INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Parabolas

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

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

20m 15s

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

19m 10s

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

17m 31s

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

17m 14s

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

25m

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

32m 5s

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

14m 46s

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

23m 7s

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

23m 5s

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

31m 5s

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

21m 42s

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

17m 13s

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

23m 53s

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

27m 12s

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

28m 53s

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

11m 34s

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

21m 36s

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

29m 36s

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

33m 13s

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

28m 25s

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

22m 25s

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

22m 32s

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

31m 48s

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

27m 3s

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

19m 53s

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

35m 45s

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

27m 11s

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

22m 48s

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

30m 7s

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

27m 5s

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

19m 29s

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

13m 27s

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

31m 11s

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

22m 30s

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

33m 29s

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

21m 10s

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

31m 21s

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

31m 27s

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

31m 16s

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

34m 30s

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

22m 42s

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

30m 4s

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

20m 46s

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

41m 11s

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

30m 45s

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

31m 27s

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

40m 54s

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

55m 4s

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

57m 13s

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

20m 21s

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

55m 14s

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

35m 58s

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

45m 54s

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

28m 43s

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

25m 23s

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

21m 14s

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

24m 30s

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

32m 42s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

47m 4s

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

21m 16s

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

21m 36s

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

23m 3s

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

22m 43s

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

18m 32s

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

14m 34s

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

48m 30s

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

0 answers

Post by Hannah Yao on August 9, 2022

In example 1, shouldn't the vertex be 2, -2 not -2, 2?

1 answer

Last reply by: Dr Carleen Eaton
Sun Jun 1, 2014 10:00 PM

Post by Sophie Zhong on April 26, 2014

Does the equation h=-b/2a and k=-(b^2-4ac)/4a also apply in horizontal parabolas? Because when I used it for example 1, I came up with h=-(-12/6)=2, k=-2, which is wrong.

1 answer

Last reply by: Dr Carleen Eaton
Thu May 16, 2013 11:10 PM

Post by Saki Amagai on May 16, 2013

I don't know why but.. I'm having trouble watching this entire conic section due to technical issues. I don't have any problem for the other ones. It's just this section that I get "network failure". I really need to watch this... Can you please check if there's nothing wrong with the server? Thank you.

1 answer

Last reply by: Dr Carleen Eaton
Sun Jan 27, 2013 1:01 PM

Post by Monis Mirza on January 26, 2013

how do you find the maximum and minimum of a parabola using the equation?
i have a test on this on monday and i really need the answer!

0 answers

Post by julius mogyorossy on January 11, 2013

Merc, I think you are correct. I dig it that Educator is being advertised on my blog page. Educator said, learn like you are going to live forever, it seems somebody there knows who I am.

1 answer

Last reply by: Dr Carleen Eaton
Thu Feb 9, 2012 7:43 PM

Post by Edmund Mercado on February 9, 2012

For Horizontal Parabolas at 18:57, should the standard form say x = a(y-k)^2 +h instead of
y = a(x-k)^2 +h with the x and y in opposite positions?

0 answers

Post by norman stradleigh on June 21, 2011

thanks really helped me out

1 answer

Last reply by: Dr Carleen Eaton
Mon Jul 5, 2010 5:15 PM

Post by Timothy miranda on June 21, 2010

thanks that cleared it up for me

Parabolas

  • Understand the geometric significance of the sign of the coefficient of the squared term in the equation of a parabola.
  • Use the axis of symmetry to help you graph a parabola.
  • Know the standard formula for a parabola.
  • Review how to complete the square.
  • If the coefficient of the squared term is not 1, then before completing the square, you must first factor this coefficient out of both the squared term and the linear term.

Parabolas

Write in standard form and identify the key features
x = y2 − 4y + 1
  • Write in standard form by completing the square, isolate the y on the right.
  • x − 1 = y2 − 4y
  • Add [(b2)/4] on both sides
  • x − 1 + [(b2)/4] = y2 − 4y + [(b2)/4]
  • x − 1 + [(( − 4)2)/4] = y2 − 4y + [(( − 4)2)/4]
  • x − 1 + 4 = (y − 2)2
  • x + 3 = (y − 2)2
  • x = (y − 2)2 − 3
  • Identify the key features:
a) Horizontal Parabola
b) Vertex = (h,k) = ( − 3,2)
c) a = 1,a > 0; Parabola opens to the right
d) Axis of symmetry = k = 2 = y = 2
Graph: x = y2 − 4y + 1
  • Notice this is the same problem as question 1 Notice this is the same problem as question # 1
  • x = (y − 2)2 − 3
  • Sketch the graph with the following information
  • a) Horizontal Parabola
  • b) Vertex = (h,k) = ( − 3,2)
  • c) a = 1,a > 0 Parabola opens to the right
  • d) Axis of symmetry = k = 2 = y = 2
Write in standard form and identify the key features
x = − 7y2 − 14y − 11
  • Write in standard form by completing the square, isolate the y on the right.
  • x + 11 = − 7y2 − 14y
  • Factor out a − 7 in order to complete the square
  • x + 11 = − 7(y2 + 2y)
  • Add [(b2)/4] on both sides. When adding to the left side, multiply by − 7
  • x + 11 + − 7( [(b2)/4] ) = − 7(y2 + 2y + [(b2)/4])
  • x + 11 + − 7( [(22)/4] ) = − 7(y2 + 2y + [(22)/4])
  • x + 11 − 7 = − 7(y + 1)2
  • x + 4 = − 7(y + 1)2
  • x = − 7(y + 1)2 − 4
  • Identify the key features:
a) Horizontal Parabola
b) Vertex = (h,k) = ( − 4, − 1)
c) a = − 7,a < 0 Parabola opens to the left
d) Axis of symmetry = k = − 4 = y = − 4
Graph: x = − 7y2 − 14y − 111
  • Notice this is the same problem as question 3 Notice this is the same problem as question # 3
  • x = − 7(y + 1)2 − 4
  • Sketch the graph with the following information
  • a) Horizontal Parabola
  • b) Vertex = (h,k) = ( − 4, − 1)
  • c) a = − 7,a < 0 Parabola opens to the left
  • d) Axis of symmetry = k = − 4 = y = − 4
  • Since a is a whole number, your parabola will be thinner. Use couple of points to get the right shape.
Write in standard form and identify the key features
x = 3y2 − 24y + 43
  • Write in standard form by completing the square, isolate the y on the right.
  • x − 43 = 3y2 − 24y
  • Factor out a 3 in order to complete the square
  • x − 43 = 3(y2 − 8y)
  • Add [(b2)/4] on both sides. When adding to the left side, multiply by 3
  • x − 43 + 3( [(b2)/4] ) = 3(y2 − 8y + [(b2)/4])
  • x − 43 + 3( [( − 82)/4] ) = 3(y2 − 8y + [( − 82)/4])
  • x − 43 + 48 = 3(y − 4)2
  • x + 5 = 3(y − 4)2
  • x = 3(y − 4)2 − 5
  • Identify the key features:
a) Horizontal Parabola
b) Vertex = (h,k) = ( − 5,4)
c) a = 3,a > 0 Parabola opens to the right
d) Axis of symmetry = k = 4 = y = 4
Graph: x = 3y2 − 24y + 43
  • Notice this is the same problem as question 5 Notice this is the same problem as question # 5
  • x = 3(y − 4)2 − 5
  • Sketch the graph with the following information
  • a) Horizontal Parabola
  • b) Vertex = (h,k) = ( − 5,4)
  • c) a = 3,a > 0 Parabola opens to the right
  • d) Axis of symmetry = k = 4 = y = 4
  • Since a is a whole number, your parabola will be somewhat thinner. Use couple of points to get the right shape.
Write in standard form and identify the key features
x = 2y2 + 8y + 6
  • Write in standard form by completing the square, isolate the y on the right.
  • x − 6 = 2y2 + 8y
  • Factor out a 2 in order to complete the square
  • x − 6 = 2(y2 + 4y)
  • Add [(b2)/4] on both sides. When adding to the left side, multiply by 2
  • x − 6 + 2( [(b2)/4] ) = 2(y2 + 4y + [(b2)/4])
  • x − 6 + 2( [(42)/4] ) = 2(y2 + 4y + [(42)/4])
  • x − 6 + 8 = 2(y + 2)2
  • x + 2 = 2(y + 2)2
  • x = 2(y + 2)2 − 2
  • Identify the key features:
a) Horizontal Parabola
b) Vertex = (h,k) = ( − 2, − 2)
c) a = 3,a > 0 Parabola opens to the right
d) Axis of symmetry = k = − 2 = y = − 2
Graph: x = 2y2 + 8y + 6
  • Notice this is the same problem as question 7 Notice this is the same problem as question # 7
  • x = 2(y + 2)2 − 2
  • Sketch the graph with the following information
  • a) Horizontal Parabola
  • b) Vertex = (h,k) = ( − 2, − 2)
  • c)a = 3,a > 0 Parabola opens to the right
  • d) Axis of symmetry = k = − 2 = y = − 2
  • Since a is a whole number, your parabola will be somewhat thinner. Use couple of points to get the right shape.
Find the equation of the parabola with vertex ( − 7,5) and focus ( − 7,4). Draw the graph.
  • Standard form of a parabola is y = a(x − h)2 + k where vertex = (h,k)
  • Focus is located at (h,k + [1/4a])
  • Find a given the focus ( − 7,4)
  • k + [1/4a] = 4
  • 5 + [1/4a] = 4
  • [1/4a] = − 1
  • [1/a] = − 4
  • a = − [1/4]
  • Write the quation given a and vertex
  • y = − [1/4](x + 7)2 + 5
Find the equation of the parabola with vertex ( − 2,1) and focus ( − 2,2). Draw the graph.
  • Standard form of a parabola is y = a(x − h)2 + k where vertex = (h,k)
  • Focus is located at (h,k + [1/4a])
  • Find a given the focus ( − 2,2)
  • k + [1/4a] = 2
  • 1 + [1/4a] = 2
  • [1/4a] = 1
  • [1/a] = 4
  • a = [1/4]
  • Write the quation given a and vertex
  • y = [1/4](x + 2)2 + 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

Parabolas

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

  • Intro 0:00
  • What is a Parabola? 0:20
    • Definition of a Parabola
    • Focus
    • Directrix
    • Axis of Symmetry
  • Vertex 3:33
    • Minimum or Maximum
  • Standard Form 4:59
    • Horizontal Parabolas
    • Vertex Form
    • Upward or Downward
    • Example: Standard Form
  • Graphing Parabolas 8:31
    • Shifting
    • Example: Completing the Square
    • Symmetry and Translation
    • Example: Graph Parabola
  • Latus Rectum 17:13
    • Length
    • Example: Latus Rectum
  • Horizontal Parabolas 18:57
    • Not Functions
    • Example: Horizontal Parabola
  • Focus and Directrix 24:11
    • Horizontal
  • 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

Transcription: Parabolas

Welcome to Educator.com.0000

Today, we are going to talk about parabolas.0002

And in some earlier lectures in this series on quadratic equations, we talked about parabolas and did some graphing.0004

But now, we are going to go on and give a specific definition to parabolas, and learn about some other features of parabolas.0011

Although you have seen parabolas previously, when we graphed, we didn't form a specific definition of them.0021

So, the definition of a parabola is that it is the set of points in the plane whose distance from a given point,0027

called the focus, is equal to its distance from a given line, called the directrix.0033

Let's talk about that before we go on to talk about the axis of symmetry.0041

So, if you had a parabola (let's say right here; and we will do an upward-facing parabola), you would have some point,0045

which is known as the focus, and a line (I'm going to put that right about here) called the directrix.0059

By definition, every point on this parabola is equidistant from the focus and the directrix.0080

So, if I took a point right here, and I measured the distance from the focus, it would be equal to the distance from the directrix.0085

And this is just a very rough sketch; but these distances actually would be equal; they are theoretically equal.0095

Looking right here at the vertex, these distances would be equal; so that would be, say, y.0108

If I took some other point, say here, and I measured here to here, these two distances would be equal.0117

So, a couple things to note: the focus is inside the parabola; the directrix is outside.0128

And this is because the focus and the directrix are on the opposite sides of the vertex.0150

So, you could have a parabola facing downward, and then it would have a focus here and a directrix up here.0155

We are also going to talk, today, about parabolas that face to the left and right--horizontal parabolas.0165

But right now, we are going to stick with just (for this discussion) focusing on vertical ones,0173

the definition being that every point in the parabola equidistant between the focus and the directrix.0180

The axis of symmetry of the parabola passes through the focus; and it is perpendicular to the directrix.0189

In this case, the y-axis is the axis of symmetry; it is right here.0195

And you see that it passes through the focus, and it forms a right angle; it is perpendicular to the directrix.0204

Again, we talked about some of these concepts in earlier lectures.0214

But to review, vertex: the vertex of a parabola is the point at which the axis of symmetry intersects the parabola.0216

And it is a maximum or minimum point on the parabola, if the axis of symmetry is vertical.0224

If the axis of symmetry is horizontal (say we have a parabola like this, then the axis of symmetry would be horizontal),0230

we still have a vertex, but it is not a maximum or minimum.0242

And again, we are going to focus a little more on vertical parabolas right now, and then we will talk about horizontal parabolas.0247

So, if I have a downward-facing parabola, the vertex is here; the axis of symmetry is right here.0254

And this vertex is the maximum; this is as large as y gets--it is the largest value that the function attains.0265

If I am looking at a vertex that is upward-facing, then the axis of symmetry...we will put it right here; and the vertex is here.0274

In this case, the vertex is a minimum; this is the smallest value that the function will attain.0286

The standard form of a parabola with vertex at (h,k) is y = a(x - h)2 + k.0299

And this is for vertical parabolas; there is a slightly different form when we are talking about horizontal parabolas.0308

And you might recall this form of the equation that we covered earlier on, under the lecture on quadratic equations.0314

And we called this the vertex form of the equation; now we are going to refer to it as standard form.0319

And it is a very useful form, because it tells you a lot about the parabola.0324

The axis of symmetry is x = h: so I know a few things just from looking at this.0330

I know the vertex, because it is (h,k); I know the axis of symmetry--it is at x = h;0335

and if I look at a, I will know if the parabola is upward- or downward-facing.0341

If a is greater than 0, the parabola will open upward; and k gives you the minimum.0348

If a is a negative value--if it is less than 0--the parabola opens downward, and k is the maximum value of the function.0357

Let's look at an example: y = 2(x - 1)2 + 4.0367

So, this is in standard form: this means that I have h = 1, k = 4, and a = 2.0374

So, I know that my vertex is going to be at (1,4); the axis of symmetry is going to be at x = h, so at x = 1.0384

And since a is greater than 0, this opens upward.0404

So, I can sketch this out: I have a vertex at (1,4), right here, and it opens upward.0411

And the axis of symmetry is going to be right here at x = 1.0421

Here is my vertex at (1,4); and this vertex is a minimum, because this opens upward.0428

The minimum value is k, which is 4.0433

If I were to take a similar situation, but say y = -2(x - 1)2 + 4,0441

I would have, again, an h equal to 1 and a k equal to 4, but this time a would be -2, so this would open downward.0453

What I would end up with would be a parabola here, again, with the vertex at (1,4).0465

But it would open downward, and therefore, this would be a maximum.0473

Also, if the absolute value of a is greater than 1, you end up with a relatively narrow parabola.0481

If the absolute value of a is less than 1, you end up with a relatively wide parabola.0490

So, this form is very useful, because just by having the equation in this form, we can at least sketch the graph.0500

Let's talk a little bit more about graphing parabolas.0508

You can use symmetry and translations to graph a parabola: and by translations, we mean a shift.0511

Looking at the standard form: what this really is: if you took a graph of y = ax2, this is letting h equal 0 and k equal 0.0519

And then, if you altered what h is, it is going to shift the graph horizontally by that number of units.0530

If you alter what k is, it is going to translate or shift that graph upward and downward by a certain number of units.0538

In order to graph a parabola, you often need to put it in standard form.0547

Let's start out by just talking about putting an equation or a parabola in standard form.0552

And then we will go on and look at some graphs, and how different values of h and k can affect the graph.0556

So, in order to put the equation into standard form...let's say you are given an equation such as this, y = x2 + 6x - 8,0562

and I want it in this standard form, y = a(x - h)2 + k.0572

The first thing to do (and this is, again, review from an earlier lesson--you can go back and look at the lesson0580

on completing the square as part of this lecture series, but we will review it again now): first, I am going0586

to isolate the x variable terms on the right side of the equation.0592

I am going to add 8 to both sides: now I am going to complete the square.0596

I am going to focus on this, and I need to add a term to it to make this a perfect square trinomial.0602

The term I am going to add is going to be b2/4.0608

In this case, b is 6, so this is going to give me 62/4, which is 36/4, which is equal to 9.0613

So, that is what I need to add in here: y + 8, plus I need to add 9 to both sides.0627

It is easy to forget to add it to the other side, because you get so focused on completing the square.0640

But if you don't, the equation will no longer be balanced.0645

So, I am going to add 9 to both sides.0648

And I want this to end up in this form; so I am going to rewrite this.0653

First I will add these two together to simplify to get y + 17 =...well, this is a perfect square trinomial, so I just take (x + 3)2.0657

And I look at what I have, and it is almost in this form, but not quite.0668

I want to isolate y on the left, so I am going to subtract 17 from both sides to get y = (x + 3)2 - 17.0671

And this is in this form: a happens to be equal to 1 in this case.0679

And so, if you are given an equation that is not in standard form, and you want to get it in standard form,0683

isolate the x variable values on the right (although if we are working with horizontal parabolas,0690

it is going to be the other way around, as we will see in a minute--we are actually going to end up0697

getting the y variable terms on the right; but for now, the x variable terms on the right); complete the square0701

by adding the b2/4 term to both sides of the equation; and then simplify;0707

shift things around as needed to get it in this form.0716

Remember, also, that if you have a leading coefficient that is something other than 1,0718

when you get to this step after isolating the x variable terms, you are going to need to factor out that term before completing the square.0724

All right, assuming that you have gotten your equation in standard form, and you are ready to graph the parabola, you are going to use symmetry.0734

The two halves of the parabola are symmetrical; if you graph half the points, you can use reflection across the axis of symmetry to graph the other points.0741

And translation is knowing how h and k, and changes in h and k, affect the graph, in order to graph.0749

All right, so let's just start out with something in this form--a very basic equation for a parabola.0760

Let's let f(x) equal x2, so it is in this form: y = ax2.0767

And so, here, what is happening is: if you think about what we have, we have a = 1, and then h is 0 and k is 0.0775

What this tells me is that the vertex is going to be at (0,0), and the axis of symmetry is going to be at x = 0.0784

And you can also very easily find some points to graph this.0796

right now, I am just going to sketch it out, and not worry about exact points, just so you get the idea.0800

So, since a = 1, this is going to open upward; this is going to be upward-opening, so the vertex is here at (0,0);0805

it is upward-opening; and it is going to look something like this.0819

So, this is my graph here of y, or f(x), = x2.0833

Now, let's say I change this slightly: let's say I have another function, g(x) = x2 + 2.0838

So, looking at this form, h is still 0; but now I have k = 2.0847

And according to this, this is going to shift the graph up 2 units; so k is going to translate this graph up 2 units.0855

I have a similar graph, but it is going to be with the vertex right here at (0,2).0867

And remember that the axis of symmetry is at x = h, so the axis of symmetry is going to still be at x = 0; right here--this is the axis of symmetry.0880

This is shifted upward; it still opens upward, because a is positive.0888

So, now I am just going to have a similar idea, but shifted upward by 2.0892

So here, I have y = x2 + 2.0901

If this had been a -2, then it would have been shifted down by 2, and I would have had a graph right here.0907

So, let's see what happens when I change h.0912

Let's get a third function: we will call it h(x) = (x - 1)2.0917

OK, now what I have here is h = 1; k, if I look here, is 0.0935

Therefore, the vertex (this is the vertex right here) equals (1,0), and the axis of symmetry is going to be at x = 1.0947

So, this is going to be shifted to the right; so I am going to have a graph something...let me move this out of the way...like this.0962

So, this one is y = x2, and this is y = (x - 1)2.0979

Important take-home points: a change in h will shift the graph horizontally, to the right or left.0991

A change in k will shift the basic graph either up or down, by k number of units.1001

Using symmetry: if I were to graph these out exactly, I would need to find points.1009

And I don't need to find all of the points: for example, if I had a parabola that was a downward-facing parabola1013

somewhere, then I could use the axis of symmetry, and I could just find the points over here and reflect across that axis in order to graph.1021

All right, this concept is another one adding on to our knowledge of parabolas from prior lessons.1034

And it is defining a segment called the latus rectum.1040

The latus rectum is the segment passing through the focus and perpendicular to the axis of symmetry.1044

Let's see what that means--let's visualize that.1051

Let's say I have a parabola like this, and let's say the focus is here.1054

So, this passes through the focus, and is perpendicular to the axis of symmetry.1065

This is the focus, and here we have the axis of symmetry.1072

That means the latus rectum is going to pass through here, and it is going to be perpendicular to the axis of symmetry.1083

So, that is this line; this is the latus rectum.1090

The equation for its length is the absolute value of 1/a; and if you have the equation of the parabola in standard form,1095

then this a is the same a as you will see in that formula.1108

So, this is something you might occasionally need to use.1112

For example, if I were given an equation of a parabola y = 2(x - 3)2 + 5,1115

and I was asked to find the length of the latus rectum of this parabola, then I would just say,1122

"OK, a equals 2; therefore, the length equals the absolute value of 1/2."1127

Horizontal parabolas: I mentioned that you can also have parabolas that open to the right or left, not just up and down,1138

although up to this point in the course, we have just talked about vertical parabolas, or parabolas that open upward or downward.1143

For parabolas whose axis of symmetry is horizontal, we end up with equations in this form: y = a(x - k)2 + h.1150

So, one thing to note: the positions of the x's and y's are reversed, but so are the h's and k's.1160

In the vertical formula, the h was in here, and the k was out here.1168

So, be careful when you are working with this formula to notice that the positions of h and k are reversed.1171

And there are translations of x = ay2, and then again, h and k shift this graph around horizontally and vertically.1177

So, it would look something like this, for example: the axis of symmetry would be right here;1189

and it would be a horizontal axis of symmetry; or maybe I have one that opens to the left, and it has an axis of symmetry right here.1197

These do not represent functions; and you can see that they don't represent functions1208

by trying to pass a vertical line through them: they fail the vertical line test.1212

Remember: with a function, the vertical line test tells us that a vertical line drawn...you could try1216

any possible area of the curve, and the vertical line will only cross the curve once.1224

If the vertical line crosses the curve more than once, it is not a function.1230

So, this fails the vertical line test.1233

It is not a function; it is still an equation--you can still make a graph of it; but horizontal parabolas do not represent functions.1241

I am working on graphing some horizontal parabolas.1250

When you look at the equation in standard form, y = a...and remember, the k and h are in opposite positions;1253

they are reversed...looking at a, if a is positive (if a is greater than 0), then the parabola is going to open to the right.1263

If a is negative, then the parabola is going to open to the left.1273

So, let's look at a very simple horizontal parabola, x = y2.1278

OK, the vertex is at (h,k); and I can see that h and k are both 0, so the vertex equals (0,0).1284

The axis of symmetry is at y = k, so that is going to be at y = 0.1294

And the a here is 1: a = 1, so this opens to the right.1298

So, you are going to have a parabola that looks something like this.1308

You could have another parabola, x = -y2.1321

Here we would have the same vertex and the same axis of symmetry; here the x-axis is actually the axis of symmetry.1325

And I look at a now, and a equals -1, so this parabola is going to open to the left.1333

So, I am going to end up with a parabola like this.1343

Now again, change in h or change in k is going to shift this parabola a bit.1350

Let's change h and see what happens: let's let x equal y2 + 2.1358

Here I have h = 2, k = 0; so (2,0) is the vertex; a = 1, so it is positive, so this still opens to the right.1366

If I look at this, x = y2...here is my graph of x = y2; over here is x = -y2.1379

Now, I am going to have h = 2, so that is going to shift horizontally by 2.1386

(2,0) will be the vertex; and it is going to open to the right.1397

So, this is x = y2 over here; right here, this is actually x = y2 + 2 now.1402

And k, as discussed before, shifts the graph of a parabola vertically.1416

The same idea here: if I were to change k, then I would shift this graph up or down by k units.1423

So, with horizontal parabolas, you need to be familiar with this equation.1430

You need to know that they open to the right if a is greater than 0; they open to the left if a is less than 0.1435

The vertex is at (h,k), and the axis of symmetry is y = k.1441

And you also need to keep in mind that these do not represent functions.1446

In the beginning of today's lesson, we talked about the focus and directrix.1452

And here are formulas to allow you to find those if you need to.1455

If you have a vertical parabola, the coordinates of the focus are h for the x-coordinate, and k plus 1/4a.1460

And the equation for the directrix is y = k - 1/4a; remember that the directrix is a line, so this is giving you the equation for that line.1473

And this would be for a vertical parabola; for a horizontal parabola, the focus is found at the coordinates h + 1/4a;1485

and then the y-coordinate is k, so the focus is a point, and this gives the coordinates of that point.1493

The directrix is a line, and the equation for this line for a horizontal parabola is x = h - 1/4a.1498

And you might need to occasionally use these when we are working problems.1505

And we will see that in one of the examples, actually, shortly.1508

Starting out with Example 1: Write in standard form and identify the key features: x = 3y2 - 12y + 10.1513

We have x equal to all of this; so this tells me, since I have x set equal to this y2 term, that I am looking at a horizontal parabola.1524

So, the standard form of this equation is going to be x = a(y - k)2 + h.1536

Remember, h and k are going to be in opposite positions.1547

In order to get this equation in standard form, we need to complete the square.1550

This time, since I am working with a horizontal parabola, I am going to isolate all of the y variable terms on the right.1554

And I am going to do that by subtracting 10 from both sides to get x - 10 = 3y2 - 12y.1561

This leading coefficient is not 1, so I have to factor it out.1569

And then, I have to be really careful when I am adding to both sides of the equation, because this is factored out.1573

So, factor out a 3 to get y2 - 4y.1580

I need to complete the square: that means I need to add something over here.1585

And the term that I need to add is going to be b2/4.1589

b is actually 4; so this is going to be 42/4, equals 16/4, equals 4.1594

Here is where I need to be careful: on the right, I am adding 4 inside these parentheses, which is pretty straightforward.1604

But what I need to do on the left is realize that I am actually going to be adding 3 times 4, which is 12.1613

So, if I were just to add 4, this equation would not be balanced,1628

because in reality, what I am doing over here is adding 3 times 4.1631

So, on the right, I am going to add 12; and I got that from 3 times 4.1635

Simplifying the left: 12 - 10 is 2; on the right, inside here, I now have a perfect square.1640

And I want this to end up in this form, so I am going to write this as (y - 2) (and it is negative, because I end up with a negative sign in here) squared.1649

I am almost done; I just need to move this constant over to the right to have it in this form.1661

x = 3 times (y - 2)2, minus 2.1666

So, now that I have this in standard form, I can identify key features.1671

Key features: 1: this is a horizontal parabola, as you can see from looking at this equation.1677

2: The vertex is at (h,k); h is 2, and k is also 2.1686

Actually, being careful with the signs, h is actually -2, because remember, standard form has a plus here.1703

I don't have a plus here; I could rewrite this so that I do, and that would give me + -2.1710

And it is good practice, actually, to write it exactly in this form, although this is correct--you could leave it like this.1718

By writing it in this form...and the same thing if I had ended up with a plus here--then I would need to rewrite that,1725

because here I need a negative to be in standard form; if I ended up with a plus here,1736

then I would have needed to rewrite that, as well, which would have been equal to minus -2.1741

Standard form, just like this, looking here, gives me a vertex at (-2,2).1748

And because a equals 3, that means that a is greater than 0; a is positive, so the parabola opens to the right.1755

OK, so key features: horizontal parabola; it has a vertex at (-2,2); a = 3, so this tells me that the parabola opens to the right.1773

We can also say that the axis of symmetry is at y = k, and therefore the axis of symmetry is at y = 2.1784

OK, in Example 2, we are asked to graph.1797

And you will notice that this is the same equation that we worked with in Example 1.1802

We already figured out standard form: and standard form is x = 3(y - 2)2 - 2.1806

And for clarity, we can actually write this as I did at the end, which is 3(y - 2)2 + -2,1816

so that we truly have it in standard form, with the plus here to make it easy to see what is going on.1826

To graph this, I want to know the vertex: the vertex is (h,k): h here is -2; k is 2.1831

The axis of symmetry is going to be at y = k; k is 2, so it is going to be at y = 2.1840

I know that this opens to the right, so I have a general sense of this graph.1854

But I can also just find a few points.1859

And we are used to working with a situation where x is the input and y is the output.1868

It is the opposite here, so we need to be really careful.1873

I also want to note that, since the vertex is here at (-2,2), and this opens to the right,1876

for this graph, we are not going to have values of x that are smaller than -2.1881

So, if I end up with something where an x is smaller than -2, then it is going to be off the graph.1885

Let's let y equal 1: if y is 1, 1 - 2 is -1, squared gives me 1; 1 times 3 is 3, minus 2 is 1; so, when y is 1, x is 1.1891

Let's let y equal 3: when y is 3, 3 minus 2 is 1, squared is 1; 1 times 3 is 3, minus 2 is 1.1906

And you can see, as I mentioned, that this is not a function; it failed the vertical line test (as horizontal parabolas do).1915

And you can see that there is an x-value, 1, that is assigned 2 values of y; so it does not meet the definition of a function.1921

So, just a couple of points...let's do one more: 0...0 minus 2 is -2; squared is 4; 4 times 3 is 12; 12 minus 2 is 10.1929

So, that is off this graph; but it gives us an idea of the shape.1941

So, I know that my axis of symmetry is going to be here; and I have a point at (1,1);1945

I have another point at (1,3); and then I have a point way out here at (10,0).1954

I know that this is going to be a fairly narrow graph, because a equals 3.1960

This is the graph of the horizontal parabola described by this equation; and here it is, written in standard form.1971

So, it opens to the right; it is fairly narrow, because a equals 3.1979

It has a vertex at (-2,2), and it has an axis of symmetry at y = 2.1984

Example 3: we are asked to graph; this is also going to be a horizontal parabola.1994

We are going to start out by putting it in the standard form, x = (y - k)2 + h.1999

We need to complete the square; start out by isolating the y variable terms on the right.2009

So, I am going to add 6 to both sides to get -2y2 + 8y.2014

Since the leading coefficient is not 1, I need to factor it out; so I am going to factor this -2 to get y2.2021

Factoring a -2 from here would give me a -4.2032

And I need to add something to this to complete the square.2035

What I need to add is b2/4.2040

b is 4, so I am going to be adding 42/4; that is 16, divided by 4; that is 4.2044

So, I am going to be adding 4 to the right; but to the left, I am actually adding -2 times 4, which is -8.2061

So, we subtract 8 from that side; to this side, since I am adding inside the parentheses, I am just adding 4.2073

But then, 4 times -2--that is how I got the -8 on the left.2082

This gives me x - 2 = -2; and I want it in this form, so I am going to rewrite this as (y - 2)2.2085

The last thing I need to do is add 2 to both sides; and I have it in standard form.2097

Now that I have this in standard form, it is much easier to graph.2106

The vertex is going to be at (h,k); so h is here; k is here; the vertex is at (2,2).2110

There is going to be an axis of symmetry at y = k, and so that is going to be at y = 2; my axis of symmetry is going to be at y = 2.2119

Now, to finish out graphing this, I am going to find a few points.2142

I have the vertex at (2,2); I also know that a is less than 0 (a is negative), so I know this is going to open to the left.2146

So, I know it is going to look something like this; but I will find a couple of points.2155

And I know that x is (actually, (2,2) is right here)...I know that this opens to the left, and that x is not going to get any larger than that.2158

The graph is just going to go this way.2172

So, I can't use values that end up giving me an x that is greater than 2.2174

Let's try some simple values: I am going to try 1 for y, and looking in standard form, 1 - 2 gives me -1, squared is 1, times -2 is -2, plus 2 is 0.2181

And 3: 3 minus 2 is 1, squared is 1; 1 times -2 is -2, plus 2 is 0.2195

So, I have a couple of points here: this is at 0...when x is 0, y is 1; when x is 0, y is 3.2204

And this is going to give me a parabola shaped like this, opening to the left with a vertex at (2,2).2214

The axis of symmetry would be right through here; and I have a couple of points, just to make it a bit more precise.2226

So, the first step in graphing a parabola is always to get it into this form by completing the square.2235

And then, using the features you can see from here, sketch it out, and finding a few points, make the graph more accurate.2240

Find the equation of the parabola with a vertex of (2,3) and focus at (2,7); draw the graph.2250

This is a very challenging problem: we are not given an equation--we actually have to find the equation based on some key points that we are given.2256

Well, I am given that the vertex is at (2,3); so I know that the vertex is right here; that is the vertex.2266

This time, I am also given the focus; the focus is at (2,7), which is going to be up here somewhere...5, 6, 7...about here.2278

So, the vertex is (2,3); the focus is (2,7).2290

Remember, in the beginning of this lesson, I mentioned that the focus is always inside the parabola.2296

Since the focus is inside the parabola, I already know that this has to open upward.2301

So, I know something about the shape of the graph.2306

Let's find the equation: now, I know that this is a vertical parabola, because the focus is inside the parabola.2311

That told me that this has to open upward, so I know I am dealing with a vertical parabola.2316

And that helps me to find the equation, because the standard form is going to be y = a(x - h)2 + k.2320

I am given the vertex, so I am given h and k: I know that h = 2 and k = 3.2330

In order to write this equation, I need a, h, and k; all I am missing is a.2340

I am given the piece of information, though, that the focus is (2,7).2347

And that is going to allow me to find a.2350

You will recall that I mentioned the formulas for focus and directrix.2353

And for a vertical parabola, the focus is at h...the x-coordinate is h, which we see here; and the y-coordinate is k + 1/4a.2359

And I know the focus is at (2,7): so 2 = h, and 7 = k + 1/4a, according to this definition.2377

Well, since I know that k is 3, then I can solve this.2392

So, I know k; so I can solve for a.2401

Subtract 3 from both sides to get 1/4a; 1/a equals 16; multiply both sides by a, and then divide both sides by 16,2408

or just take the reciprocal of each side (essentially, that is what you are doing) to get a = 1/16.2422

Now, I have h and k given; I was able to figure out a, based on the definition of focus.2427

So, I end up with the equation y = 1/16(x - 2)2 + 3.2433

So, this is the equation.2444

And as you know, once we have the equation, the graphing is pretty easy.2446

I know that this opens upward; and since I know what a is, I know that this is going to be a pretty wide parabola; the a is a small value.2451

I am going to have a parabola that opens upward, with a vertex of (2,3), and fairly wide in shape.2461

That was a pretty challenging problem, because you had to go back2470

and think about how you could use a formula to find the focus; and knowing the focus allowed you to find a.2473

That concludes this lesson on parabolas at Educator.com; thanks for visiting!2483

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