Graphing Rational Functions, Algebra 2

Graphing Rational Functions

VIII. Radical Expressions and Equations: Lecture 3 | 45:13 min

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Graphing Rational Functions

If the numerator and denominator have a common binomial factor, the graph will have a hole at the point where this factor is 0. If the denominator has a binomial factor that is not in the numerator, the graph will have a vertical asymptote at the point where this factor is 0.
The graph of a rational function often has a horizontal asymptote.

Graphing Rational Functions

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

Rational Functions

Example

Breaks in Continuity

Graphs and Discontinuities

Horizontal Asymptotes

Example

Lecture Example 1

Lecture Example 2

Additional Example 3

Additional Example 4

I. Equations and Inequalities
	Expressions and Formulas	28:56
	Properties of Real Numbers	23:45
	Solving Equations	24:41
	Solving Absolute Value Equations	17:36
	Solving Inequalities	19:27
	Solving Compound and Absolute Value Inequalities	24:08
II. Linear Relations and Functions
	Relations and Functions	38:15
	Linear Equations	12:50
	Slope	20:07
	Writing Linear Functions	27:36
	Special Functions	24:28
	Graphing Inequalities	30:37
III. Systems of Equations and Inequalities
	Solving Systems of Equations by Graphing	21:27
	Solving Systems of Equations Algebraically	31:26
	Solving Systems of Inequalities by Graphing	20:43
	Solving Systems of Equations in 3 Variables	21:27
IV. Matrices
	Basic Matrix Concepts	14:08
	Matrix Operations	16:40
	Matrix Multiplication	22:47
	Determinants	25:47
	Cramer's Rule	25:42
	Identity and Inverse Matrices	27:01
	Solving Systems of Equations with Matrices	28:40
V. Quadratic Functions and Inequalities
	Graphing Quadratic Equations	26:36
	Solving Quadratic Equations by Graphing	19:26
	Solving Quadratic Equations by Factoring	17:46
	Imaginary and Complex Numbers	37:41
	Completing the Square	16:42
	Quadratic Formula and the Discriminant	17:44
	Analyzing the Graphs of Quadratic Functions	23:00
	Graphing and Solving Quadratic Inequalities	34:38
VI. Polynomial Functions
	Properties of Exponents	20:28
	Operations on Polynomials	16:13
	Dividing Polynomials	29:26
	Polynomial Functions	29:34
	Analyzing Graphs of Polynomials	34:36
	Solving Polynomial Equations	19:23
	Remainder and Factor Theorems	27:52
	Roots and Zeros	31:04
	Rational Zero Theorem	29:27
VII. Rational Equations and Inequalities
	Operations on Functions	35:12
	Inverse Functions and Relations	18:12
	Square Root Functions and Inequalities	26:24
	nth Roots	24:06
	Operations with Radical Expressions	34:38
	Rational Exponents	24:36
	Solving Radical Equations and Inequalities	38:46
VIII. Radical Expressions and Equations
	Multiplying and Dividing Rational Expressions	30:11
	Adding and Subtracting Rational Exprsesions	51:53
	Graphing Rational Functions	45:13
	Direct, Joint, and Inverse Variation	21:49
	Solving Rational Equations and Inequalities	53:21
IX. Exponential and Logarithmic Relations
	Exponential Functions	28:22
	Logarithms and Logarithmic Functions	36:31
	Properties of Logarithms	29:50
	Common Logarithms	27:10
	Base 'e' and Natural Logarithms	19:52
	Exponential Growth and Decay	28:10
X. Conic Sections
	Midpoint and Distance Formulas	29:35
	Parabolas	26:11
	Circles	17:33
	Ellipses	38:57
	Hyperbolas	37:59
	Conic Sections	23:10
	Solving Quadratic Systems	28:18
XI. Sequences and Series
	Arithmetic Sequences	27:44
	Arithmetic Series	29:12
	Geometric Sequences	24:52
	Geometric Series	27:02
	Infinite Geometric Series	24:01
	Recursion and Special Sequences	17:11
	Binomial Theorem	38:25
	Proof and Mathematical Induction	19:53

Mathematics: Algebra 2

I. Equations and Inequalities
	Expressions and Formulas			28:56
		Intro		0:00
		Order of Operations		0:51
			Variables and Algebraic Expressions	0:57
			Order of Operations	3:05
		Monomials		5:25
			Examples	5:37
			Constant, Coefficient, Degree, Power	6:27
		Polynomials		8:29
			Examples	8:42
			Terms, Like Terms, Binomial, Trinomial	8:59
		Formulas		12:35
			Examples: Area, Volume, Surface Area	12:50
		Lecture Example 1		15:50
		Lecture Example 2		21:31
		Additional Example 3		1:17
		Additional Example 4		2:38
	Properties of Real Numbers			23:45
		Intro		0:00
		Real Numbers		0:15
			Rational Numbers	0:40
			Irrational Numbers	1:38
		Venn Diagram of the Real Numbers		2:55
		Properties of Real Numbers		6:49
			Commutative Property	7:06
			Associative Property	7:27
			Identity Property	8:01
			Inverse Property	8:42
			Distributive Property	10:05
		Lecture Example 1		10:43
		Lecture Example 2		13:08
		Additional Example 3		1:47
		Additional Example 4		5:22
	Solving Equations			24:41
		Intro		0:00
		Translations		0:11
			Example: Verbal to Algebraic Expressions	0:44
		Properties of Equality		2:51
			Reflexive, Symmetric, Transitive Properties	2:58
			Addition, Subtraction, Multiplication, Division	3:32
		Solving Equations		6:09
			Example	6:23
		Solving for a Variable		8:49
			Example: Surface Area of a Cone	8:58
		Lecture Example 1		11:06
		Lecture Example 2		12:39
		Additional Example 3		4:36
		Additional Example 4		2:09
	Solving Absolute Value Equations			17:36
		Intro		0:00
		Absolute Value Expressions		0:10
			Example: Positive Distance	0:15
		Absolute Value Equations		1:07
			Examples	1:18
		No Solutions		2:54
			Example: Empty Set	2:58
		Number of Solutions		3:56
			Examples	4:42
		Lecture Example 1		6:42
		Lecture Example 2		8:54
		Additional Example 3		1:55
		Additional Example 4		5:32
	Solving Inequalities			19:27
		Intro		0:00
		Properties of Inequality		0:07
			Addition Property	0:21
			Subtraction Property	0:48
			Example	1:02
		Multiplication Properties		1:44
			Multiplying by a Positive Number	1:48
			Example: Positive	2:17
			Multiplying by a Negative Number	2:25
			Example: Negative	2:35
		Division Properties		3:23
			Example: Positive	3:32
			Example: Negative	4:04
		Describing the Solution Set		6:00
			Set Builder Notation	6:15
			Graphing	7:15
		Lecture Example 1		8:04
		Lecture Example 2		9:09
		Additional Example 3		3:06
		Additional Example 4		5:32
	Solving Compound and Absolute Value Inequalities			24:08
		Intro		0:00
		Compound Inequalities		0:11
			Example	0:33
		'And' Inequality		3:41
			Example: Set Intersection	4:00
		'Or' Inequality		6:01
			Example: Set Union	6:15
		Absolute Value Inequalities		8:19
			Examples	8:37
		Lecture Example 1		11:43
		Lecture Example 2		14:47
		Additional Example 3		2:33
		Additional Example 4		4:49
II. Linear Relations and Functions
	Relations and Functions			38:15
		Intro		0:00
		Coordinate Plane		0:38
			Example: Origin and Quadrants	0:44
		Relations		4:08
			Example: Ordered Pairs	4:14
			Domain and Range	5:05
		Functions		5:57
			Example: Mapping	6:11
		One-to-One Functions		9:58
			Example	10:05
		Graphs of Relations		13:42
			Example: Discrete and Continuous	13:55
		Vertical Line Test		16:26
			Examples	16:38
		Equations, Relations, Functions		19:38
			Example: Independent and Dependent Variables	19:45
		Function Notation		21:51
			Examples	22:27
		Lecture Example 1		24:39
		Lecture Example 2		28:29
		Additional Example 3		3:47
		Additional Example 4		2:03
	Linear Equations			12:50
		Intro		0:00
		Linear Equations and Functions		0:12
			Example: Linear Equation	0:21
			Example: Linear Function	1:16
		Standard Form		2:13
			Examples	2:43
		Graphing with Intercepts		3:26
			Example: Intercepts	3:51
		Lecture Example 1		6:25
		Lecture Example 2		7:53
		Additional Example 3		2:06
		Additional Example 4		2:09
	Slope			20:07
		Intro		0:00
		Definition of Slope		0:23
		Interpretation of Slope		2:19
			Example: 0 Slope and Undefined Slope	2:25
			Example: Positive Slope	4:04
			Example: Negative Slope	4:43
		Parallel Lines		6:16
		Perpendicular Lines		7:15
		Lecture Example 1		8:20
		Lecture Example 2		10:45
		Additional Example 3		2:39
		Additional Example 4		4:14
	Writing Linear Functions			27:36
		Intro		0:00
		Slope Intercept Form		0:08
			Origin of Form	0:21
			Example	2:08
		Point Slope Form		3:47
			Origin of Form	4:01
		Parallel and Perpendicular Lines		5:36
			Example: Find Parallel Line	5:58
		Lecture Example 1		8:27
		Lecture Example 2		12:08
		Additional Example 3		4:35
		Additional Example 4		7:05
	Special Functions			24:28
		Intro		0:00
		Step Functions		0:13
			Graph	0:21
			Example: Birthday Function	2:32
		Absolute Value Functions		5:21
			Graph	5:27
		Piecewise Functions		7:34
			Example	7:38
		Lecture Example 1		10:20
		Lecture Example 2		14:38
		Additional Example 3		3:10
		Additional Example 4		4:25
	Graphing Inequalities			30:37
		Intro		0:00
		Graphing Linear Inequalities		0:11
			Example: Linear Inequalities	0:20
			Half Plane	2:04
			Test Point	2:53
		Graphing Absolute Value Inequalities		5:38
			Example: Linear Inequalities	5:49
			Example: Absolute Value	8:23
		Lecture Example 1		11:39
		Lecture Example 2		14:50
		Additional Example 3		4:13
		Additional Example 4		7:06
III. Systems of Equations and Inequalities
	Solving Systems of Equations by Graphing			21:27
		Intro		0:00
		Systems of Equations		0:14
		Solving by Graphing		0:34
		Types of Systems		1:07
			Independent (One Solution)	2:02
			Dependent (Infinite Solutions)	2:30
			Inconsistent (No Solutions, Parallel)	3:37
		Lecture Example 1		4:52
		Lecture Example 2		8:42
		Additional Example 3		3:29
		Additional Example 4		3:33
	Solving Systems of Equations Algebraically			31:26
		Intro		0:00
		Solving by Substitution		0:15
			Examples	0:50
		Solving by Elimination		4:19
			Examples	4:27
		Solving by Multiplication		7:24
			Examples	7:37
		Inconsistent and Dependent Systems		11:42
			Example: Spotting Differences	12:07
		Lecture Example 1		15:00
		Lecture Example 2		17:35
		Additional Example 3		6:21
		Additional Example 4		3:50
	Solving Systems of Inequalities by Graphing			20:43
		Intro		0:00
		Solving by Graphing		0:10
			Example: Single Inequality	0:14
		No Solution		4:16
			Example: No Solution	4:25
		Lecture Example 1		6:25
		Lecture Example 2		8:23
		Additional Example 3		4:57
		Additional Example 4		3:31
	Solving Systems of Equations in 3 Variables			21:27
		Intro		0:00
		Solving Systems in Three Variables		0:15
			Ordered Triple	0:36
		Number of Solutions		1:32
		Lecture Example 1		2:19
		Lecture Example 2		6:14
		Additional Example 3		6:53
		Additional Example 4		3:17
IV. Matrices
	Basic Matrix Concepts			14:08
		Intro		0:00
		What is a Matrix?		0:33
			Example: Rectangular Array	0:41
			Element	1:52
			Examples: More Matrices	2:04
		Dimensions		3:40
			Examples	4:53
		Special Matrices		6:31
			(m x 1) Matrix	6:36
			Square Matrix	7:01
			Zero Matrix	7:38
		Equal Matrices		8:23
			Examples	8:32
		Lecture Example 1		10:56
		Lecture Example 2		11:28
		Additional Example 3		0:18
		Additional Example 4		1:31
	Matrix Operations			16:40
		Intro		0:00
		Matrix Addition		0:10
			Example	1:07
		Matrix Subtraction		2:12
			Example	2:31
		Scalar Multiplication		3:23
			Example	4:05
		Properties of Matrix Operations		5:31
			Commutative Property	5:48
			Associative Property	5:59
			Distributive Property	6:34
		Lecture Example 1		7:03
		Lecture Example 2		8:15
		Additional Example 3		1:07
		Additional Example 4		6:04
	Matrix Multiplication			22:47
		Intro		0:00
		Dimension Requirement		0:19
			Example	0:45
		Matrix Multiplication		1:35
			Example	2:21
		Properties of Matrix Multiplication		6:46
			Associative Property	6:59
			Distributive Property	7:15
			Commutative Property	7:39
		Lecture Example 1		8:49
		Lecture Example 2		11:43
		Additional Example 3		3:49
		Additional Example 4		4:31
	Determinants			25:47
		Intro		0:00
		What is a Determinant		0:15
		Determinant of a 2x2 Matrix		0:56
			Difference from Matrices	1:16
			Second Order Determinant	1:38
			Example	2:06
		Determinant of a 3x3 Matrix		3:20
			Third Order Determinants	3:25
			Origin of Equation (Minors)	3:38
			Expansion by Minors	6:05
			Example: 3x3 Matrix	8:55
		Diagonal Method for 3x3 Matrix		12:45
			Example	12:55
		Lecture Example 1		17:03
		Lecture Example 2		17:42
		Additional Example 3		2:32
		Additional Example 4		3:15
	Cramer's Rule			25:42
		Intro		0:00
		System of 2 Equations in 2 Variables		0:27
			Example	1:20
		System of 3 Equations in 3 Variables		3:10
			Example	3:51
		Lecture Example 1		6:45
		Lecture Example 2		10:22
		Additional Example 3		5:44
		Additional Example 4		6:23
	Identity and Inverse Matrices			27:01
		Intro		0:00
		Identity Matrix		0:10
			Example: 2x2 Matrix	2:18
		Matrix Inverses		4:40
			Example: Does Not Exist	6:04
		Inverse of a 2x2 Matrix		8:17
			Example	9:38
		Lecture Example 1		13:19
		Lecture Example 2		15:57
		Additional Example 3		3:45
		Additional Example 4		3:37
	Solving Systems of Equations with Matrices			28:40
		Intro		0:00
		Matrix Equations		0:22
			Example	0:40
		Solving Systems of Equations		4:20
			Example	5:58
		Lecture Example 1		9:11
		Lecture Example 2		15:09
		Additional Example 3		4:31
		Additional Example 4		4:18
V. Quadratic Functions and Inequalities
	Graphing Quadratic Equations			26:36
		Intro		0:00
		Quadratic Functions		0:10
			Parabola	0:50
			Example: Opens Upward	1:03
			Example: Opens Downward	1:54
		Properties of Parabolas		3:17
			Axis of Symmetry	3:26
			Vertex	4:05
			Example	4:28
		Maximum and Minimum Values		7:10
			Example: Upwards/Minimum	7:32
			Example: Downwards/Maximum	8:19
		Lecture Example 1		9:09
		Lecture Example 2		13:05
		Additional Example 3		4:42
		Additional Example 4		5:10
	Solving Quadratic Equations by Graphing			19:26
		Intro		0:00
		Quadratic Equations		0:18
			Example: Standard Form	0:55
		Solving by Graphing		1:39
			Roots	1:48
			Example: 2 Solutions	1:56
			Example: 1 Solution	2:39
			Example: 0 Solutions	3:10
		Estimating Solutions		3:55
			Example	4:07
		Lecture Example 1		5:16
		Lecture Example 2		7:51
		Additional Example 3		4:21
		Additional Example 4		4:37
	Solving Quadratic Equations by Factoring			17:46
		Intro		0:00
		Factoring Techniques		0:16
			Greatest Common Factor (GCF)	0:29
			Difference of Two Squares	1:45
			Perfect Square Trinomials	2:07
			General Trinomials	3:16
		Zero Product Rule		4:50
			Example	5:01
		Lecture Example 1		6:19
		Lecture Example 2		8:13
		Additional Example 3		2:20
		Additional Example 4		5:11
	Imaginary and Complex Numbers			37:41
		Intro		0:00
		Properties of Square Roots		0:17
			Example: Product and Quotient Rules	0:33
		Imaginary Numbers		4:04
			Powers of Imaginary Numbers	5:06
			Example	6:27
		Complex Numbers		7:21
			Real and Complex Numbers	8:19
		Equality		9:04
			Example	9:17
		Addition and Subtraction		9:43
			Example	9:55
		Complex Plane		11:38
			Example	11:52
		Multiplication		13:34
			Example	13:43
		Division		16:36
			Complex Conjugates	16:45
			Example	18:16
		Lecture Example 1		23:40
		Lecture Example 2		26:34
		Additional Example 3		2:10
		Additional Example 4		6:57
	Completing the Square			16:42
		Intro		0:00
		Square Root Property		0:22
			Examples	0:33
		Completing the Square		1:48
			Example: Making into Perfect Square	1:50
		Solve Equations		3:43
			Example	3:53
		Equations Where 'a' Not Equal to 1		6:47
			Example	6:57
		Complex Solutions		10:14
			Example	10:22
		Lecture Example 1		11:30
		Lecture Example 2		12:34
		Additional Example 3		1:27
		Additional Example 4		2:09
	Quadratic Formula and the Discriminant			17:44
		Intro		0:00
		Quadratic Formula		0:37
			Example	0:56
		One Rational Root		3:10
			Why It Works	3:26
			Repeated/Double Root	3:49
		Complex Solutions		4:31
			Example	4:50
		Discriminant		7:19
			Discriminant Value and Root Type	8:50
		Lecture Example 1		12:08
		Lecture Example 2		14:15
		Additional Example 3		2:11
		Additional Example 4		2:27
	Analyzing the Graphs of Quadratic Functions			23:00
		Intro		0:00
		Vertex Form		0:24
			Example	1:56
		Significance of Coefficient 'a'		3:15
			Example	3:39
		Writing Quadratic Equations in Vertex Form		4:51
			Examples	5:19
		Lecture Example 1		8:14
		Lecture Example 2		10:22
		Additional Example 3		3:38
		Additional Example 4		4:40
	Graphing and Solving Quadratic Inequalities			34:38
		Intro		0:00
		Graphing Quadratic Inequalities		1:14
			Example: Linear Inequality	1:29
			Example: Quadratic Inequality	3:11
		Solving Quadratic Inequalities		6:32
			Example	6:38
		Lecture Example 1		11:50
		Lecture Example 2		15:09
		Additional Example 3		5:37
		Additional Example 4		8:07
VI. Polynomial Functions
	Properties of Exponents			20:28
		Intro		0:00
		Simplifying Exponential Expressions		0:32
		Negative Exponents		0:54
			Example: Base 0	1:16
			Examples	1:30
		Properties of Exponents		2:22
			Base and Exponent	2:52
		Lecture Example 1		8:29
		Lecture Example 2		10:58
		Additional Example 3		5:05
		Additional Example 4		2:01
	Operations on Polynomials			16:13
		Intro		0:00
		Adding and Subtracting Polynomials		0:24
			Example: Signs	0:34
		Multiplying Polynomials		3:04
			Example	3:12
		Lecture Example 1		6:40
		Lecture Example 2		7:21
		Additional Example 3		3:06
		Additional Example 4		3:50
	Dividing Polynomials			29:26
		Intro		0:00
		Dividing by a Monomial		0:16
			Example	0:28
		Long Division		3:24
			Example: Missing Terms, Remainder	3:49
		Example: Long Division		6:51
		Synthetic Division		10:13
			Example	10:44
		Divisor in Synthetic Division		13:18
			Example: Coefficient Not 1	13:30
		Lecture Example 1		16:41
		Lecture Example 2		18:22
		Additional Example 3		2:08
		Additional Example 4		6:27
	Polynomial Functions			29:34
		Intro		0:00
		Polynomial in One Variable		0:17
			Degree n	0:30
			Descending Order	0:43
			Example: Leading Coefficient	1:04
		Function Values		3:31
			Example	3:42
		Zeros of Polynomial Functions		5:45
			Example: Zeros	6:04
		End Behavior		9:51
			Example: 4 Situations	10:51
		Lecture Example 1		17:30
		Lecture Example 2		19:11
		Additional Example 3		3:43
		Additional Example 4		4:27
	Analyzing Graphs of Polynomials			34:36
		Intro		0:00
		Graphing Polynomial Functions		0:09
			End Behavior	0:19
			Examples: Degree and Sign of Polynomials	1:13
		Location Principle		4:50
			Example	6:19
		Maximum and Minimum Points		7:34
			Example: Relative Maximum and Relative Minimum	7:44
		Lecture Example 1		10:17
		Lecture Example 2		15:13
		Additional Example 3		6:11
		Additional Example 4		7:28
	Solving Polynomial Equations			19:23
		Intro		0:00
		Factoring Polynomials		0:08
			Example: Greatest Common Factor (GCF)	0:40
			Example: Perfect Square Trinomials	1:30
			Example: General Trinomials	2:48
		Sum and Difference of Two Cubes		3:25
			Example	4:18
		Quadratic Form		6:20
		Lecture Example 1		7:30
		Lecture Example 2		10:43
		Additional Example 3		2:27
		Additional Example 4		4:06
	Remainder and Factor Theorems			27:52
		Intro		0:00
		Remainder Theorem		0:04
			Quotient and Remainder	0:30
			Examples	1:34
		Synthetic Substitution		5:04
			Example	5:28
		Factor Theorem		10:00
		Factoring Polynomials		11:21
			Example	11:51
		Lecture Example 1		16:38
		Lecture Example 2		18:41
		Additional Example 3		1:52
		Additional Example 4		3:06
	Roots and Zeros			31:04
		Intro		0:00
		Numbers of Roots		0:10
			Example: Real and Complex Roots	0:23
		Descartes' Rule of Signs		3:43
			Example: Positive Real Roots	4:58
			Example: Negative Real Roots	8:00
		Finding the Roots		12:11
		Conjugate Roots		13:24
		Lecture Example 1		15:41
		Lecture Example 2		19:41
		Additional Example 3		4:04
		Additional Example 4		5:04
	Rational Zero Theorem			29:27
		Intro		0:00
		Equation		0:14
			Leading Coefficient and Constant Term	0:30
			Example	2:15
		Leading Coefficient Equal to 1		8:08
			Example	9:20
		Finding Rational Zeros		11:48
		Lecture Example 1		12:10
		Lecture Example 2		15:59
		Additional Example 3		3:15
		Additional Example 4		5:17
VII. Rational Equations and Inequalities
	Operations on Functions			35:12
		Intro		0:00
		Arithmetic Operations		0:12
			Example: Domain	0:25
		Composition of Functions		7:35
			Example	7:55
		Composition is Not Commutative		17:13
			Example	18:18
		Lecture Example 1		21:51
		Lecture Example 2		24:25
		Additional Example 3		3:12
		Additional Example 4		4:18
	Inverse Functions and Relations			18:12
		Intro		0:00
		Inverse of a Relation		0:24
			Example: Ordered Pairs	0:33
		Inverse of a Function		2:15
		Procedure to Construct an Inverse Function		4:28
			Example: Inverse Function	4:58
			Example: Inverse Function 2	7:31
		Inverses and Compositions		8:41
		Lecture Example 1		9:59
		Lecture Example 2		10:45
		Additional Example 3		1:48
		Additional Example 4		4:32
	Square Root Functions and Inequalities			26:24
		Intro		0:00
		Square Root Functions		0:06
			Example: Not Square Root Function	0:23
			Example: Square Root Function	1:17
		Graphing Square Root Functions		3:11
			Example: Radicand	3:21
		Square Root Inequalities		6:51
			Example	7:13
		Lecture Example 1		11:27
		Lecture Example 2		14:05
		Additional Example 3		3:07
		Additional Example 4		4:36
	nth Roots			24:06
		Intro		0:00
		Definition of the nth Root		0:13
			Example	0:36
		Principal nth Root		2:18
			Index	3:04
			Examples	3:20
		Using Absolute Values		6:25
			Examples	6:52
		Lecture Example 1		11:26
		Lecture Example 2		13:17
		Additional Example 3		3:05
		Additional Example 4		2:54
	Operations with Radical Expressions			34:38
		Intro		0:00
		Properties of Radicals		0:22
			Example	1:37
		Simplifying Radical Expressions		2:58
			Examples	3:24
		Rationalizing Denominators		4:08
			Examples	4:18
		Conjugate Radical Expressions		8:01
			Example	8:09
		Adding and Subtracting Radicals		11:23
			Examples	11:44
		Multiplying Radicals		12:57
			Examples	13:03
		Lecture Example 1		16:53
		Lecture Example 2		20:11
		Additional Example 3		4:16
		Additional Example 4		6:49
	Rational Exponents			24:36
		Intro		0:00
		Definition 1		0:24
			nth Root	0:44
			Example: Even	1:29
		Definition 2		2:55
		Simplifying Expressions		3:20
			Examples	3:40
		Simplified Form		7:07
			Example	7:32
		Lecture Example 1		8:18
		Lecture Example 2		10:20
		Additional Example 3		6:35
		Additional Example 4		5:57
	Solving Radical Equations and Inequalities			38:46
		Intro		0:00
		Radical Equations		0:23
			Examples	0:34
		Example: Radical Equation		4:47
		Extraneous Roots		12:29
		Eliminating nth Roots		14:28
			Examples	14:54
		Radical Inequalities		16:38
			Example	17:18
		Lecture Example 1		20:28
		Lecture Example 2		22:57
		Additional Example 3		4:27
		Additional Example 4		7:27
VIII. Radical Expressions and Equations
	Multiplying and Dividing Rational Expressions			30:11
		Intro		0:00
		Simplifying Rational Expressions		0:12
			Examples: Rational Expressions	0:31
		Factoring -1		3:26
			Example	3:33
		Multiplying and Dividing Rational Expressions		4:50
			Multiplying	5:08
			Dividing	5:16
			Example	6:10
		Factoring		9:13
			Example	9:33
		Complex Fractions		13:15
			Example	13:27
		Lecture Example 1		15:36
		Lecture Example 2		18:25
		Additional Example 3		4:30
		Additional Example 4		5:02
	Adding and Subtracting Rational Exprsesions			51:53
		Intro		0:00
			Example: Fractions	0:22
		Least Common Multiple (LCM)		1:36
			Example	2:07
		Adding and Subtracting		7:56
			Least Common Denominator (LCD)	8:01
			Example: Fractions	8:14
			Example: Rational Expression	10:23
			Equivalent Fractions	13:45
			Example	14:20
		Simplifying Complex Fractions		20:03
			Example	20:28
		Lecture Example 1		26:34
		Lecture Example 2		31:06
		Additional Example 3		7:30
		Additional Example 4		5:46
	Graphing Rational Functions			45:13
		Intro		0:00
		Rational Functions		0:35
			Example	0:57
		Breaks in Continuity		2:48
			Discontinuities	3:19
			Example: Excluded Values	3:52
		Graphs and Discontinuities		4:36
			Example: Hole Discontinuity	6:07
			Example: Asymptote	8:53
		Horizontal Asymptotes		13:34
			Example	13:54
		Lecture Example 1		17:58
		Lecture Example 2		20:29
		Additional Example 3		9:04
		Additional Example 4		8:38
	Direct, Joint, and Inverse Variation			21:49
		Intro		0:00
		Direct Variation		0:16
			Constant of Variation	0:44
		Graph of Direct Variation		1:28
			Example: Straight Line	1:36
		Joint Variation		2:55
		Inverse Variation		4:17
			Example	4:50
		Graph of Inverse Variation		5:35
			Example	6:00
		Proportions		8:00
			Example	9:28
		Lecture Example 1		12:32
		Lecture Example 2		14:26
		Additional Example 3		2:46
		Additional Example 4		2:32
	Solving Rational Equations and Inequalities			53:21
		Intro		0:00
		Rational Equations		0:15
			Example: Not Rational Equation	0:26
			Example: X in Denominator	0:38
			Example: LCD	1:08
		Example: Rational Equations		5:19
		Extraneous Solutions		12:08
			Example	12:42
		Rational Inequalities		15:31
			Example	15:45
		Example: Rational Inequalities		12:05
		Lecture Example 1		32:06
		Lecture Example 2		35:18
		Additional Example 3		6:38
		Additional Example 4		6:35
IX. Exponential and Logarithmic Relations
	Exponential Functions			28:22
		Intro		0:00
		What is an Exponential Function?		0:11
			Exponent and Base	0:38
		Graphing Exponential Functions		1:31
			Example	1:34
		Properties		4:05
		Growth and Decay		9:38
		Equations		10:32
			Example	11:05
		Inequalities		13:00
			Example	14:29
		Lecture Example 1		16:48
		Lecture Example 2		18:50
		Additional Example 3		3:16
		Additional Example 4		3:48
	Logarithms and Logarithmic Functions			36:31
		Intro		0:00
		What are Logarithms?		0:17
			Examples	1:30
		Logarithmic Functions		4:09
		Graph of the Logarithmic Function		4:52
		Properties		9:08
		Inverse Property		10:47
		Equations		11:44
			Example	12:11
		Inequalities		14:45
		Equations with Logarithms on Both Sides		17:00
			Example	17:18
		Inequalities with Logarithms on Both Sides		19:17
			Example	19:32
		Lecture Example 1		20:31
		Lecture Example 2		22:38
		Additional Example 3		4:03
		Additional Example 4		7:53
	Properties of Logarithms			29:50
		Intro		0:00
		Product Property		0:08
			Example	0:26
		Quotient Property		1:06
			Example	1:12
		Power Rule		3:29
			Example	3:33
		Equations		5:43
			Example	6:19
		Lecture Example 1		12:19
		Lecture Example 2		16:13
		Additional Example 3		4:43
		Additional Example 4		4:56
	Common Logarithms			27:10
		Intro		0:00
		What are Common Logarithms?		0:54
			Base 10	0:58
		Equations		2:00
			Examples	2:22
		Inequalities		5:35
			Example	5:42
		Change of Base		9:23
			Example	10:09
		Lecture Example 1		12:04
		Lecture Example 2		15:16
		Additional Example 3		4:52
		Additional Example 4		5:49
	Base 'e' and Natural Logarithms			19:52
		Intro		0:00
		The Number 'e'		0:32
			Natural Base	0:44
			Euler	1:12
			Natural Exponential Function	1:38
			Natural Log Function	2:44
			Growth and Decay	2:55
		Natural Logarithms		3:16
			Graph (Inverse)	3:34
		Equations and Inequalities		4:49
		Lecture Example 1		7:21
		Lecture Example 2		9:10
		Additional Example 3		2:05
		Additional Example 4		3:58
	Exponential Growth and Decay			28:10
		Intro		0:00
		Decay		0:15
			Fixed Percentage	0:24
			Rate of Decay	2:35
		Scientific Model of Decay (Exponential Decay)		4:17
			Graph	5:19
		Growth		6:19
			Rate of Growth	6:36
		Scientific Model of Growth (Exponential Growth)		6:41
			Graph	6:48
		Lecture Example 1		7:48
		Lecture Example 2
		Additional Example 3		5:07
		Additional Example 4		6:38
X. Conic Sections
	Midpoint and Distance Formulas			29:35
		Intro		0:00
		Midpoint Formula		0:35
		Distance Formula		1:42
			Example	2:52
		Lecture Example 1		3:40
		Lecture Example 2		6:37
		Additional Example 3		4:26
		Additional Example 4		6:27
	Parabolas			26:11
		Intro		0:00
		What is a Parabola?		0:21
			Focus and Directrix	0:33
			Axis of Symmetry	1:41
		Vertex		2:03
			Example	2:15
		Standard Form		3:11
			Upward and Downward	4:07
		Graphing Parabolas		5:24
			Example	6:32
		Latus Rectum		7:37
		Horizontal Parabolas		9:10
		Focus and Direction		12:31
		Lecture Example 1		13:11
		Lecture Example 2		16:46
		Additional Example 3		2:35
		Additional Example 4		4:48
	Circles			17:33
		Intro		0:00
		What are Circles		0:17
			Center, Radius	0:37
		Equation (Standard Form)		0:46
		Graphing		1:21
		Center Not at Origin		1:53
			Example	2:06
		Lecture Example 1		4:16
		Lecture Example 2		8:22
		Additional Example 3		2:25
		Additional Example 4		4:00
	Ellipses			38:57
		Intro		0:00
		What are Ellipses?		0:59
			Foci	1:04
		Properties		3:47
			Major Axis, Minor Axis	4:03
		Standard Form		7:22
			Example	8:05
		Vertical Major Axis		10:12
			Example	10:40
		Graphing Ellipses		13:33
			Example: Completing the Square	14:04
		Equation with Center at (h,k)		17:25
			Example	17:53
		Lecture Example 1		19:36
		Lecture Example 2		23:52
		Additional Example 3		4:58
		Additional Example 4		5:50
	Hyperbolas			37:59
		Intro		0:00
		What are Hyperbolas?		1:09
		Properties		2:35
			Transverse Axis, Conjugate Axis	2:57
			Center, Vertices	3:54
		Standard Form		4:33
		Vertical Transverse Axis		6:35
		Asymptotes		10:17
		Graphing Hyperbolas		13:44
			Example	17:23
		Equation with Center at (h,k)		18:20
		Lecture Example 1		20:19
		Lecture Example 2		23:25
		Additional Example 3		8:20
		Additional Example 4		2:59
	Conic Sections			23:10
		Intro		0:00
		What are Conic Sections?		2:16
		Standard Form		2:58
			Example	5:29
		Identifying Conic Sections		6:14
			Example	6:55
		Lecture Example 1		8:55
		Lecture Example 2		11:18
		Additional Example 3		6:26
		Additional Example 4		4:15
	Solving Quadratic Systems			28:18
		Intro		0:00
		Linear Quadratic Systems		0:04
			Example	0:28
		Solutions		3:13
		Quadratic Quadratic System		3:36
			Example: Elimination	3:45
		Solutions		7:34
		Systems of Quadratic Inequalities		7:55
			Example	8:07
		Lecture Example 1		11:10
		Lecture Example 2		16:12
		Additional Example 3		4:29
		Additional Example 4		3:34
XI. Sequences and Series
	Arithmetic Sequences			27:44
		Intro		0:00
		Sequences		0:27
			Example: Term	0:36
		Arithmetic Sequence		2:13
			Common Difference	2:22
			Example	2:35
		Formula for nth Term		3:39
			Example	4:29
		Equation for nth Term		5:58
			Example	6:10
		Arithmetic Means		7:40
			Example	8:13
		Lecture Example 1		14:08
		Lecture Example 2		15:35
		Additional Example 3		3:36
		Additional Example 4		5:01
	Arithmetic Series			29:12
		Intro		0:00
		What are Arithmetic Series?		0:22
			Example: Sequence	0:29
			Example: Common Difference (d)	0:35
		Sum of Arithmetic Series		2:52
			Example	3:44
		Sigma Notation		6:10
			Example	6:48
		Lecture Example 1		8:32
		Lecture Example 2		12:39
		Additional Example 3		4:07
		Additional Example 4		5:01
	Geometric Sequences			24:52
		Intro		0:00
		What are Geometric Sequences?		0:20
			Common Ratio	1:03
			Example	1:20
		nth Term of a Geometric Sequence		3:39
		Geometric Means		4:16
			Example: Missing Term	5:06
		Lecture Example 1		8:09
		Lecture Example 2		11:42
		Additional Example 3		2:33
		Additional Example 4		5:18
	Geometric Series			27:02
		Intro		0:00
		What are Geometric Series?		0:20
			Example: Common Ratio	1:00
		Sum of Geometric Series		2:27
			Example	4:01
		Sigma Notation		4:56
			Example: Index	5:24
			Example	6:20
		Another Sum Formula		7:51
		Specific Terms		9:19
		Lecture Example 1		11:15
		Lecture Example 2		14:30
		Additional Example 3		4:06
		Additional Example 4		4:37
	Infinite Geometric Series			24:01
		Intro		0:00
		What are Infinite Geometric Series?		0:35
			Partial Sums of the Infinite Series	1:17
			Example	1:24
		Sum of an Infinite Geometric Series		3:16
			Convergent Series	3:25
			Example	4:17
		Sigma Notation		5:31
			Example	5:43
		Repeating Decimals		6:38
			Example	6:48
		Lecture Example 1		9:33
		Lecture Example 2		11:20
		Additional Example 3		2:21
		Additional Example 4		3:21
	Recursion and Special Sequences			17:11
		Intro		0:00
		Fibonacci Sequence		0:17
			Example: Fibonacci Sequence	0:36
			Example: Recursive Formula	2:38
		Iteration		3:40
			Example	4:57
		Lecture Example 1		7:10
		Lecture Example 2		9:03
		Additional Example 3		3:06
		Additional Example 4		2:52
	Binomial Theorem			38:25
		Intro		0:00
		Pascal's Triangle		0:11
			General Form	2:43
		Properties		7:01
		Binomial Theorem		9:20
			Example	11:47
		Finding a Specific Term		16:24
			Example	16:32
		Lecture Example 1		20:35
		Lecture Example 2		23:30
		Additional Example 3		5:24
		Additional Example 4		5:29
	Proof and Mathematical Induction			19:53
		Intro		0:00
		Math Induction Principle		0:19
			Example	0:29
		Counter Examples		5:00
			Example	5:14
		Lecture Example 1		7:16
		Lecture Example 2		10:53
		Additional Example 3		5:07
		Additional Example 4		1:31

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Graphing Rational Functions

Graphing Rational Functions

Mathematics: Algebra 2

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