INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Circles

Dr. Carleen Eaton

Dr. Carleen Eaton

Circles

Slide Duration:

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
Loading...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Algebra 2
Bookmark & Share Embed

Share this knowledge with your friends!

Copy & Paste this embed code into your website’s HTML

Please ensure that your website editor is in text mode when you paste the code.
(In Wordpress, the mode button is on the top right corner.)
  ×
  • - Allow users to view the embedded video in full-size.
Since this lesson is not free, only the preview will appear on your website.

  • Discussion

  • Answer Engine

  • Study Guides

  • Practice Questions

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Lecture Comments (14)

1 answer

Last reply by: Dr Carleen Eaton
Sun Feb 11, 2018 9:45 PM

Post by Peggy Chen on July 20, 2017

Hi. For the last example. I think there can be two answers.

(X-1)^2+(Y-6)^2=36
(X-1)^2+(Y+6)^2=36

as the center could be both above and below the x axis

1 answer

Last reply by: Dr Carleen Eaton
Sat Apr 19, 2014 12:47 AM

Post by Robert Monett on March 31, 2014

How do you which quadrants to plot the points. Couldn't the circle be below the x axis?

1 answer

Last reply by: Dr Carleen Eaton
Sun Mar 11, 2012 7:28 PM

Post by Jeff Mitchell on March 9, 2012

In the "Center not at origin" lecture section, I believe you forgot to square the (y-2) part of the equation at the bottom right side of the slide presentation.

1 answer

Last reply by: Dr Carleen Eaton
Mon Nov 7, 2011 8:40 PM

Post by Jonathan Taylor on November 4, 2011

(x-1)squared+(Y-3)SQUARED 3SQUARE IS 3*3=9 ARE AM I CONFUSED BY THE FORMULA

1 answer

Last reply by: Dr Carleen Eaton
Thu Oct 13, 2011 8:54 PM

Post by Manuela Fridman on October 9, 2011

Can you please explain how you put the other 2 points on the graph after you plotted (1,3) in example:circle. Also is there a way for us to reach the teacher better because i notice it takes weeks for anyone to respond.

1 answer

Last reply by: Dr Carleen Eaton
Fri Aug 12, 2011 6:57 PM

Post by Lee Fulton on July 29, 2011

Your demonstration was impeccable! I have chosen certain lectures from you in preparation for my GRE's to enter Temple University! Thanks! This was much better than that boring GRE Manual!
Lee

0 answers

Post by aloosh aloosh on March 20, 2011

help please how can we say in the last example that circle is tangent to two points on the x axis i think unless the circle tangent to the lines x=7 x=-5 not the x axis any one know if there is a mistake in the example ???? other wise the points could be at the bottom of the circle and not the diameter

0 answers

Post by Mohammed Jaweed on August 18, 2010

Great teaching style,

Way better than my teacher.

I like the step by step explanations and examples. Very productive lecture.

Thanks

Circles

  • Use symmetry to help you graph a circle.
  • Review completing the square and do so to put the equation of a circle in standard form.

Circles

Find the equation of a circle which has a diameter with end points ( − 1, − 10), ( − 11, 16)
  • Recall that the equation of a circle is (x − h)2 + (y − k)2 = r2 where the center is (h,k)
  • Step 1 - Find the center using the midpoint formula
  • C = ( [( − 1 + − 11)/2],[( − 10 + 16)/2] ) = ( − 6,3)
  • Step 2 - Find the radius of the circle.This is the distanc between the center and one of the end - points
  • r = √{(x2 − x1)2 + (y2 − y1)}
  • r = √{( − 6 − ( − 1)2 + (3 − ( − 10)2} = √{( − 5)2 + (13)2} = √{194}
  • r2 = ( √{194} )2 = 194
  • Step 3 - Write the equation given the center = ( − 6,3) and r2 = 194
  • (x − h)2 + (y − k)2 = r2
(x + 6)2 + (y − 3)2 = 194
Find the equation of a circle which has a diameter with end points (10, − 8), (10, 0)
  • Recall that the equation of a circle is (x − h)2 + (y − k)2 = r2 where the center is (h,k)
  • Step 1 - Find the center using the midpoint formula
  • C = ( [(10 + 10)/2],[( − 8 + 0)/2] ) = (10, − 4)
  • Step 2 - Find the radius of the circle.This is the distanc between the center and one of the end - points
  • r = √{(x2 − x1)2 + (y2 − y1)}
  • r = √{(10 − (10))2 + ( − 4 − (0)2} = √{(0)2 + ( − 4)2} = √{16}
  • r2 = ( √{16} )2 = 16
  • Step 3 - Write the equation given the center = (10, − 4) and r2 = 16
  • (x − h)2 + (y − k)2 = r2
(x − 10)2 + (y + 4)2 = 16
Find the equation of a circle which has a diameter with end points (9,9), ( − 7, − 9)
  • Recall that the equation of a circle is (x − h)2 + (y − k)2 = r2 where the center is (h,k)
  • Step 1 - Find the center using the midpoint formula
  • C = ( [(9 − 7)/2],[(9 − 9)/2] ) = (1,0)
  • Step 2 - Find the radius of the circle.This is the distanc between the center and one of the end - points
  • r = √{(x2 − x1)2 + (y2 − y1)}
  • r = √{(1 − (9))2 + (0 − (9)2} = √{( − 8)2 + ( − 9)2} = √{145}
  • r2 = ( √{145} )2 = 145
  • Step 3 - Write the equation given the center = (1,0) and r2 = 145
  • (x − h)2 + (y − k)2 = r2
  • (x − 1)2 + (y + 0)2 = 145
(x − 1)2 + y2 = 145
Find the equation of a circle which has a diameter with end points (9,9), ( − 7, − 9)
  • Recall that the equation of a circle is (x − h)2 + (y − k)2 = r2 where the center is (h,k)
  • Step 1 - Find the center using the midpoint formula
  • C = ( [(9 − 7)/2],[(9 − 9)/2] ) = (1,0)
  • Step 2 - Find the radius of the circle.This is the distanc between the center and one of the end - points
  • r = √{(x2 − x1)2 + (y2 − y1)}
  • r = √{(1 − (9))2 + (0 − (9)2} = √{( − 8)2 + ( − 9)2} = √{145}
  • r2 = ( √{145} )2 = 145
  • Step 3 - Write the equation given the center = (1,0) and r2 = 145
  • (x − h)2 + (y − k)2 = r2
  • (x − 1)2 + (y + 0)2 = 145
(x − 1)2 + y2 = 145
Find the center and radius of the circle with equation:
x2 + y2 + 2x − 8y + 16 = 0
  • Step 1: Group together the x's and y's inside parenthesis, constant temrs need to be moved to the right side of the equation.
  • (x2 + 2x) + (y2 − 8y) = − 16
  • Step 2: Complete the square for x′ and for y′ terms by using the formula [(b2)/4]. Whatever is added to the left must be added to the right of the equation.
  • (x2 + 2x + [(b2)/4]) + (y2 − 8y + [(b2)/4]) = − 16 + [(b2)/4] + [(b2)/4]
  • (x2 + 2x + [(22)/4]) + (y2 − 8y + [(( − 8)2)/4]) = − 16 + [(22)/4] + [(( − 8)2)/4]
  • Factor both the x′ and ′y terms using the Perfect Square Trinomial equation x2 + bx + [(b2)/4] = (x + [b/2])2
  • (x + 1)2 + (y − 4)2 = − 16 + 1 + 16
  • (x + 1)2 + (y − 4)2 = 1
  • Step 3 - Find the center = (h,k) and radius
Center = (h,k) = ( − 1,4)
r = 1
Find the center and radius of the circle with equation:
x2 + y2 + 2x + 6y + 9 = 0
  • Step 1: Group together the x's and y's inside parenthesis, constant temrs need to be moved to the right side of the equation.
  • (x2 + 2x) + (y2 + 6y) = − 9
  • Step 2: Complete the square for x′ and for y′ terms by using the formula [(b2)/4]. Whatever is added to the left must be added to the right of the equation.
  • (x2 + 2x + [(b2)/4]) + (y2 + 6y + [(b2)/4]) = − 9 + [(b2)/4] + [(b2)/4]
  • (x2 + 2x + [(22)/4]) + (y2 + 6y + [(62)/4]) = − 9 + [(22)/4] + [(62)/4]
  • Factor both the x′ and ′y terms using the Perfect Square Trinomial equation x2 + bx + [(b2)/4] = (x + [b/2])2
  • (x + 1)2 + (y + 3)2 = − 9 + 1 + 9
  • (x + 1)2 + (y + 3)2 = 1
  • Step 3 - Find the center = (h,k) and radius
Center = (h,k) = ( − 1, − 3)
r = 1
Find the center and radius of the circle with equation:
x2 + y2 + 2y − 25 = 0
  • Step 1: Group together the x's and y's inside parenthesis, constant temrs need to be moved to the right side of the equation.
  • (x2 + 0x) + (y2 + 2y) = 25
  • Step 2: Complete the square for x' and for y' terms by using the formula [(b2)/4]. Whatever is added to the left must be added to the right of the equation.
  • (x2 + 0x + [(b2)/4]) + (y2 + 2y + [(b2)/4]) = 25 + [(b2)/4] + [(b2)/4]
  • (x2 + 0x + [(02)/4]) + (y2 + 2y + [(22)/4]) = 25 + [(02)/4] + [(22)/4]
  • Factor both the x′ and ′y terms using the Perfect Square Trinomial equation x2 + bx + [(b2)/4] = (x + [b/2])2
  • (x + 0)2 + (y + 1)2 = 25 + 0 + 1
  • (x + 0)2 + (y + 1)2 = 26
  • Step 3 - Find the center = (h,k) and radius
Center = (h,k) = (0, − 1)
r2 = 26 = √{r2} = √{26}
r = √{26}
Find the radius of the circle with center ( − 15, − 12) and tangent to x = − 18
  • Step 1 - Plot the Center and draw line through x = − 18
  • Step 2 - Count the number of units the center is away from the tangent line. Label it r.
r = 3
Find the radius of the circle with center ( − 12,12) and tangent to y = 17
  • Step 1 - Plot the Center and draw line through y = 17
  • Step 2 - Count the number of units the center is away from the tangent line. Label it r.
r = 5
Find the radius of the circle with center ( − 2, − 16) and tangent to x = − 4
  • Step 1 - Plot the Center and draw line through x = − 4
  • Step 2 - Count the number of units the center is away from the tangent line. Label it r.
r = 2
Find the equation of the circle that is tangent to the y = 0, y = 6, x = − 2 and x = 4.
  • We're going to solve this problem by inspection.
  • Step 1 - graph all four lines
  • y = 0
  • y = 6
  • x = − 2
  • x = 4
  • Step 2 - Find the center of the resulting box
  • C = (1,3)
  • Step 3 - Find the distance between the center and any of the 4 tangent points, label this distance r
  • Step 4 - Write the equation
  • (x − 1)2 + (y − 3)3 = 32
(x − 1)2 + (y − 3)3 = 9

*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

Circles

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 Circles? 0:08
    • Example: Equidistant
    • Radius
  • Equation of a Circle 0:44
    • Example: Standard Form
  • Graphing Circles 1:47
    • Example: Circle
  • Center Not at Origin 3:07
    • Example: Completing the Square
  • 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

Transcription: Circles

Welcome to Educator.com.0000

Today we are going to talk about circles, beginning with the definition of a circle.0002

A circle is defined as the set of points in the plane equidistant from a given point, called the center.0008

For example, if you had a center of a circle here, and you measured any point's distance from the center, these would all be equal.0018

And the radius is the segment with endpoints at the center and at a point on the circle.0031

The equation for the circle is given as follows: if the center is at (h,k) and the radius is r,0044

then the equation is (x - h)2 + (y - k)2 = r2.0052

And this is the standard form; and just as with parabolas, the standard form gives you a lot of useful information0060

and allows you to graph what you are trying to graph.0067

For example, if I were given (x - 4)2 + (y - 5)2 = 9, then I would have a lot of information.0071

I would know that my center is at (h,k), so it is at (4,5).0084

And the radius...r2 = 9; therefore, r = √9, which is 3.0091

So, based on this information, I could work on graphing out my circle.0099

Use symmetry to graph a circle, as well as what you discover from looking at the equation in standard form.0108

Looking at a different equation, (x - 1)2 + (y - 3)2 = 4: this equation describes the circle0114

with the center at (h,k), which is (1,3); r2 = 4; therefore, r = 2.0124

So, I have a circle with a radius of 2 and the center at (1,3).0134

So, if this is (1,3) up here, and I know that the radius is 2, I would have a point here; I would have a point up here.0139

Symmetry: I know that, if I divide a circle up, I could divide it into four symmetrical quarters, for example.0156

So, if I have this graphed, I could use symmetry to find the other three sections of this circle.0169

All right, if the center is not at the origin, then we need to use completing the square to get the equation in standard form.0187

Remember that standard form of a circle is (x - h)2 + (y - k)2 = r2.0198

With parabolas, we put those in standard form by completing the square.0210

But at that time, we were just having to complete the square of either the x variable terms or the y variable terms.0213

Now, we are going to be working with both; and as always, we need to remember to add the same thing to both sides to keep the equation balanced.0219

If I was looking at something such as x2 + y2 - 4x - 8y - 5 = 0,0226

what I am going to do is keep all the x variable and y variable terms together, and then just move the constant over to the right.0241

The other thing I am going to do is group the x variable terms together: x2 - 4x is grouped together, just like up here.0249

And then, I am going to have y2 - 8y grouped together, and add 5 to both sides.0258

Now, I have to complete the square for both of these.0266

This is going to give me x2 - 4x, and then here I need to have b2/4.0269

Since b is 4, that is going to give me 42/4 = 16/4 = 4; so, I am going to put a 4 in here.0277

For the y expression, I am going to have...let's do this up here...b2/4 = 82/4, which is going to come out to 16.0292

So, I am going to add 16 here; and I need to make sure I do the same thing on the right.0307

So, I need to add 4, and I need to add 16; if I don't, this won't end up being balanced.0312

Now, I want this in this form; so let's change it to (x - 2)2 +...here it is going to be (y - 4)2 =...0321

4 and 16 is 20, plus 5; so that is 25.0335

This gives me the equation in standard form; and the center is at (h,k), (2,4).0341

And the radius...well, r2 is 25; therefore, the radius equals 5.0350

And as always, you need to be careful: let's say I ended up with something in this form, (x + 3)2 + (y - 2) = 9.0355

The temptation for the center might be just to put (3,2); but standard form says that this should be a negative.0365

So, I may even want to rewrite this as (x - (-3))2 + (y - 2) = 9, just to make it clear that the center is actually at (-3,2).0372

And then, the radius is going to be the square root of 9, which is 3.0387

So, be careful that you look at the signs; and if the signs aren't exactly the same as standard form,0390

you need to compensate for that, or even just write it out--because -(-3) would give me +3, so these two are interchangeable.0396

All right, in this example, we are asked to find the equation of the circle which has a diameter with the endpoints (-3,-7) and (9,-1).0405

So, let's see what we are working with.0413

Just sketch this out at (-3,-7), right about there; over here is (9,-1); there we have the diameter of the circle.0418

We have a circle like this, and we want to find the equation.0433

Recall that the formula for the equation of a circle is (x - h)2 + (y - k)2 = r2.0439

Therefore, I need to find h; I need to find k; and I need to find the radius.0449

Recall that (h,k) gives you the center of the circle.0453

Since this is the diameter of the circle, the center of this line segment is going to be the center of the circle.0460

So, (-3,-7)...over here I have (9,-1); and here I have the center--the center is going to be equal to the midpoint of this segment.0467

Recall the midpoint formula equals (x1 + x2)/2, and then (y1 + y2)/2.0477

So, the center--the coordinates for that are going to be equal to (-3 + 9)/2, and then (-7 + -1)/2,0491

which is going to be equal to 6/2...-7 and -1 is -8/2; which is equal to (3,-4).0507

This means that h and k are 3 and -4; so I have h and k; I need to find the radius.0520

Well, half of the diameter...this is all the diameter; this is my midpoint; and I know that this is at (3,-4).0526

So, I just need to find this length--this is the radius.0535

And I now have endpoints; so I can use either one of these--I have a set of endpoints here and here, and here and here.0539

I am going to go ahead and use these two, and put them into the distance formula.0549

(-3,-7) and (3,-4)--I can use these in the distance formula: the distance here equals the radius,0553

which is the square root of...I am going to make this (x1,y1), and then this (x2,y2).0562

So, this is going to give me...x2 is 3, minus -3, squared, plus...y2 is -4, minus -7, squared.0574

So, the radius equals the square root of 3 + 3; a negative and a negative is a positive; all of this squared,0589

plus -4...and a negative and a negative is a positive, so -4 + 7, squared.0599

So, the radius equals the square root of...3 + 3 gives me 6, squared, plus...7 - 4 gives me 3, squared.0608

So, the radius equals √(36 + 9); 36 plus 9 is 45, so the radius equals √45.0619

But what I really want for this is r2, so r2 is going to equal (√45)2, which is going to equal 45.0632

Putting this all together, I can write my equation, because I now have h; I have k; and I have r.0651

So, writing the equation up here gives me (x - 3)2 + (y...I can either write this as - -4,0658

or I can rewrite this as (y + 4)2 = r2, which is 45.0672

So, this, or a little more neatly, like this: (y + 4)2 = 45--this is the equation for the circle.0679

And I found that information based on simply knowing the diameter.0689

Knowing the diameter, I could use the midpoint formula to find the center, which gave me h and k.0693

And then, I could use the distance formula to find the distance from the center to the end of the diameter, which gave me the radius.0698

And then, I squared the radius and applied it to that formula.0707

Example 2: Find the center and radius of the circle with this equation.0711

In order to achieve that, we need to put this equation in standard form.0716

And recall that standard form of a circle is (x - h)2 + (y - k)2 = r2.0719

So, we need to complete the square: group the x variable terms together on the left; also group the y variable terms on the left.0730

Add 8 to both sides to move the constant over.0742

I need x2 - 8x + something to complete the square, and y2 + 10y + something to complete the square, equals 8.0746

So, for the x variable terms, I want b2/4, and this is going to be 82/4, or 64/4, equals 16.0759

So, I am going to add 16 here.0773

For the y variable terms, b2/4 is going to equal 102/4, which is 100/4, which is 25.0774

I need to be careful that I add the same thing to both sides to keep this equation balanced, so I am also going to add 16 and 10 to the right side.0785

Correction: it is 16 and 25--there we go.0805

(x - 4)2 equals this perfect square trinomial, and I am trying to get it in this form; that is what I want it to look like.0809

Plus...(y + 5)2 comes out to this perfect square trinomial.0820

On the right, if I add 8 and 16 and 25, I am going to end up with 49.0830

8 and 16 is going to give me 24, plus 25 is going to give me 49.0840

Now, I have this in standard form: because the center of a circle is (h,k), I know I have h here,0848

and I know I have k here, this is going to give me...h is 4; k is -5.0861

Be careful with the sign here, because notice: this is (y + 5), but standard form is - 5, so this is equal to (y - -5)2--the same thing.0869

It is just simpler to write it like this; but make sure you are careful with that.0880

The radius here: well, I have r2; the radius, r2, I know, is equal to 49.0884

Therefore, r = √49, so the radius equals 7.0892

Therefore, the center of this circle is at (4,-5), and the radius is equal to 7.0898

Example 3: Find the radius of the circle with the center at (-3,-4) and tangent to the y-axis.0909

This one takes more drawing and just thinking, versus calculating.0919

The center is at (-3,-4), right about here.0924

The other thing I know about this circle is that it is tangent to the y-axis.0933

That means that, if I drew the circle, it is going to extend around, and it is going to touch this y-axis.0939

Well, the radius is going to have one endpoint on the circle, and the other endpoint at the center.0948

Therefore, the radius has to extend from (-3,-4) over here.0954

And at this point, we are able to then find the length, because, since x is -3, and it has to go all the way to x = 0,0965

then this distance must be 3; therefore, the radius equals 3.0976

And again, that is because I know the center is here at -3, and I know0984

that the other endpoint of the radius is going to be out here, forming the circle,0988

and that, because it is tangent to the y-axis, x is going to be equal to 0 right here.0996

So, I know that x is equal to 0 here; and I know that x is equal to -3 over here; so it is just 1, 2, 3 over--this distance here is going to be 3.1003

And that is going to be the same as the radius, so the radius is equal to 3.1013

Example 4: Find the equation of the circle that is tangent to the x-axis, to x = 7, and to x = -5.1018

We are given a bunch of information about this circle and told to put it in standard form.1026

The first thing we are told is that it is tangent to the x-axis; so this circle is touching the x-axis; let's just draw a line here to emphasize that.1040

It is also tangent to x = 7; x = 7 is going to be right here--it is tangent to this.1049

It is also tangent to x = -5, out here.1061

I am going to end up with a circle that is touching, that is tangent to, these three things.1068

Let's think about what that tells me.1074

I need to find h and k (I need to find the center).1077

I also need to find the radius, so I can find r2.1081

If this extends from -5 to 7, that gives me the diameter.1086

So, the diameter goes from 7 all the way to -5; so if I just add 7 and 5 (the distance from here to here,1097

plus the distance from here to here), I am going to get that the diameter equals 12.1104

The radius is 1/2 the diameter, so the radius equals 6.1110

I found that the radius equals 6.1117

The radius is going to extend from these endpoints to the center.1122

And I know that it is 6, so I know that the radius is going to go from 7 over here, 6 away from that.1127

7 - 6 is 1; it is going to go up to x = 1.1136

Again, that is because the length of the radius is 6, so the distance between the center and this endpoint has to be 6.1141

7 - 6 is 1; the radius is going to extend from there to there.1152

Therefore, the x-value of the center has to be 1.1156

Now, what is the y-value of the radius? The other thing I know is that this circle is tangent to this x-axis.1164

So, I know that it is going to have an endpoint on the x-axis.1171

And then, if it is going to extend from here to here, it is going to have to go up to 6; therefore, the center is at (1,6).1175

All right, so the radius equals 6, and the center is at (1,6); and that gives me an equation:1193

(x - 1)2 + (y - 6)2 = the radius squared.1200

If r = 6, then r2 = 36.1208

Again, that is based on knowing that this is tangent to x = -5, x = 7, and the x-axis.1212

So, I had the diameter, 12; I divided that by 2 to get the radius; I know that I have an endpoint here and an endpoint at the center.1221

So, that gives me the x-value of the center, which is 1.1231

I know I have an endpoint here, and also an endpoint at the center; so it has to be up at 6.1233

That gives me (1,6) for my value.1238

OK, and this is just drawn schematically, because the center would actually be higher up here.1243

This is just to give you...the center is actually going to be up here, now that I have my value: it is going to be at (1,6).1247

OK, that concludes this lesson of Educator.com on circles; thanks for visiting!1258

Related Books

Related Books

Educator®

Please sign in to participate in this lecture discussion.

Resetting Your Password?
or
Still having trouble? Contact us
OR

Start Learning Now

Our free lessons will get you started (Adobe Flash® required).
 Get immediate access to our entire library.

Membership Overview

  • Available 24/7. Unlimited Access to Our Entire Library.
  • Search and jump to exactly what you want to learn.
  • *Ask questions and get answers from the community and our teachers!
  • Practice questions with step-by-step solutions.
  • Download lecture slides for taking notes.
  • Track your course viewing progress.
  • Accessible anytime, anywhere with our Android and iOS apps.