

And what is going to happen is that this hyperbola is actually going to approach, but it is never going to intersect with, the asymptote.0960

So again, write the equation in standard form, which might require completing the square.0972

I gave it to you in standard form already; use that to figure out this rectangle.0977

And you are going to need to know A and B to figure out this rectangle.0982

Draw the asymptotes, and draw then the hyperbola approaching (but not reaching) those asymptotes.0985

So far, we have been talking about hyperbolas with a center at the origin (0,0).0993

However, that is not going to always be the case.1000

If the center is at another point, (h,k), that is not (0,0), then standard form looks like this.1002

It is very similar to what we saw with the origin of the center, except instead of just x2/A2, we now have an h and a k.1008

For a horizontal transverse axis, you are going to have (x - h)2/A2 - (y - k)2/B2.1015

For a vertical transverse axis, this term is going to be first; it will be positive.1027

And then, you are going to subtract (x - h)2/B2.1033

But the k stays associated with the y term.1037

For example, given this equation, (y - 3)2 - (x - 2)2...and we are going to divide that by 16,1041

and divide this by 9, and set it all equal to 1: what this is telling me is that the center is at (2,3), because this is h;1053

that A2 = 16, so A = 4; and that B2 = 9, so B = 3.1065

From that, I can graph out this hyperbola.1072

And this has a vertical transverse axis; something else to be careful of--let's say I had something like this:1076

(y + 5)/10, the quantity squared, plus (x + 4), the quantity squared, divided by 12, equals 1.1087

The center is actually at (that actually should be a negative right here--this is a difference) (-4,-5).1101

And the reason for that is that this is the same as (y - -5)2, and then (x - -4)2.1112

A negative and a negative is a positive.1125

So, you need to be careful: even though it is acceptable to write it like this, it is good practice,1128

if you are trying to figure out what the center is, to maybe write it out like this,1134

so that you have a negative here, so that whatever is in here is already k, or already h.1139

You don't have to say, "Oh, I need to make that a negative; I need to change the sign."1145

So, that is just something to be careful of.1148

Here, I already had negative signs in here; they are completely in standard form--I have h and k here; h and k is (-4,-5).1151

All right, to get some practice, we are going to first find the equation of a hyperbola that I am going to give you some information on.1160

I will give you that one of the vertices is at (0,2); the other vertex is at (0,-2).1168

The other piece of information is that you have a focus at (0,4), and a focus called f2 at (0,-4).1178

So, looking at this, I can see that this is the transverse axis, and then the center is right there.1188

So, I have a horizontal transverse axis.1196

I can also see that the midpoint right here, the center, is at the origin; so the center equals (0,0).1207

So, this is actually vertical--correction--a vertical transverse axis, going up and down: a vertical transverse axis.1217

Since this is actually a vertical transverse axis with a center at (0,0), I am working with this standard form:1227

y2/A2 - x2/B2 = 1.1234

So, the A2 term is with the y2 term, since this is a vertical transverse axis.1240

All right, in order to find the equation, I need to find A2.1247

This distance, from 0 to the vertex, is 2, because this is at (0,2).1251

Therefore, A equals 2; since A = 2, A2 = 22, or 4.1258

I have A2; I need to find B2; I am not given that.1267

But what I am given is an additional piece of information, and that is that there is a focus here and a focus here.1270

This allows me to find C: the distance from the center to either focus (let's look at this one)--from the center, 0, down to -4--1280

the absolute value of that is 4; therefore, C = 4; the distance is 4.1291

C2, therefore, equals 42, or 16.1298

Recall the relationship: C2 = A2 + B2 for a hyperbola.1303

So, I have C2, which is 16, equals A2, which is 4, plus B2.1309

16 - 4 is 12; 12 = B2; therefore, B = √12, which is about 3.5.1315

If you wanted to draw B, then you could, because that is right here at (3.5,0).1330

But what we are just asked to do is write the equation; and we have enough information to do that,1338

because I have that y2 divided by A2; I determined that A2 is 4;1342

minus x2/B2; I determined that that is 12; equals 1.1348

So, this is the equation for this hyperbola, with a vertical transverse axis and a center at (0,0) in standard form.1355

The next example: Find the equation of the hyperbola satisfying vertices at (-5,0) and (5,0) and a conjugate axis that has a length of 12.1368

Just sketching this out to get a general idea of what we are looking at--just a rough sketch--vertices are at (-5,0) and (5,0).1380

That means that the center is going to be right here at (0,0).1400

So, the center is at the origin; since the vertices are here and here, then I have a horizontal transverse axis;1406

this is going to go through like this, and then like this.1424

So, my second piece of information is that I have a horizontal transverse axis.1432

Since I have a horizontal transverse axis, then I am going to have an equation in the form x2/A2 - y2/B2 = 1.1440

The center is at the origin; it has a horizontal transverse axis; this is a standard form that I am working with.1451

I need to find A: well, I know that the center is here, and that A is this length; so from this point to the center,1457

or from this point (the vertex) to the center, is 5: A = 5.1466

Since A = 5, A2 = 52; it equals 25.1473

The other information I have is that the conjugate axis has a length of 12.1486

So, the length of the conjugate axis, recall, is 2B; here they are telling me that that length is 12.1490

Therefore, 12/2 gives me B; B = 6; so, that would be up here and here: (0,6) and then (0,-6).1503

This would be the conjugate axis; so this is B = 6.1517

Since B equals 6, I want B2 that equals 62, which equals 36.1523

Now, I can write this equation: I have (this is my final one) x2/A2, which is 25,1533

minus y2/B2, and I determined that that is 36, equals 1.1545

So, this is a hyperbola with a center at the origin.1551

And A2 is 25; B2 is 36; and it has a horizontal transverse axis.1554

We are asked to graph this equation; and it is not in standard form.1566

But when I look at it, I see that I have a y2 term and an x2 term, and they have opposite signs.1571

So, I am working with the difference between a y2 term1577

and an x2 term, which tells me that this is the equation for a hyperbola.1579

If they were a sum, this would have been an ellipse, since they have different coefficients.1585

But it is a difference, so it is a graph of a hyperbola.1589

What I need to do is complete the square to get this in standard form.1591

OK, so first I am grouping y terms and x terms: y2 + 12y - 6x2 + 12x - 36 = 0.1597

What I am going to do is move this 36 to the other side and get that out of the way for a moment by adding 36 to both sides.1617

The next thing I need to do with completing the square is factor out the leading coefficient, since it is something other than 1.1627

So, from the y terms, I will factor out a 2; that is going to leave me with y2 + 6y.1634

You have to be careful here, because you are factoring out a -6, so I need to make sure that I worry about the signs.1640

And that is going to leave behind an x2 here; here, it is going to leave behind, actually, -2x.1649

So, checking that, -6 times x2 is -6x2--I got that back.1656

-6 times -2x is + 12x; equals 36.1661

Now, to complete the square, I have to add b2/4 in here, which equals...b is 6; 62/4 is 36/4; that is 9.1669

I need to be careful to keep this equation balanced.1686

Now, this is really 9 times 2 that I am adding; 9(2) = 18--I need to add that to the right.1688

Working with the x terms: b2/4 = 22/4, which is 4/4; that is 1, so I am going to add 1 here.1697

-6 times 1 needs to be added to the other side; so I am going to actually subtract 6 from the right to keep it balanced.1714

Now, I am rewriting this as (y + 3)2 - 6(x - 1)2 =...18 - 6 is 12; 36 + 12 gives me 48.1726

The next step, because standard form would have a 1 on this side, is: I need to set all this equal to 1.1747

I need to divide both sides of the equation by 48.1753

This cancels, so it becomes y + 32; the 2 is gone; this becomes a 24; minus...6 cancels out,1769

and that leaves me with (x - 1)2; 6 goes into 48 eight times; and this is a 1.1780

OK, so it is a lot of work just to get this to the point where it is in standard form.1788

But once it is in standard form, we can do the graph, because now I know the center; I know A2 and B2.1792

We have this in standard form; so now we are going to go ahead and graph it.1799

I will rewrite the standard form that we came up with, (y + 3)2/24 - (x - 1)2/8 = 1.1803

Looking at this; since this is positive, I see that I have a vertical transverse axis.1815

The other thing to note is this plus here: recall that, if you have (y + 3)2, this is the same as (y - -3)2.1824

And when we look at standard form, we actually have a negative here.1835

So, you need to be careful to realize that the center is at (1,-3), not at (1,3).1838

Let's make this 2, 4, 6, 8, -2, -4, -6, -8; the center, then, is going to be at (1,-3).1846

The next piece of information: A2 = 24; therefore, A = √24, which is approximately 4.9.1856

B2 = 8; therefore, B = √8; therefore, if you figure that out on your calculator, that is approximately 2.8.1871

Since I have the center, and I have A and B, I can draw the rectangle that will allow me to extend diagonals out to form the asymptotes.1883

The goal is to write this in standard form, find A and B, find the center, make the rectangle, and make the asymptotes;1892

and then, you can finally draw both branches of the hyperbola.1902

All right, so if the center is here at (1,3), then I am going to have 2 vertices.1906

And what is going to happen, since this is a vertical transverse axis, is: one vertex is going to be up here; the other is going to be down here.1912

The center is at (1,3); that means I am going to have a vertex at 1, and then it is going to start at the center,1926

and then it is going to be 4.9 directly above that center; so that means this is going to be at -3 + 4.9.1933

The y-coordinate will be at -3 + 4.9, which equals (1,1.9).1944

Therefore, at (1,1.9) (that is right there)--that is where there is going to be one vertex.1950

And this is A--this is the length of A.1958

The second vertex is going to be at (1,-3); that is the center; and then I am going to go down 4.9--that is the length, again, of A.1963

That is -3 - 4.9 (or + -4.9; you can look at it that way) = (1,-7.9), down here.1979

OK, vertices are at (1,-7.9) and (1,1.9).1999

Now, I need to find where B is--where that endpoint over here is, horizontally--so that I can make this rectangle.2014

I know that B equals approximately 2.8; that means that I am going to have a point over here at 1 + 2.8...-3.2024

Well, 1 + 2.8 is (3.8,-3); so (3.8,-3) is right there.2040

I can reflect across; and I am going to have a point at 1 - 2.8, -3, which is going to give me...1 - 2.8 is (-1.8,-3).2054

That is going to be...this is 2...-2 is right here; so that is going to be right about there.2068

I now have these points; and recall that I can then extend out to make a box.2076

There is going to be a vertex here; I am going to extend across; there is going to be a vertex here.2083

Bring this directly down; there is a vertex here, and then another vertex right here.2090

Again, I got these points by knowing where the center is, knowing where the vertices of the hyperbola are, and then knowing the length of B.2096

This is B; then this length is A.2109

Once I have this rectangle, I can go ahead and draw the asymptotes by extending diagonals out.2113

Another way to approach this, recall, would have been to use the formula for the slope that we discussed, for the slope of the asymptotes.2130

Either method works.2140

I know that I am going to have a hyperbola branch up here; the vertex is right here, and it is going to approach, but never reach, this asymptote.2143

It is going to do the same thing with the other branch: a vertex is here; it is going to approach, but never reach, the asymptote.2156

OK, so this was a difficult problem; we were given an equation in this form.2170

We had to do a lot of work just to get it in standard form.2175

And then, once we did, we were able to find the center and form this rectangle, draw the asymptotes, and then (at last) graph the hyperbola.2178

Example 4: We don't need to do graphing on this one.2189

We are just finding the equation of a hyperbola with the center here, (0,0), and a horizontal transverse axis.2192

I am going to stop right there and think, "OK, I have a center at (0,0) and a horizontal transverse axis."2200

So, the standard form is going to be x2/A2 - y2/B2 = 1.2206

Since the center is at (0,0), I don't have to worry about h and k.2215

The horizontal transverse axis has a length of 12; well, the transverse axis length, recall, is equal to 2A.2219

I am given that that length is 12; if I take 12/2, that is going to give me A = 6.2231

The conjugate axis--recall that the length of the conjugate axis is equal to 2B, which is 6: B = 3.2240

Now, I need to find A2, which is 62, or 36, to put in here.2257

B2 is 32, which is 9.2264

Now, x2/36 - y2/B2 (which is 9) = 1.2267

So, this is the equation for a hyperbola with the center at the origin, a horizontal transverse axis, and a conjugate axis with a length of 6.2279

That concludes this lesson on hyperbolas; thanks for visiting Educator.com!2290

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