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Dr. Carleen Eaton

Graphing Rational Functions

Slide Duration:Table of Contents

I. Equations and Inequalities

Expressions and Formulas

22m 23s

- Intro0:00
- Order of Operations0:19
- Variable0:27
- Algebraic Expression0:46
- Term0:57
- Example: Algebraic Expression1:25
- Evaluate Inside Grouping Symbols1:55
- Evaluate Powers2:30
- Multiply/Divide Left to Right2:55
- Add/Subtract Left to Right3:35
- Monomials4:40
- Examples of Monomials4:52
- Constant5:27
- Coefficient5:46
- Degree6:25
- Power7:15
- Polynomials8:02
- Examples of Polynomials8:24
- Binomials, Trinomials, Monomials8:53
- Term9:21
- Like Terms10:02
- Formulas11:00
- Example: Pythagorean Theorem11:15
- Example 1: Evaluate the Algebraic Expression11:50
- Example 2: Evaluate the Algebraic Expression14:38
- Example 3: Area of a Triangle19:11
- Example 4: Fahrenheit to Celsius20:41

Properties of Real Numbers

20m 15s

- Intro0:00
- Real Numbers0:07
- Number Line0:15
- Rational Numbers0:46
- Irrational Numbers2:24
- Venn Diagram of Real Numbers4:03
- Irrational Numbers5:00
- Rational Numbers5:19
- Real Number System5:27
- Natural Numbers5:32
- Whole Numbers5:53
- Integers6:19
- Fractions6:46
- Properties of Real Numbers7:15
- Commutative Property7:34
- Associative Property8:07
- Identity Property9:04
- Inverse Property9:53
- Distributive Property11:03
- Example 1: What Set of Numbers?12:21
- Example 2: What Properties Are Used?13:56
- Example 3: Multiplicative Inverse16:00
- Example 4: Simplify Using Properties17:18

Solving Equations

19m 10s

- Intro0:00
- Translations0:06
- Verbal Expressions and Algebraic Expressions0:13
- Example: Sum of Two Numbers0:19
- Example: Square of a Number1:33
- Properties of Equality3:20
- Reflexive Property3:30
- Symmetric Property3:42
- Transitive Property4:01
- Addition Property5:01
- Subtraction Property5:37
- Multiplication Property6:02
- Division Property6:30
- Solving Equations6:58
- Example: Using Properties7:18
- Solving for a Variable8:25
- Example: Solve for Z8:34
- Example 1: Write Algebraic Expression10:15
- Example 2: Write Verbal Expression11:31
- Example 3: Solve the Equation14:05
- Example 4: Simplify Using Properties17:26

Solving Absolute Value Equations

17m 31s

- Intro0:00
- Absolute Value Expressions0:09
- Distance from Zero0:18
- Example: Absolute Value Expression0:24
- Absolute Value Equations1:50
- Example: Absolute Value Equation2:00
- Example: Isolate Expression3:13
- No Solution3:46
- Empty Set3:58
- Example: No Solution4:12
- Number of Solutions4:46
- Check Each Solution4:57
- Example: Two Solutions5:05
- Example: No Solution6:18
- Example: One Solution6:28
- Example 1: Evaluate for X7:16
- Example 2: Write Verbal Expression9:08
- Example 3: Solve the Equation12:18
- Example 4: Simplify Using Properties13:36

Solving Inequalities

17m 14s

- Intro0:00
- Properties of Inequalities0:08
- Addition Property0:17
- Example: Using Numbers0:30
- Subtraction Property1:03
- Example: Using Numbers1:19
- Multiplication Properties1:44
- C>0 (Positive Number)1:50
- Example: Using Numbers2:05
- C<0 (Negative Number)2:40
- Example: Using Numbers3:10
- Division Properties4:11
- C>0 (Positive Number)4:15
- Example: Using Numbers4:27
- C<0 (Negative Number)5:21
- Example: Using Numbers5:32
- Describing the Solution Set6:10
- Example: Set Builder Notation6:26
- Example: Graph (Closed Circle)7:08
- Example: Graph (Open Circle)7:30
- Example 1: Solve the Inequality7:58
- Example 2: Solve the Inequality9:06
- Example 3: Solve the Inequality10:10
- Example 4: Solve the Inequality13:12

Solving Compound and Absolute Value Inequalities

25m

- Intro0:00
- Compound Inequalities0:08
- And and Or0:13
- Example: And0:22
- Example: Or1:12
- And Inequality1:41
- Intersection1:49
- Example: Numbers2:08
- Example: Inequality2:43
- Or Inequality4:35
- Example: Union4:45
- Example: Inequality5:53
- Absolute Value Inequalities7:19
- Definition of Absolute Value7:33
- Examples: Compound Inequalities8:30
- Example: Complex Inequality12:21
- Example 1: Solve the Inequality12:54
- Example 2: Solve the Inequality17:21
- Example 3: Solve the Inequality18:54
- Example 4: Solve the Inequality22:15

II. Linear Relations and Functions

Relations and Functions

32m 5s

- Intro0:00
- Coordinate Plane0:20
- X-Coordinate and Y-Coordinate0:30
- Example: Coordinate Pairs0:37
- Quadrants1:20
- Relations2:14
- Domain and Range2:19
- Set of Ordered Pairs2:29
- As a Table2:51
- Functions4:21
- One Element in Range4:32
- Example: Mapping4:43
- Example: Table and Map6:26
- One-to-One Functions8:01
- Example: One-to-One8:22
- Example: Not One-to-One9:18
- Graphs of Relations11:01
- Discrete and Continuous11:12
- Example: Discrete11:22
- Example: Continous12:30
- Vertical Line Test14:09
- Example: S Curve14:29
- Example: Function16:15
- Equations, Relations, and Functions17:03
- Independent Variable and Dependent Variable17:16
- Function Notation19:11
- Example: Function Notation19:23
- Example 1: Domain and Range20:51
- Example 2: Discrete or Continous23:03
- Example 3: Discrete or Continous25:53
- Example 4: Function Notation30:05

Linear Equations

14m 46s

- Intro0:00
- Linear Equations and Functions0:07
- Linear Equation0:19
- Example: Linear Equation0:29
- Example: Linear Function1:07
- Standard Form2:02
- Integer Constants with No Common Factor2:08
- Example: Standard Form2:27
- Graphing with Intercepts4:05
- X-Intercept and Y-Intercept4:12
- Example: Intercepts4:26
- Example: Graphing5:14
- Example 1: Linear Function7:53
- Example 2: Linear Function9:10
- Example 3: Standard Form10:04
- Example 4: Graph with Intercepts12:25

Slope

23m 7s

- Intro0:00
- Definition of Slope0:07
- Change in Y / Change in X0:26
- Example: Slope of Graph0:37
- Interpretation of Slope3:07
- 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 Lines7:18
- Example: Not Vertical7:30
- Example: Vertical7:58
- Perpendicular Lines8:31
- Example: Perpendicular8:42
- Example 1: Slope of Line10:32
- Example 2: Graph Line11:45
- Example 3: Parallel to Graph13:37
- Example 4: Perpendicular to Graph17:57

Writing Linear Functions

23m 5s

- Intro0:00
- Slope Intercept Form0:11
- m and b0:28
- Example: Graph Using Slope Intercept0:43
- Point Slope Form2:41
- Relation to Slope Formula3:03
- Example: Point Slope Form4:36
- Parallel and Perpendicular Lines6:28
- Review of Parallel and Perpendicular Lines6:31
- Example: Parallel7:50
- Example: Perpendicular9:58
- Example 1: Slope Intercept Form11:07
- Example 2: Slope Intercept Form13:07
- Example 3: Parallel15:49
- Example 4: Perpendicular18:42

Special Functions

31m 5s

- Intro0:00
- Step Functions0:07
- Example: Apple Prices0:30
- Absolute Value Function4:55
- Example: Absolute Value5:05
- Piecewise Functions9:08
- Example: Piecewise9:27
- Example 1: Absolute Value Function14:00
- Example 2: Absolute Value Function20:39
- Example 3: Piecewise Function22:26
- Example 4: Step Function25:25

Graphing Inequalities

21m 42s

- Intro0:00
- Graphing Linear Inequalities0:07
- Shaded Region0:19
- Using Test Points0:32
- Graph Corresponding Linear Function0:46
- Dashed or Solid Lines0:59
- Use Test Point1:21
- Example: Linear Inequality1:58
- Graphing Absolute Value Inequalities4:50
- Graph Corresponding Equations4:59
- Use Test Point5:20
- Example: Absolute Value Inequality5:38
- Example 1: Linear Inequality9:17
- Example 2: Linear Inequality11:56
- Example 3: Linear Inequality14:29
- Example 4: Absolute Value Inequality17:06

III. Systems of Equations and Inequalities

Solving Systems of Equations by Graphing

17m 13s

- Intro0:00
- Systems of Equations0:09
- Example: Two Equations0:24
- Solving by Graphing0:53
- Point of Intersection1:09
- Types of Systems2:29
- Independent (Single Solution)2:34
- Dependent (Infinite Solutions)3:05
- Inconsistent (No Solution)4:23
- Example 1: Solve by Graphing5:20
- Example 2: Solve by Graphing9:10
- Example 3: Solve by Graphing12:27
- Example 4: Solve by Graphing14:54

Solving Systems of Equations Algebraically

23m 53s

- Intro0:00
- Solving by Substitution0:08
- Example: System of Equations0:36
- Solving by Multiplication7:22
- Extra Step of Multiplying7:38
- Example: System of Equations8:00
- Inconsistent and Dependent Systems11:14
- Variables Drop Out11:48
- Inconsistent System (Never True)12:01
- Constant Equals Constant12:53
- Dependent System (Always True)13:11
- Example 1: Solve Algebraically13:58
- Example 2: Solve Algebraically15:52
- Example 3: Solve Algebraically17:54
- Example 4: Solve Algebraically21:40

Solving Systems of Inequalities By Graphing

27m 12s

- Intro0:00
- Solving by Graphing0:08
- Graph Each Inequality0:25
- Overlap0:35
- Corresponding Linear Equations1:03
- Test Point1:23
- Example: System of Inequalities1:51
- No Solution7:06
- Empty Set7:26
- Example: No Solution7:34
- Example 1: Solve by Graphing10:27
- Example 2: Solve by Graphing13:30
- Example 3: Solve by Graphing17:19
- Example 4: Solve by Graphing23:23

Solving Systems of Equations in Three Variables

28m 53s

- Intro0:00
- Solving Systems in Three Variables0:17
- Triple of Values0:31
- Example: Three Variables0:56
- Number of Solutions5:55
- One Solution6:08
- No Solution6:24
- Infinite Solutions7:06
- Example 1: Solve 3 Variables7:59
- Example 2: Solve 3 Variables13:50
- Example 3: Solve 3 Variables19:54
- Example 4: Solve 3 Variables25:50

IV. Matrices

Basic Matrix Concepts

11m 34s

- Intro0:00
- What is a Matrix0:26
- Brackets0:46
- Designation1:21
- Element1:47
- Matrix Equations1:59
- Dimensions2:27
- Rows (m) and Columns (n)2:37
- Examples: Dimensions2:43
- Special Matrices4:22
- Row Matrix4:32
- Column Matrix5:00
- Zero Matrix6:00
- Equal Matrices6:30
- Example: Corresponding Elements6:36
- Example 1: Matrix Dimension8:12
- Example 2: Matrix Dimension9:03
- Example 3: Zero Matrix9:38
- Example 4: Row and Column Matrix10:26

Matrix Operations

21m 36s

- Intro0:00
- Matrix Addition0:18
- Same Dimensions0:25
- Example: Adding Matrices1:04
- Matrix Subtraction3:42
- Same Dimensions3:48
- Example: Subtracting Matrices4:04
- Scalar Multiplication6:08
- Scalar Constant6:24
- Example: Multiplying Matrices6:32
- Properties of Matrix Operations8:23
- Commutative Property8:41
- Associative Property9:08
- Distributive Property9:44
- Example 1: Matrix Addition10:24
- Example 2: Matrix Subtraction11:58
- Example 3: Scalar Multiplication14:23
- Example 4: Matrix Properties16:09

Matrix Multiplication

29m 36s

- Intro0:00
- Dimension Requirement0:17
- n = p0:24
- Resulting Product Matrix (m x q)1:21
- Example: Multiplication1:54
- Matrix Multiplication3:38
- Example: Matrix Multiplication4:07
- Properties of Matrix Multiplication10:46
- Associative Property11:00
- Associative Property (Scalar)11:28
- Distributive Property12:06
- Distributive Property (Scalar)12:30
- Example 1: Possible Matrices13:31
- Example 2: Multiplying Matrices17:08
- Example 3: Multiplying Matrices20:41
- Example 4: Matrix Properties24:41

Determinants

33m 13s

- Intro0:00
- What is a Determinant0:13
- Square Matrices0:23
- Vertical Bars0:41
- Determinant of a 2x2 Matrix1:21
- Second Order Determinant1:37
- Formula1:45
- Example: 2x2 Determinant1:58
- Determinant of a 3x3 Matrix2:50
- Expansion by Minors3:08
- Third Order Determinant3:19
- Expanding Row One4:06
- Example: 3x3 Determinant6:40
- Diagonal Method for 3x3 Matrices13:24
- Example: Diagonal Method13:36
- Example 1: Determinant of 2x218:59
- Example 2: Determinant of 3x320:03
- Example 3: Determinant of 3x325:35
- Example 4: Determinant of 3x329:22

Cramer's Rule

28m 25s

- Intro0:00
- System of Two Equations in Two Variables0:16
- One Variable0:50
- Determinant of Denominator1:14
- Determinants of Numerators2:23
- Example: System of Equations3:34
- System of Three Equations in Three Variables7:06
- Determinant of Denominator7:17
- Determinants of Numerators7:52
- Example 1: Two Equations8:57
- Example 2: Two Equations13:21
- Example 3: Three Equations17:11
- Example 4: Three Equations23:43

Identity and Inverse Matrices

22m 25s

- Intro0:00
- Identity Matrix0:13
- Example: 2x2 Identity Matrix0:30
- Example: 4x4 Identity Matrix0:50
- Properties of Identity Matrices1:24
- Example: Multiplying Identity Matrix2:52
- Matrix Inverses5:30
- Writing Matrix Inverse6:07
- Inverse of a 2x2 Matrix6:39
- Example: 2x2 Matrix7:31
- Example 1: Inverse Matrix10:18
- Example 2: Find the Inverse Matrix13:04
- Example 3: Find the Inverse Matrix17:53
- Example 4: Find the Inverse Matrix20:44

Solving Systems of Equations Using Matrices

22m 32s

- Intro0:00
- Matrix Equations0:11
- Example: System of Equations0:21
- Solving Systems of Equations4:01
- Isolate x4:16
- Example: Using Numbers5:10
- Multiplicative Inverse5:54
- Example 1: Write as Matrix Equation7:18
- Example 2: Use Matrix Equations9:12
- Example 3: Use Matrix Equations15:06
- Example 4: Use Matrix Equations19:35

V. Quadratic Functions and Inequalities

Graphing Quadratic Functions

31m 48s

- Intro0:00
- Quadratic Functions0:12
- A is Zero0:27
- Example: Parabola0:45
- Properties of Parabolas2:08
- Axis of Symmetry2:11
- Vertex2:32
- Example: Parabola2:48
- Minimum and Maximum Values9:02
- Positive or Negative9:28
- Upward or Downward9:58
- Example: Minimum10:31
- Example: Maximum11:16
- Example 1: Axis of Symmetry, Vertex, Graph12:41
- Example 2: Axis of Symmetry, Vertex, Graph17:25
- Example 3: Minimum or Maximum21:47
- Example 4: Minimum or Maximum27:09

Solving Quadratic Equations by Graphing

27m 3s

- Intro0:00
- Quadratic Equations0:16
- Standard Form0:18
- Example: Quadratic Equation0:47
- Solving by Graphing1:41
- Roots (x-Intercepts)1:48
- Example: Number of Solutions2:12
- Estimating Solutions9:23
- Example: Integer Solutions9:30
- Example: Estimating9:53
- Example 1: Solve by Graphing10:52
- Example 2: Solve by Graphing15:10
- Example 1: Solve by Graphing17:50
- Example 1: Solve by Graphing20:54

Solving Quadratic Equations by Factoring

19m 53s

- Intro0:00
- Factoring Techniques0:15
- Greatest Common Factor (GCF)0:37
- Difference of Two Squares1:48
- Perfect Square Trinomials2:30
- General Trinomials3:09
- Zero Product Rule5:22
- Example: Zero Product5:53
- Example 1: Solve by Factoring7:46
- Example 1: Solve by Factoring9:48
- Example 1: Solve by Factoring12:34
- Example 1: Solve by Factoring15:28

Imaginary and Complex Numbers

35m 45s

- Intro0:00
- Properties of Square Roots0:10
- Product Property0:26
- Example: Product Property0:56
- Quotient Property2:17
- Example: Quotient Property2:35
- Imaginary Numbers3:12
- Imaginary i3:51
- Examples: Imaginary Number4:22
- Complex Numbers7:23
- Real Part and Imaginary Part7:33
- Examples: Complex Numbers7:57
- Equality9:37
- Example: Equal Complex Numbers9:52
- Addition and Subtraction10:12
- Examples: Adding Complex Numbers10:25
- Complex Plane13:32
- Horizontal Axis (Real)13:49
- Vertical Axis (Imaginary)13:59
- Example: Labeling14:11
- Multiplication15:57
- Example: FOIL Method16:03
- Division18:37
- Complex Conjugates18:45
- Conjugate Pairs19:10
- Example: Dividing Complex Numbers20:00
- Example 1: Simplify Complex Number24:50
- Example 2: Simplify Complex Number27:56
- Example 3: Multiply Complex Numbers29:27
- Example 3: Dividing Complex Numbers31:48

Completing the Square

27m 11s

- Intro0:00
- Square Root Property0:12
- Example: Perfect Square0:38
- Example: Perfect Square Trinomial3:00
- Completing the Square4:39
- Constant Term4:50
- Example: Complete the Square5:04
- Solve Equations6:42
- Add to Both Sides6:59
- Example: Complete the Square7:07
- Equations Where a Not Equal to 110:58
- Divide by Coefficient11:08
- Example: Complete the Square11:24
- Complex Solutions14:05
- Real and Imaginary14:14
- Example: Complex Solution14:35
- Example 1: Square Root Property18:31
- Example 2: Complete the Square19:15
- Example 3: Complete the Square20:40
- Example 4: Complete the Square23:56

Quadratic Formula and the Discriminant

22m 48s

- Intro0:00
- Quadratic Formula0:21
- Standard Form0:29
- Example: Quadratic Formula0:57
- One Rational Root3:00
- Example: One Root3:31
- Complex Solutions6:16
- Complex Conjugate6:28
- Example: Complex Solution7:15
- Discriminant9:42
- Positive Discriminant10:03
- Perfect Square (Rational)10:51
- Not Perfect Square (2 Irrational)11:27
- Negative Discriminant12:28
- Zero Discriminant12:57
- Example 1: Quadratic Formula13:50
- Example 2: Quadratic Formula16:03
- Example 3: Quadratic Formula19:00
- Example 4: Discriminant21:33

Analyzing the Graphs of Quadratic Functions

30m 7s

- Intro0:00
- Vertex Form0:12
- H and K0:32
- Axis of Symmetry0:36
- Vertex0:42
- Example: Origin1:00
- Example: k = 22:12
- Example: h = 14:27
- Significance of Coefficient a7:13
- Example: |a| > 17:25
- Example: |a| < 18:18
- Example: |a| > 08:51
- Example: |a| < 09:05
- Writing Quadratic Equations in Vertex Form10:22
- Standard Form to Vertex Form10:35
- Example: Standard Form11:02
- Example: a Term Not 114:42
- Example 1: Vertex Form19:47
- Example 2: Vertex Form22:09
- Example 3: Vertex Form24:32
- Example 4: Vertex Form28:23

Graphing and Solving Quadratic Inequalities

27m 5s

- Intro0:00
- Graphing Quadratic Inequalities0:11
- Test Point0:18
- Example: Quadratic Inequality0:29
- Solving Quadratic Inequalities3:57
- Example: Parameter4:24
- Example 1: Graph Inequality11:16
- Example 2: Solve Inequality14:27
- Example 3: Graph Inequality19:14
- Example 4: Solve Inequality23:48

VI. Polynomial Functions

Properties of Exponents

19m 29s

- Intro0:00
- Simplifying Exponential Expressions0:09
- Monomial Simplest Form0:19
- Negative Exponents1:07
- Examples: Simple1:34
- Properties of Exponents3:06
- Negative Exponents3:13
- Mutliplying Same Base3:24
- Dividing Same Base3:45
- Raising Power to a Power4:33
- Parentheses (Multiplying)5:11
- Parentheses (Dividing)5:47
- Raising to 0th Power6:15
- Example 1: Simplify Exponents7:59
- Example 2: Simplify Exponents10:41
- Example 3: Simplify Exponents14:11
- Example 4: Simplify Exponents18:04

Operations on Polynomials

13m 27s

- Intro0:00
- Adding and Subtracting Polynomials0:13
- Like Terms and Like Monomials0:23
- Examples: Adding Monomials1:14
- Multiplying Polynomials3:40
- Distributive Property3:44
- Example: Monomial by Polynomial4:06
- Example 1: Simplify Polynomials5:47
- Example 2: Simplify Polynomials6:28
- Example 3: Simplify Polynomials8:38
- Example 4: Simplify Polynomials10:47

Dividing Polynomials

31m 11s

- Intro0:00
- Dividing by a Monomial0:13
- Example: Numbers0:26
- Example: Polynomial by a Monomial1:18
- Long Division2:28
- Remainder Term2:41
- Example: Dividing with Numbers3:04
- Example: With Polynomials5:01
- Example: Missing Terms7:58
- Synthetic Division11:44
- Restriction12:04
- Example: Divisor in Form12:20
- Divisor in Synthetic Division15:54
- Example: Coefficient to 116:07
- Example 1: Divide Polynomials17:10
- Example 2: Divide Polynomials19:08
- Example 3: Synthetic Division21:42
- Example 4: Synthetic Division25:09

Polynomial Functions

22m 30s

- Intro0:00
- Polynomial in One Variable0:13
- Leading Coefficient0:27
- Example: Polynomial1:18
- Degree1:31
- Polynomial Functions2:57
- Example: Function3:13
- Function Values3:33
- Example: Numerical Values3:53
- Example: Algebraic Expressions5:11
- Zeros of Polynomial Functions5:50
- Odd Degree6:04
- Even Degree7:29
- End Behavior8:28
- Even Degrees9:09
- Example: Leading Coefficient +/-9:23
- Odd Degrees12:51
- Example: Leading Coefficient +/-13:00
- Example 1: Degree and Leading Coefficient15:03
- Example 2: Polynomial Function15:56
- Example 3: Polynomial Function17:34
- Example 4: End Behavior19:53

Analyzing Graphs of Polynomial Functions

33m 29s

- Intro0:00
- Graphing Polynomial Functions0:11
- Example: Table and End Behavior0:39
- Location Principle4:43
- Zero Between Two Points5:03
- Example: Location Principle5:21
- Maximum and Minimum Points8:40
- Relative Maximum and Relative Minimum9:16
- Example: Number of Relative Max/Min11:11
- Example 1: Graph Polynomial Function11:57
- Example 2: Graph Polynomial Function16:19
- Example 3: Graph Polynomial Function23:27
- Example 4: Graph Polynomial Function28:35

Solving Polynomial Functions

21m 10s

- Intro0:00
- Factoring Polynomials0:06
- Greatest Common Factor (GCF)0:25
- Difference of Two Squares1:14
- Perfect Square Trinomials2:07
- General Trinomials2:57
- Grouping4:32
- Sum and Difference of Two Cubes6:03
- Examples: Two Cubes6:14
- Quadratic Form8:22
- Example: Quadratic Form8:44
- Example 1: Factor Polynomial12:03
- Example 2: Factor Polynomial13:54
- Example 3: Quadratic Form15:33
- Example 4: Solve Polynomial Function17:24

Remainder and Factor Theorems

31m 21s

- Intro0:00
- Remainder Theorem0:07
- Checking Work0:22
- Dividend and Divisor in Theorem1:12
- Example: f(a)2:05
- Synthetic Substitution5:43
- Example: Polynomial Function6:15
- Factor Theorem9:54
- Example: Numbers10:16
- Example: Confirm Factor11:27
- Factoring Polynomials14:48
- Example: 3rd Degree Polynomial15:07
- Example 1: Remainder Theorem19:17
- Example 2: Other Factors21:57
- Example 3: Remainder Theorem25:52
- Example 4: Other Factors28:21

Roots and Zeros

31m 27s

- Intro0:00
- Number of Roots0:08
- Not Nature of Roots0:18
- Example: Real and Complex Roots0:25
- Descartes' Rule of Signs2:05
- Positive Real Roots2:21
- Example: Positve2:39
- Negative Real Roots5:44
- Example: Negative6:06
- Finding the Roots9:59
- Example: Combination of Real and Complex10:07
- Conjugate Roots13:18
- Example: Conjugate Roots13:50
- Example 1: Solve Polynomial16:03
- Example 2: Solve Polynomial18:36
- Example 3: Possible Combinations23:13
- Example 4: Possible Combinations27:11

Rational Zero Theorem

31m 16s

- Intro0:00
- Equation0:08
- List of Possibilities0:16
- Equation with Constant and Leading Coefficient1:04
- Example: Rational Zero2:46
- Leading Coefficient Equal to One7:19
- Equation with Leading Coefficient of One7:34
- Example: Coefficient Equal to 18:45
- Finding Rational Zeros12:58
- Division with Remainder Zero13:32
- Example 1: Possible Rational Zeros14:20
- Example 2: Possible Rational Zeros16:02
- Example 3: Possible Rational Zeros19:58
- Example 4: Find All Zeros22:06

VII. Radical Expressions and Inequalities

Operations on Functions

34m 30s

- Intro0:00
- Arithmetic Operations0:07
- Domain0:16
- Intersection0:24
- Denominator is Zero0:49
- Example: Operations1:02
- Composition of Functions7:18
- Notation7:48
- Right to Left8:18
- Example: Composition8:48
- Composition is Not Commutative17:23
- Example: Not Commutative17:51
- Example 1: Function Operations20:55
- Example 2: Function Operations24:34
- Example 3: Compositions27:51
- Example 4: Function Operations31:09

Inverse Functions and Relations

22m 42s

- Intro0:00
- Inverse of a Relation0:14
- Example: Ordered Pairs0:56
- Inverse of a Function3:24
- Domain and Range Switched3:52
- Example: Inverse4:28
- Procedure to Construct an Inverse Function6:42
- f(x) to y6:42
- Interchange x and y6:59
- Solve for y7:06
- Write Inverse f(x) for y7:14
- Example: Inverse Function7:25
- Example: Inverse Function 28:48
- Inverses and Compositions10:44
- Example: Inverse Composition11:46
- Example 1: Inverse Relation14:49
- Example 2: Inverse of Function15:40
- Example 3: Inverse of Function17:06
- Example 4: Inverse Functions18:55

Square Root Functions and Inequalities

30m 4s

- Intro0:00
- Square Root Functions0:07
- Examples: Square Root Function0:16
- Example: Not Square Root Function0:46
- Radicand1:12
- Example: Restriction1:31
- Graphing Square Root Functions3:42
- Example: Graphing3:49
- Square Root Inequalities8:47
- Same Technique9:00
- Example: Square Root Inequality9:20
- Example 1: Graph Square Root Function15:19
- Example 2: Graph Square Root Function18:03
- Example 3: Graph Square Root Function22:41
- Example 4: Square Root Inequalities25:37

nth Roots

20m 46s

- Intro0:00
- Definition of the nth Root0:07
- Example: 5th Root0:20
- Example: 6th Root0:51
- Principal nth Root1:39
- Example: Principal Roots2:06
- Using Absolute Values5:58
- Example: Square Root6:18
- Example: 6th Root8:40
- Example: Negative10:15
- Example 1: Simplify Radicals12:23
- Example 2: Simplify Radicals13:29
- Example 3: Simplify Radicals16:07
- Example 4: Simplify Radicals18:18

Operations with Radical Expressions

41m 11s

- Intro0:00
- Properties of Radicals0:16
- Quotient Property0:29
- Example: Quotient1:00
- Example: Product Property1:47
- Simplifying Radical Expressions3:24
- Radicand No nth Powers3:47
- Radicand No Fractions6:33
- No Radicals in Denominator7:16
- Rationalizing Denominators8:27
- Example: Radicand nth Power9:05
- Conjugate Radical Expressions11:47
- Conjugates12:07
- Example: Conjugate Radical Expression13:11
- Adding and Subtracting Radicals16:12
- Same Index, Same Radicand16:20
- Example: Like Radicals16:28
- Multiplying Radicals19:04
- Distributive Property19:10
- Example: Multiplying Radicals19:20
- Example 1: Simplify Radical24:11
- Example 2: Simplify Radicals28:43
- Example 3: Simplify Radicals32:00
- Example 4: Simplify Radical36:34

Rational Exponents

30m 45s

- Intro0:00
- Definition 10:20
- Example: Using Numbers0:39
- Example: Non-Negative2:46
- Example: Odd3:34
- Definition 24:32
- Restriction4:52
- Example: Relate to Definition 15:04
- Example: m Not 15:31
- Simplifying Expressions7:53
- Multiplication8:31
- Division9:29
- Multiply Exponents10:08
- Raised Power11:05
- Zero Power11:29
- Negative Power11:49
- Simplified Form13:52
- Complex Fraction14:16
- Negative Exponents14:40
- Example: More Complicated15:14
- Example 1: Write as Radical19:03
- Example 2: Write with Rational Exponents20:40
- Example 3: Complex Fraction22:09
- Example 4: Complex Fraction26:22

Solving Radical Equations and Inequalities

31m 27s

- Intro0:00
- Radical Equations0:11
- Variables in Radicands0:22
- Example: Radical Equation1:06
- Example: Complex Equation2:42
- Extraneous Roots7:21
- Squaring Technique7:35
- Double Check7:44
- Example: Extraneous8:21
- Eliminating nth Roots10:04
- Isolate and Raise Power10:14
- Example: nth Root10:27
- Radical Inequalities11:27
- Restriction: Index is Even11:53
- Example: Radical Inequality12:29
- Example 1: Solve Radical Equation15:41
- Example 2: Solve Radical Equation17:44
- Example 3: Solve Radical Inequality20:24
- Example 4: Solve Radical Equation24:34

VIII. Rational Equations and Inequalities

Multiplying and Dividing Rational Expressions

40m 54s

- Intro0:00
- Simplifying Rational Expressions0:22
- Algebraic Fraction0:29
- Examples: Rational Expressions0:49
- Example: GCF1:33
- Example: Simplify Rational Expression2:26
- Factoring -14:04
- Example: Simplify with -14:19
- Multiplying and Dividing Rational Expressions6:59
- Multiplying and Dividing7:28
- Example: Multiplying Rational Expressions8:36
- Example: Dividing Rational Expressions11:20
- Factoring14:01
- Factoring Polynomials14:19
- Example: Factoring14:35
- Complex Fractions18:22
- Example: Numbers18:37
- Example: Algebraic Complex Fractions19:25
- Example 1: Simplify Rational Expression25:56
- Example 2: Simplify Rational Expression29:34
- Example 3: Simplify Rational Expression31:39
- Example 4: Simplify Rational Expression37:50

Adding and Subtracting Rational Expressions

55m 4s

- Intro0:00
- Least Common Multiple (LCM)0:27
- Examples: LCM of Numbers0:43
- Example: LCM of Polynomials4:02
- Adding and Subtracting7:55
- Least Common Denominator (LCD)8:07
- Example: Numbers8:17
- Example: Rational Expressions11:03
- Equivalent Fractions15:22
- Simplifying Complex Fractions21:19
- Example: Previous Lessons21:36
- Example: More Complex22:53
- Example 1: Find LCM28:30
- Example 2: Add Rational Expressions31:44
- Example 3: Subtract Rational Expressions39:18
- Example 4: Simplify Rational Expression38:26

Graphing Rational Functions

57m 13s

- Intro0:00
- Rational Functions0:18
- Restriction0:34
- Example: Rational Function0:51
- Breaks in Continuity2:52
- Example: Continuous Function3:10
- Discontinuities3:30
- Example: Excluded Values4:37
- Graphs and Discontinuities5:02
- Common Binomial Factor (Hole)5:08
- Example: Common Factor5:31
- Asymptote10:06
- Example: Vertical Asymptote11:08
- Horizontal Asymptotes20:00
- Example: Horizontal Asymptote20:25
- Example 1: Holes and Vertical Asymptotes26:12
- Example 2: Graph Rational Faction28:35
- Example 3: Graph Rational Faction39:23
- Example 4: Graph Rational Faction47:28

Direct, Joint, and Inverse Variation

20m 21s

- Intro0:00
- Direct Variation0:07
- Constant of Variation0:25
- Graph of Constant Variation1:26
- Slope is Constant k1:35
- Example: Straight Lines1:41
- Joint Variation2:48
- Three Variables2:52
- Inverse Variation3:38
- Rewritten Form3:52
- Examples in Biology4:22
- Graph of Inverse Variation4:51
- Asymptotes are Axes5:12
- Example: Inverse Variation5:40
- Proportions10:11
- Direct Variation10:25
- Inverse Variation11:32
- Example 1: Type of Variation12:42
- Example 2: Direct Variation14:13
- Example 3: Joint Variation16:24
- Example 4: Graph Rational Faction18:50

Solving Rational Equations and Inequalities

55m 14s

- Intro0:00
- Rational Equations0:15
- Example: Algebraic Fraction0:26
- Least Common Denominator0:49
- Example: Simple Rational Equation1:22
- Example: Solve Rational Equation5:40
- Extraneous Solutions9:31
- Doublecheck10:00
- No Solution10:38
- Example: Extraneous10:44
- Rational Inequalities14:01
- Excluded Values14:31
- Solve Related Equation14:49
- Find Intervals14:58
- Use Test Values15:25
- Example: Rational Inequality15:51
- Example: Rational Inequality 217:07
- Example 1: Rational Equation28:50
- Example 2: Rational Equation33:51
- Example 3: Rational Equation38:19
- Example 4: Rational Inequality46:49

IX. Exponential and Logarithmic Relations

Exponential Functions

35m 58s

- Intro0:00
- What is an Exponential Function?0:12
- Restriction on b0:31
- Base0:46
- Example: Exponents as Bases0:56
- Variables as Exponents1:12
- Example: Exponential Function1:50
- Graphing Exponential Functions2:33
- Example: Using Table2:49
- Properties11:52
- Continuous and One to One12:00
- Domain is All Real Numbers13:14
- X-Axis Asymptote13:55
- Y-Intercept14:02
- Reflection Across Y-Axis14:31
- Growth and Decay15:06
- Exponential Growth15:10
- Real Life Examples15:41
- Example: Growth15:52
- Example: Decay16:12
- Real Life Examples16:30
- Equations17:32
- Bases are Same18:05
- Examples: Variables as Exponents18:20
- Inequalities21:29
- Property21:51
- Example: Inequality22:37
- Example 1: Graph Exponential Function24:05
- Example 2: Growth or Decay27:50
- Example 3: Exponential Equation29:31
- Example 4: Exponential Inequality32:54

Logarithms and Logarithmic Functions

45m 54s

- Intro0:00
- What are Logarithms?0:08
- Restrictions0:15
- Written Form0:26
- Logarithms are Exponents0:52
- Example: Logarithms1:49
- Logarithmic Functions5:14
- Same Restrictions5:30
- Inverses5:53
- Example: Logarithmic Function6:24
- Graph of the Logarithmic Function9:20
- Example: Using Table9:35
- Properties15:09
- Continuous and One to One15:14
- Domain15:36
- Range15:56
- Y-Axis is Asymptote16:02
- X Intercept16:12
- Inverse Property16:57
- Compositions of Functions17:10
- Equations18:30
- Example: Logarithmic Equation19:13
- Inequalities20:36
- Properties20:47
- Example: Logarithmic Inequality21:40
- Equations with Logarithms on Both Sides24:43
- Property24:51
- Example: Both Sides25:23
- Inequalities with Logarithms on Both Sides26:52
- Property27:02
- Example: Both Sides28:05
- Example 1: Solve Log Equation31:52
- Example 2: Solve Log Equation33:53
- Example 3: Solve Log Equation36:15
- Example 4: Solve Log Inequality39:19

Properties of Logarithms

28m 43s

- Intro0:00
- Product Property0:08
- Example: Product0:46
- Quotient Property2:40
- Example: Quotient2:59
- Power Property3:51
- Moved Exponent4:07
- Example: Power4:37
- Equations5:15
- Example: Use Properties5:58
- Example 1: Simplify Log11:17
- Example 2: Single Log15:54
- Example 3: Solve Log Equation18:48
- Example 4: Solve Log Equation22:13

Common Logarithms

25m 23s

- Intro0:00
- What are Common Logarithms?0:10
- Real World Applications0:16
- Base Not Written0:27
- Example: Base 100:39
- Equations1:47
- Example: Same Base1:56
- Example: Different Base2:37
- Inequalities6:07
- Multiplying/Dividing Inequality6:21
- Example: Log Inequality6:54
- Change of Base12:45
- Base 1013:24
- Example: Change of Base14:05
- Example 1: Log Equation15:21
- Example 2: Common Logs17:13
- Example 3: Log Equation18:22
- Example 4: Log Inequality21:52

Base e and Natural Logarithms

21m 14s

- Intro0:00
- Number e0:09
- Natural Base0:21
- Growth/Decay0:33
- Example: Exponential Function0:53
- Natural Logarithms1:11
- ln x1:19
- Inverse and Identity Function1:39
- Example: Inverse Composition1:55
- Equations and Inequalities4:39
- Extraneous Solutions5:30
- Examples: Natural Log Equations5:48
- Example 1: Natural Log Equation9:08
- Example 2: Natural Log Equation10:37
- Example 3: Natural Log Inequality16:54
- Example 4: Natural Log Inequality18:16

Exponential Growth and Decay

24m 30s

- Intro0:00
- Decay0:17
- Decreases by Fixed Percentage0:23
- Rate of Decay0:56
- Example: Finance1:34
- Scientific Model of Decay3:37
- Exponential Decay3:45
- Radioactive Decay4:13
- Example: Half Life5:33
- Growth9:06
- Increases by Fixed Percentage9:18
- Example: Finance10:09
- Scientific Model of Growth11:35
- Population Growth12:04
- Example: Growth12:20
- Example 1: Computer Price14:00
- Example 2: Stock Price15:46
- Example 3: Medicine Disintegration19:10
- Example 4: Population Growth22:33

X. Conic Sections

Midpoint and Distance Formulas

32m 42s

- Intro0:00
- Midpoint Formula0:15
- Example: Midpoint0:30
- Distance Formula2:30
- Example: Distance2:52
- Example 1: Midpoint and Distance4:58
- Example 2: Midpoint and Distance8:07
- Example 3: Median Length18:51
- Example 4: Perimeter and Area23:36

Parabolas

41m 27s

- Intro0:00
- What is a Parabola?0:20
- Definition of a Parabola0:29
- Focus0:59
- Directrix1:15
- Axis of Symmetry3:08
- Vertex3:33
- Minimum or Maximum3:44
- Standard Form4:59
- Horizontal Parabolas5:08
- Vertex Form5:19
- Upward or Downward5:41
- Example: Standard Form6:06
- Graphing Parabolas8:31
- Shifting8:51
- Example: Completing the Square9:22
- Symmetry and Translation12:18
- Example: Graph Parabola12:40
- Latus Rectum17:13
- Length18:15
- Example: Latus Rectum18:35
- Horizontal Parabolas18:57
- Not Functions20:08
- Example: Horizontal Parabola21:21
- Focus and Directrix24:11
- Horizontal24:48
- Example 1: Parabola Standard Form25:12
- Example 2: Graph Parabola30:00
- Example 3: Graph Parabola33:13
- Example 4: Parabola Equation37:28

Circles

21m 3s

- Intro0:00
- What are Circles?0:08
- Example: Equidistant0:17
- Radius0:32
- Equation of a Circle0:44
- Example: Standard Form1:11
- Graphing Circles1:47
- Example: Circle1:56
- Center Not at Origin3:07
- Example: Completing the Square3:51
- Example 1: Equation of Circle6:44
- Example 2: Center and Radius11:51
- Example 3: Radius15:08
- Example 4: Equation of Circle16:57

Ellipses

46m 51s

- Intro0:00
- What Are Ellipses?0:11
- Foci0:23
- Properties of Ellipses1:43
- Major Axis, Minor Axis1:47
- Center1:54
- Length of Major Axis and Minor Axis3:21
- Standard Form5:33
- Example: Standard Form of Ellipse6:09
- Vertical Major Axis9:14
- Example: Vertical Major Axis9:46
- Graphing Ellipses12:51
- Complete the Square and Symmetry13:00
- Example: Graphing Ellipse13:16
- Equation with Center at (h, k)19:57
- Horizontal and Vertical20:14
- Difference20:27
- Example: Center at (h, k)20:55
- Example 1: Equation of Ellipse24:05
- Example 2: Equation of Ellipse27:57
- Example 3: Equation of Ellipse32:32
- Example 4: Graph Ellipse38:27

Hyperbolas

38m 15s

- Intro0:00
- What are Hyperbolas?0:12
- Two Branches0:18
- Foci0:38
- Properties2:00
- Transverse Axis and Conjugate Axis2:06
- Vertices2:46
- Length of Transverse Axis3:14
- Distance Between Foci3:31
- Length of Conjugate Axis3:38
- Standard Form5:45
- Vertex Location6:36
- Known Points6:52
- Vertical Transverse Axis7:26
- Vertex Location7:50
- Asymptotes8:36
- Vertex Location8:56
- Rectangle9:28
- Diagonals10:29
- Graphing Hyperbolas12:58
- Example: Hyperbola13:16
- Equation with Center at (h, k)16:32
- Example: Center at (h, k)17:21
- Example 1: Equation of Hyperbola19:20
- Example 2: Equation of Hyperbola22:48
- Example 3: Graph Hyperbola26:05
- Example 4: Equation of Hyperbola36:29

Conic Sections

18m 43s

- Intro0:00
- Conic Sections0:16
- Double Cone Sections0:24
- Standard Form1:27
- General Form1:37
- Identify Conic Sections2:16
- B = 02:50
- X and Y3:22
- Identify Conic Sections, Cont.4:46
- Parabola5:17
- Circle5:51
- Ellipse6:31
- Hyperbola7:10
- Example 1: Identify Conic Section8:01
- Example 2: Identify Conic Section11:03
- Example 3: Identify Conic Section11:38
- Example 4: Identify Conic Section14:50

Solving Quadratic Systems

47m 4s

- Intro0:00
- Linear Quadratic Systems0:22
- Example: Linear Quadratic System0:45
- Solutions2:49
- Graphs of Possible Solutions3:10
- Quadratic Quadratic System4:10
- Example: Elimination4:21
- Solutions11:39
- Example: 0, 1, 2, 3, 4 Solutions11:50
- Systems of Quadratic Inequalities12:48
- Example: Quadratic Inequality13:09
- Example 1: Solve Quadratic System21:42
- Example 2: Solve Quadratic System29:13
- Example 3: Solve Quadratic System35:02
- Example 4: Solve Quadratic Inequality40:29

XI. Sequences and Series

Arithmetic Sequences

21m 16s

- Intro0:00
- Sequences0:10
- General Form of Sequence0:16
- Example: Finite/Infinite Sequences0:33
- Arithmetic Sequences0:28
- Common Difference2:41
- Example: Arithmetic Sequence2:50
- Formula for the nth Term3:51
- Example: nth Term4:32
- Equation for the nth Term6:37
- Example: Using Formula6:56
- Arithmetic Means9:47
- Example: Arithmetic Means10:16
- Example 1: nth Term12:38
- Example 2: Arithmetic Means13:49
- Example 3: Arithmetic Means16:12
- Example 4: nth Term18:26

Arithmetic Series

21m 36s

- Intro0:00
- What are Arithmetic Series?0:11
- Common Difference0:28
- Example: Arithmetic Sequence0:43
- Example: Arithmetic Series1:09
- Finite/Infinite Series1:36
- Sum of Arithmetic Series2:27
- Example: Sum3:21
- Sigma Notation5:53
- Index6:14
- Example: Sigma Notation7:14
- Example 1: First Term9:00
- Example 2: Three Terms10:52
- Example 3: Sum of Series14:14
- Example 4: Sum of Series18:13

Geometric Sequences

23m 3s

- Intro0:00
- Geometric Sequences0:11
- Common Difference0:38
- Common Ratio1:08
- Example: Geometric Sequence2:38
- nth Term of a Geometric Sequence4:41
- Example: nth Term4:56
- Geometric Means6:51
- Example: Geometric Mean7:09
- Example 1: 9th Term12:04
- Example 2: Geometric Means15:18
- Example 3: nth Term18:32
- Example 4: Three Terms20:59

Geometric Series

22m 43s

- Intro0:00
- What are Geometric Series?0:11
- List of Numbers0:24
- Example: Geometric Series1:12
- Sum of Geometric Series2:16
- Example: Sum of Geometric Series2:41
- Sigma Notation4:21
- Lower Index, Upper Index4:38
- Example: Sigma Notation4:57
- Another Sum Formula6:08
- Example: n Unknown6:28
- Specific Terms7:41
- Sum Formula7:56
- Example: Specific Term8:11
- Example 1: Sum of Geometric Series10:02
- Example 2: Sum of 8 Terms14:15
- Example 3: Sum of Geometric Series18:23
- Example 4: First Term20:16

Infinite Geometric Series

18m 32s

- Intro0:00
- What are Infinite Geometric Series0:10
- Example: Finite0:29
- Example: Infinite0:51
- Partial Sums1:09
- Formula1:37
- Sum of an Infinite Geometric Series2:39
- Convergent Series2:58
- Example: Sum of Convergent Series3:28
- Sigma Notation7:31
- Example: Sigma8:17
- Repeating Decimals8:42
- Example: Repeating Decimal8:53
- Example 1: Sum of Infinite Geometric Series12:15
- Example 2: Repeating Decimal13:24
- Example 3: Sum of Infinite Geometric Series15:14
- Example 4: Repeating Decimal16:48

Recursion and Special Sequences

14m 34s

- Intro0:00
- Fibonacci Sequence0:05
- Background of Fibonacci0:23
- Recursive Formula0:37
- Fibonacci Sequence0:52
- Example: Recursive Formula2:18
- Iteration3:49
- Example: Iteration4:30
- Example 1: Five Terms7:08
- Example 2: Three Terms9:00
- Example 3: Five Terms10:38
- Example 4: Three Iterates12:41

Binomial Theorem

48m 30s

- Intro0:00
- Pascal's Triangle0:06
- Expand Binomial0:13
- Pascal's Triangle4:26
- Properties6:52
- Example: Properties of Binomials6:58
- Factorials9:11
- Product9:28
- Example: Factorial9:45
- Binomial Theorem11:08
- Example: Binomial Theorem13:48
- Finding a Specific Term18:36
- Example: Specific Term19:26
- Example 1: Expand24:39
- Example 2: Fourth Term30:26
- Example 3: Five Terms36:13
- Example 4: Three Iterates45:07

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1 answer

Last reply by: Khanh Nguyen

Sun Jan 3, 2016 6:37 PM

Post by Khanh Nguyen on January 3, 2016

At 32:48, for x = 5, 5 - 6 = -1. So the chart should have added a - sign to the 1.

You forgot to add the - sign.

2 answers

Last reply by: Kavita Agrawal

Wed Jun 19, 2013 10:07 PM

Post by Christine Kuhlman on September 17, 2012

Is there a time that a line could cross a horizontal asymptote? If so, how can you find out when and by how much it does.

I thought that with asymptotes the line could never touch it, but today my teacher told us they can cross and I'm very confused.

1 answer

Last reply by: Dr Carleen Eaton

Sat Feb 26, 2011 6:02 PM

Post by Edgar Rariton on February 26, 2011

Regarding example IV, you say that you're going to cancel out the common factors of x+3, but then you say to graph 2x-1/x+3.

You then continue to calculate points based on 2x-1/x-5.

You seemed to have made a typo.