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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Rational Functions and Vertical Asymptotes

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

Intro
0:00
Title of the Course
0:06
Different Names for the Course
0:07
Precalculus
0:12
Math Analysis
0:14
Trigonometry
0:16
Algebra III
0:20
Geometry II
0:24
College Algebra
0:30
Same Concepts
0:36
How do the Lessons Work?
0:54
Introducing Concepts
0:56
Apply Concepts
1:04
Go through Examples
1:25
Who is this Course For?
1:38
Those Who Need eExtra Help with Class Work
1:52
Those Working on Material but not in Formal Class at School
1:54
Those Who Want a Refresher
2:00
Try to Watch the Whole Lesson
2:20
Understanding is So Important
3:56
What to Watch First
5:26
Lesson #2: Sets, Elements, and Numbers
5:30
Lesson #7: Idea of a Function
5:33
Lesson #6: Word Problems
6:04
What to Watch First, cont.
6:46
Lesson #2: Sets, Elements and Numbers
6:56
Lesson #3: Variables, Equations, and Algebra
6:58
Lesson #4: Coordinate Systems
7:00
Lesson #5: Midpoint, Distance, the Pythagorean Theorem and Slope
7:02
Lesson #6: Word Problems
7:10
Lesson #7: Idea of a Function
7:12
Lesson #8: Graphs
7:14
Graphing Calculator Appendix
7:40
What to Watch Last
8:46
Let's get Started!
9:48
Sets, Elements, & Numbers

45m 11s

Intro
0:00
Introduction
0:05
Sets and Elements
1:19
Set
1:20
Element
1:23
Name a Set
2:20
Order The Elements Appear In Has No Effect on the Set
2:55
Describing/ Defining Sets
3:28
Directly Say All the Elements
3:36
Clearly Describing All the Members of the Set
3:55
Describing the Quality (or Qualities) Each member Of the Set Has In Common
4:32
Symbols: 'Element of' and 'Subset of'
6:01
Symbol is ∈
6:03
Subset Symbol is ⊂
6:35
Empty Set
8:07
Symbol is ∅
8:20
Since It's Empty, It is a Subset of All Sets
8:44
Union and Intersection
9:54
Union Symbol is ∪
10:08
Intersection Symbol is ∩
10:18
Sets Can Be Weird Stuff
12:26
Can Have Elements in a Set
12:50
We Can Have Infinite Sets
13:09
Example
13:22
Consider a Set Where We Take a Word and Then Repeat It An Ever Increasing Number of Times
14:08
This Set Has Infinitely Many Distinct Elements
14:40
Numbers as Sets
16:03
Natural Numbers ℕ
16:16
Including 0 and the Negatives ℤ
18:13
Rational Numbers ℚ
19:27
Can Express Rational Numbers with Decimal Expansions
22:05
Irrational Numbers
23:37
Real Numbers ℝ: Put the Rational and Irrational Numbers Together
25:15
Interval Notation and the Real Numbers
26:45
Include the End Numbers
27:06
Exclude the End Numbers
27:33
Example
28:28
Interval Notation: Infinity
29:09
Use -∞ or ∞ to Show an Interval Going on Forever in One Direction or the Other
29:14
Always Use Parentheses
29:50
Examples
30:27
Example 1
31:23
Example 2
35:26
Example 3
38:02
Example 4
42:21
Variables, Equations, & Algebra

35m 31s

Intro
0:00
What is a Variable?
0:05
A Variable is a Placeholder for a Number
0:11
Affects the Output of a Function or a Dependent Variable
0:24
Naming Variables
1:51
Useful to Use Symbols
2:21
What is a Constant?
4:14
A Constant is a Fixed, Unchanging Number
4:28
We Might Refer to a Symbol Representing a Number as a Constant
4:51
What is a Coefficient?
5:33
A Coefficient is a Multiplicative Factor on a Variable
5:37
Not All Coefficients are Constants
5:51
Expressions and Equations
6:42
An Expression is a String of Mathematical Symbols That Make Sense Used Together
7:05
An Equation is a Statement That Two Expression Have the Same Value
8:20
The Idea of Algebra
8:51
Equality
8:59
If Two Things Are the Same *Equal), Then We Can Do the Exact Same Operation to Both and the Results Will Be the Same
9:41
Always Do The Exact Same Thing to Both Sides
12:22
Solving Equations
13:23
When You Are Asked to Solve an Equation, You Are Being Asked to Solve for Something
13:33
Look For What Values Makes the Equation True
13:38
Isolate the Variable by Doing Algebra
14:37
Order of Operations
16:02
Why Certain Operations are Grouped
17:01
When You Don't Have to Worry About Order
17:39
Distributive Property
18:15
It Allows Multiplication to Act Over Addition in Parentheses
18:23
We Can Use the Distributive Property in Reverse to Combine Like Terms
19:05
Substitution
20:03
Use Information From One Equation in Another Equation
20:07
Put Your Substitution in Parentheses
20:44
Example 1
23:17
Example 2
25:49
Example 3
28:11
Example 4
30:02
Coordinate Systems

35m 2s

Intro
0:00
Inherent Order in ℝ
0:05
Real Numbers Come with an Inherent Order
0:11
Positive Numbers
0:21
Negative Numbers
0:58
'Less Than' and 'Greater Than'
2:04
Tip To Help You Remember the Signs
2:56
Inequality
4:06
Less Than or Equal and Greater Than or Equal
4:51
One Dimension: The Number Line
5:36
Graphically Represent ℝ on a Number Line
5:43
Note on Infinities
5:57
With the Number Line, We Can Directly See the Order We Put on ℝ
6:35
Ordered Pairs
7:22
Example
7:34
Allows Us to Talk About Two Numbers at the Same Time
9:41
Ordered Pairs of Real Numbers Cannot be Put Into an Order Like we Did with ℝ
10:41
Two Dimensions: The Plane
13:13
We Can Represent Ordered Pairs with the Plane
13:24
Intersection is known as the Origin
14:31
Plotting the Point
14:32
Plane = Coordinate Plane = Cartesian Plane = ℝ²
17:46
The Plane and Quadrants
18:50
Quadrant I
19:04
Quadrant II
19:21
Quadrant III
20:04
Quadrant IV
20:20
Three Dimensions: Space
21:02
Create Ordered Triplets
21:09
Visually Represent This
21:19
Three-Dimension = Space = ℝ³
21:47
Higher Dimensions
22:24
If We Have n Dimensions, We Call It n-Dimensional Space or ℝ to the nth Power
22:31
We Can Represent Places In This n-Dimensional Space As Ordered Groupings of n Numbers
22:41
Hard to Visualize Higher Dimensional Spaces
23:18
Example 1
25:07
Example 2
26:10
Example 3
28:58
Example 4
31:05
Midpoints, Distance, the Pythagorean Theorem, & Slope

48m 43s

Intro
0:00
Introduction
0:07
Midpoint: One Dimension
2:09
Example of Something More Complex
2:31
Use the Idea of a Middle
3:28
Find the Midpoint of Arbitrary Values a and b
4:17
How They're Equivalent
5:05
Official Midpoint Formula
5:46
Midpoint: Two Dimensions
6:19
The Midpoint Must Occur at the Horizontal Middle and the Vertical Middle
6:38
Arbitrary Pair of Points Example
7:25
Distance: One Dimension
9:26
Absolute Value
10:54
Idea of Forcing Positive
11:06
Distance: One Dimension, Formula
11:47
Distance Between Arbitrary a and b
11:48
Absolute Value Helps When the Distance is Negative
12:41
Distance Formula
12:58
The Pythagorean Theorem
13:24
a²+b²=c²
13:50
Distance: Two Dimensions
14:59
Break Into Horizontal and Vertical Parts and then Use the Pythagorean Theorem
15:16
Distance Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
16:21
Slope
19:30
Slope is the Rate of Change
19:41
m = rise over run
21:27
Slope Between Arbitrary Points (x₁,y₁) and (x₂,y₂)
22:31
Interpreting Slope
24:12
Positive Slope and Negative Slope
25:40
m=1, m=0, m=-1
26:48
Example 1
28:25
Example 2
31:42
Example 3
36:40
Example 4
42:48
Word Problems

56m 31s

Intro
0:00
Introduction
0:05
What is a Word Problem?
0:45
Describes Any Problem That Primarily Gets Its Ideas Across With Words Instead of Math Symbols
0:48
Requires Us to Think
1:32
Why Are They So Hard?
2:11
Reason 1: No Simple Formula to Solve Them
2:16
Reason 2: Harder to Teach Word Problems
2:47
You Can Learn How to Do Them!
3:51
Grades
7:57
'But I'm Never Going to Use This In Real Life'
9:46
Solving Word Problems
12:58
First: Understand the Problem
13:37
Second: What Are You Looking For?
14:33
Third: Set Up Relationships
16:21
Fourth: Solve It!
17:48
Summary of Method
19:04
Examples on Things Other Than Math
20:21
Math-Specific Method: What You Need Now
25:30
Understand What the Problem is Talking About
25:37
Set Up and Name Any Variables You Need to Know
25:56
Set Up Equations Connecting Those Variables to the Information in the Problem Statement
26:02
Use the Equations to Solve for an Answer
26:14
Tip
26:58
Draw Pictures
27:22
Breaking Into Pieces
28:28
Try Out Hypothetical Numbers
29:52
Student Logic
31:27
Jump In!
32:40
Example 1
34:03
Example 2
39:15
Example 3
44:22
Example 4
50:24
II. Functions
Idea of a Function

39m 54s

Intro
0:00
Introduction
0:04
What is a Function?
1:06
A Visual Example and Non-Example
1:30
Function Notation
3:47
f(x)
4:05
Express What Sets the Function Acts On
5:45
Metaphors for a Function
6:17
Transformation
6:28
Map
7:17
Machine
8:56
Same Input Always Gives Same Output
10:01
If We Put the Same Input Into a Function, It Will Always Produce the Same Output
10:11
Example of Something That is Not a Function
11:10
A Non-Numerical Example
12:10
The Functions We Will Use
15:05
Unless Told Otherwise, We Will Assume Every Function Takes in Real Numbers and Outputs Real Numbers
15:11
Usually Told the Rule of a Given Function
15:27
How To Use a Function
16:18
Apply the Rule to Whatever Our Input Value Is
16:28
Make Sure to Wrap Your Substitutions in Parentheses
17:09
Functions and Tables
17:36
Table of Values, Sometimes Called a T-Table
17:46
Example
17:56
Domain: What Goes In
18:55
The Domain is the Set of all Inputs That the Function Can Accept
18:56
Example
19:40
Range: What Comes Out
21:27
The Range is the Set of All Possible Outputs a Function Can Assign
21:34
Example
21:49
Another Example Would Be Our Initial Function From Earlier in This Lesson
22:29
Example 1
23:45
Example 2
25:22
Example 3
27:27
Example 4
29:23
Example 5
33:33
Graphs

58m 26s

Intro
0:00
Introduction
0:04
How to Interpret Graphs
1:17
Input / Independent Variable
1:47
Output / Dependent Variable
2:00
Graph as Input ⇒ Output
2:23
One Way to Think of a Graph: See What Happened to Various Inputs
2:25
Example
2:47
Graph as Location of Solution
4:20
A Way to See Solutions
4:36
Example
5:20
Which Way Should We Interpret?
7:13
Easiest to Think In Terms of How Inputs Are Mapped to Outputs
7:20
Sometimes It's Easier to Think In Terms of Solutions
8:39
Pay Attention to Axes
9:50
Axes Tell Where the Graph Is and What Scale It Has
10:09
Often, The Axes Will Be Square
10:14
Example
12:06
Arrows or No Arrows?
16:07
Will Not Use Arrows at the End of Our Graphs
17:13
Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
17:18
How to Graph
19:47
Plot Points
20:07
Connect with Curves
21:09
If You Connect with Straight Lines
21:44
Graphs of Functions are Smooth
22:21
More Points ⇒ More Accurate
23:38
Vertical Line Test
27:44
If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
28:41
Every Point on a Graph Tells Us Where the x-Value Below is Mapped
30:07
Domain in Graphs
31:37
The Domain is the Set of All Inputs That a Function Can Accept
31:44
Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
33:19
Range in Graphs
33:53
Graphing Calculators: Check the Appendix!
36:55
Example 1
38:37
Example 2
45:19
Example 3
50:41
Example 4
53:28
Example 5
55:50
Properties of Functions

48m 49s

Intro
0:00
Introduction
0:05
Increasing Decreasing Constant
0:43
Looking at a Specific Graph
1:15
Increasing Interval
2:39
Constant Function
4:15
Decreasing Interval
5:10
Find Intervals by Looking at the Graph
5:32
Intervals Show x-values; Write in Parentheses
6:39
Maximum and Minimums
8:48
Relative (Local) Max/Min
10:20
Formal Definition of Relative Maximum
12:44
Formal Definition of Relative Minimum
13:05
Max/Min, More Terms
14:18
Definition of Extrema
15:01
Average Rate of Change
16:11
Drawing a Line for the Average Rate
16:48
Using the Slope of the Secant Line
17:36
Slope in Function Notation
18:45
Zeros/Roots/x-intercepts
19:45
What Zeros in a Function Mean
20:25
Even Functions
22:30
Odd Functions
24:36
Even/Odd Functions and Graphs
26:28
Example of an Even Function
27:12
Example of an Odd Function
28:03
Example 1
29:35
Example 2
33:07
Example 3
40:32
Example 4
42:34
Function Petting Zoo

29m 20s

Intro
0:00
Introduction
0:04
Don't Forget that Axes Matter!
1:44
The Constant Function
2:40
The Identity Function
3:44
The Square Function
4:40
The Cube Function
5:44
The Square Root Function
6:51
The Reciprocal Function
8:11
The Absolute Value Function
10:19
The Trigonometric Functions
11:56
f(x)=sin(x)
12:12
f(x)=cos(x)
12:24
Alternate Axes
12:40
The Exponential and Logarithmic Functions
13:35
Exponential Functions
13:44
Logarithmic Functions
14:24
Alternating Axes
15:17
Transformations and Compositions
16:08
Example 1
17:52
Example 2
18:33
Example 3
20:24
Example 4
26:07
Transformation of Functions

48m 35s

Intro
0:00
Introduction
0:04
Vertical Shift
1:12
Graphical Example
1:21
A Further Explanation
2:16
Vertical Stretch/Shrink
3:34
Graph Shrinks
3:46
Graph Stretches
3:51
A Further Explanation
5:07
Horizontal Shift
6:49
Moving the Graph to the Right
7:28
Moving the Graph to the Left
8:12
A Further Explanation
8:19
Understanding Movement on the x-axis
8:38
Horizontal Stretch/Shrink
12:59
Shrinking the Graph
13:40
Stretching the Graph
13:48
A Further Explanation
13:55
Understanding Stretches from the x-axis
14:12
Vertical Flip (aka Mirror)
16:55
Example Graph
17:07
Multiplying the Vertical Component by -1
17:18
Horizontal Flip (aka Mirror)
18:43
Example Graph
19:01
Multiplying the Horizontal Component by -1
19:54
Summary of Transformations
22:11
Stacking Transformations
24:46
Order Matters
25:20
Transformation Example
25:52
Example 1
29:21
Example 2
34:44
Example 3
38:10
Example 4
43:46
Composite Functions

33m 24s

Intro
0:00
Introduction
0:04
Arithmetic Combinations
0:40
Basic Operations
1:20
Definition of the Four Arithmetic Combinations
1:40
Composite Functions
2:53
The Function as a Machine
3:32
Function Compositions as Multiple Machines
3:59
Notation for Composite Functions
4:46
Two Formats
6:02
Another Visual Interpretation
7:17
How to Use Composite Functions
8:21
Example of on Function acting on Another
9:17
Example 1
11:03
Example 2
15:27
Example 3
21:11
Example 4
27:06
Piecewise Functions

51m 42s

Intro
0:00
Introduction
0:04
Analogies to a Piecewise Function
1:16
Different Potatoes
1:41
Factory Production
2:27
Notations for Piecewise Functions
3:39
Notation Examples from Analogies
6:11
Example of a Piecewise (with Table)
7:24
Example of a Non-Numerical Piecewise
11:35
Graphing Piecewise Functions
14:15
Graphing Piecewise Functions, Example
16:26
Continuous Functions
16:57
Statements of Continuity
19:30
Example of Continuous and Non-Continuous Graphs
20:05
Interesting Functions: the Step Function
22:00
Notation for the Step Function
22:40
How the Step Function Works
22:56
Graph of the Step Function
25:30
Example 1
26:22
Example 2
28:49
Example 3
36:50
Example 4
46:11
Inverse Functions

49m 37s

Intro
0:00
Introduction
0:04
Analogy by picture
1:10
How to Denote the inverse
1:40
What Comes out of the Inverse
1:52
Requirement for Reversing
2:02
The Basketball Factory
2:12
The Importance of Information
2:45
One-to-One
4:04
Requirement for Reversibility
4:21
When a Function has an Inverse
4:43
One-to-One
5:13
Not One-to-One
5:50
Not a Function
6:19
Horizontal Line Test
7:01
How to the test Works
7:12
One-to-One
8:12
Not One-to-One
8:45
Definition: Inverse Function
9:12
Formal Definition
9:21
Caution to Students
10:02
Domain and Range
11:12
Finding the Range of the Function Inverse
11:56
Finding the Domain of the Function Inverse
12:11
Inverse of an Inverse
13:09
Its just x!
13:26
Proof
14:03
Graphical Interpretation
17:07
Horizontal Line Test
17:20
Graph of the Inverse
18:04
Swapping Inputs and Outputs to Draw Inverses
19:02
How to Find the Inverse
21:03
What We Are Looking For
21:21
Reversing the Function
21:38
A Method to Find Inverses
22:33
Check Function is One-to-One
23:04
Swap f(x) for y
23:25
Interchange x and y
23:41
Solve for y
24:12
Replace y with the inverse
24:40
Some Comments
25:01
Keeping Step 2 and 3 Straight
25:44
Switching to Inverse
26:12
Checking Inverses
28:52
How to Check an Inverse
29:06
Quick Example of How to Check
29:56
Example 1
31:48
Example 2
34:56
Example 3
39:29
Example 4
46:19
Variation Direct and Inverse

28m 49s

Intro
0:00
Introduction
0:06
Direct Variation
1:14
Same Direction
1:21
Common Example: Groceries
1:56
Different Ways to Say that Two Things Vary Directly
2:28
Basic Equation for Direct Variation
2:55
Inverse Variation
3:40
Opposite Direction
3:50
Common Example: Gravity
4:53
Different Ways to Say that Two Things Vary Indirectly
5:48
Basic Equation for Indirect Variation
6:33
Joint Variation
7:27
Equation for Joint Variation
7:53
Explanation of the Constant
8:48
Combined Variation
9:35
Gas Law as a Combination
9:44
Single Constant
10:33
Example 1
10:49
Example 2
13:34
Example 3
15:39
Example 4
19:48
III. Polynomials
Intro to Polynomials

38m 41s

Intro
0:00
Introduction
0:04
Definition of a Polynomial
1:04
Starting Integer
2:06
Structure of a Polynomial
2:49
The a Constants
3:34
Polynomial Function
5:13
Polynomial Equation
5:23
Polynomials with Different Variables
5:36
Degree
6:23
Informal Definition
6:31
Find the Largest Exponent Variable
6:44
Quick Examples
7:36
Special Names for Polynomials
8:59
Based on the Degree
9:23
Based on the Number of Terms
10:12
Distributive Property (aka 'FOIL')
11:37
Basic Distributive Property
12:21
Distributing Two Binomials
12:55
Longer Parentheses
15:12
Reverse: Factoring
17:26
Long-Term Behavior of Polynomials
17:48
Examples
18:13
Controlling Term--Term with the Largest Exponent
19:33
Positive and Negative Coefficients on the Controlling Term
20:21
Leading Coefficient Test
22:07
Even Degree, Positive Coefficient
22:13
Even Degree, Negative Coefficient
22:39
Odd Degree, Positive Coefficient
23:09
Odd Degree, Negative Coefficient
23:27
Example 1
25:11
Example 2
27:16
Example 3
31:16
Example 4
34:41
Roots (Zeros) of Polynomials

41m 7s

Intro
0:00
Introduction
0:05
Roots in Graphs
1:17
The x-intercepts
1:33
How to Remember What 'Roots' Are
1:50
Naïve Attempts
2:31
Isolating Variables
2:45
Failures of Isolating Variables
3:30
Missing Solutions
4:59
Factoring: How to Find Roots
6:28
How Factoring Works
6:36
Why Factoring Works
7:20
Steps to Finding Polynomial Roots
9:21
Factoring: How to Find Roots CAUTION
10:08
Factoring is Not Easy
11:32
Factoring Quadratics
13:08
Quadratic Trinomials
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
Factoring Quadratics, Check Your Work
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
Factoring a Quadratic
19:04
Factoring: Roots Imply Factors
19:54
Where a Root is, A Factor Is
20:01
How to Use Known Roots to Make Factoring Easier
20:35
Not all Polynomials Can be Factored
22:30
Irreducible Polynomials
23:27
Complex Numbers Help
23:55
Max Number of Roots/Factors
24:57
Limit to Number of Roots Equal to the Degree
25:18
Why there is a Limit
25:25
Max Number of Peaks/Valleys
26:39
Shape Information from Degree
26:46
Example Graph
26:54
Max, But Not Required
28:00
Example 1
28:37
Example 2
31:21
Example 3
36:12
Example 4
38:40
Completing the Square and the Quadratic Formula

39m 43s

Intro
0:00
Introduction
0:05
Square Roots and Equations
0:51
Taking the Square Root to Find the Value of x
0:55
Getting the Positive and Negative Answers
1:05
Completing the Square: Motivation
2:04
Polynomials that are Easy to Solve
2:20
Making Complex Polynomials Easy to Solve
3:03
Steps to Completing the Square
4:30
Completing the Square: Method
7:22
Move C over
7:35
Divide by A
7:44
Find r
7:59
Add to Both Sides to Complete the Square
8:49
Solving Quadratics with Ease
9:56
The Quadratic Formula
11:38
Derivation
11:43
Final Form
12:23
Follow Format to Use Formula
13:38
How Many Roots?
14:53
The Discriminant
15:47
What the Discriminant Tells Us: How Many Roots
15:58
How the Discriminant Works
16:30
Example 1: Complete the Square
18:24
Example 2: Solve the Quadratic
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52
Properties of Quadratic Functions

45m 34s

Intro
0:00
Introduction
0:05
Parabolas
0:35
Examples of Different Parabolas
1:06
Axis of Symmetry and Vertex
1:28
Drawing an Axis of Symmetry
1:51
Placing the Vertex
2:28
Looking at the Axis of Symmetry and Vertex for other Parabolas
3:09
Transformations
4:18
Reviewing Transformation Rules
6:28
Note the Different Horizontal Shift Form
7:45
An Alternate Form to Quadratics
8:54
The Constants: k, h, a
9:05
Transformations Formed
10:01
Analyzing Different Parabolas
10:10
Switching Forms by Completing the Square
11:43
Vertex of a Parabola
16:30
Vertex at (h, k)
16:47
Vertex in Terms of a, b, and c Coefficients
17:28
Minimum/Maximum at Vertex
18:19
When a is Positive
18:25
When a is Negative
18:52
Axis of Symmetry
19:54
Incredibly Minor Note on Grammar
20:52
Example 1
21:48
Example 2
26:35
Example 3
28:55
Example 4
31:40
Intermediate Value Theorem and Polynomial Division

46m 8s

Intro
0:00
Introduction
0:05
Reminder: Roots Imply Factors
1:32
The Intermediate Value Theorem
3:41
The Basis: U between a and b
4:11
U is on the Function
4:52
Intermediate Value Theorem, Proof Sketch
5:51
If Not True, the Graph Would Have to Jump
5:58
But Graph is Defined as Continuous
6:43
Finding Roots with the Intermediate Value Theorem
7:01
Picking a and b to be of Different Signs
7:10
Must Be at Least One Root
7:46
Dividing a Polynomial
8:16
Using Roots and Division to Factor
8:38
Long Division Refresher
9:08
The Division Algorithm
12:18
How It Works to Divide Polynomials
12:37
The Parts of the Equation
13:24
Rewriting the Equation
14:47
Polynomial Long Division
16:20
Polynomial Long Division In Action
16:29
One Step at a Time
20:51
Synthetic Division
22:46
Setup
23:11
Synthetic Division, Example
24:44
Which Method Should We Use
26:39
Advantages of Synthetic Method
26:49
Advantages of Long Division
27:13
Example 1
29:24
Example 2
31:27
Example 3
36:22
Example 4
40:55
Complex Numbers

45m 36s

Intro
0:00
Introduction
0:04
A Wacky Idea
1:02
The Definition of the Imaginary Number
1:22
How it Helps Solve Equations
2:20
Square Roots and Imaginary Numbers
3:15
Complex Numbers
5:00
Real Part and Imaginary Part
5:20
When Two Complex Numbers are Equal
6:10
Addition and Subtraction
6:40
Deal with Real and Imaginary Parts Separately
7:36
Two Quick Examples
7:54
Multiplication
9:07
FOIL Expansion
9:14
Note What Happens to the Square of the Imaginary Number
9:41
Two Quick Examples
10:22
Division
11:27
Complex Conjugates
13:37
Getting Rid of i
14:08
How to Denote the Conjugate
14:48
Division through Complex Conjugates
16:11
Multiply by the Conjugate of the Denominator
16:28
Example
17:46
Factoring So-Called 'Irreducible' Quadratics
19:24
Revisiting the Quadratic Formula
20:12
Conjugate Pairs
20:37
But Are the Complex Numbers 'Real'?
21:27
What Makes a Number Legitimate
25:38
Where Complex Numbers are Used
27:20
Still, We Won't See Much of C
29:05
Example 1
30:30
Example 2
33:15
Example 3
38:12
Example 4
42:07
Fundamental Theorem of Algebra

19m 9s

Intro
0:00
Introduction
0:05
Idea: Hidden Roots
1:16
Roots in Complex Form
1:42
All Polynomials Have Roots
2:08
Fundamental Theorem of Algebra
2:21
Where Are All the Imaginary Roots, Then?
3:17
All Roots are Complex
3:45
Real Numbers are a Subset of Complex Numbers
3:59
The n Roots Theorem
5:01
For Any Polynomial, Its Degree is Equal to the Number of Roots
5:11
Equivalent Statement
5:24
Comments: Multiplicity
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
Comments: Complex Numbers Necessary
7:41
Comments: Complex Coefficients Allowed
8:55
Comments: Existence Theorem
9:59
Proof Sketch of n Roots Theorem
10:45
First Root
11:36
Second Root
13:23
Continuation to Find all Roots
16:00
IV. Rational Functions
Rational Functions and Vertical Asymptotes

33m 22s

Intro
0:00
Introduction
0:05
Definition of a Rational Function
1:20
Examples of Rational Functions
2:30
Why They are Called 'Rational'
2:47
Domain of a Rational Function
3:15
Undefined at Denominator Zeros
3:25
Otherwise all Reals
4:16
Investigating a Fundamental Function
4:50
The Domain of the Function
5:04
What Occurs at the Zeroes of the Denominator
5:20
Idea of a Vertical Asymptote
6:23
What's Going On?
6:58
Approaching x=0 from the left
7:32
Approaching x=0 from the right
8:34
Dividing by Very Small Numbers Results in Very Large Numbers
9:31
Definition of a Vertical Asymptote
10:05
Vertical Asymptotes and Graphs
11:15
Drawing Asymptotes by Using a Dashed Line
11:27
The Graph Can Never Touch Its Undefined Point
12:00
Not All Zeros Give Asymptotes
13:02
Special Cases: When Numerator and Denominator Go to Zero at the Same Time
14:58
Cancel out Common Factors
15:49
How to Find Vertical Asymptotes
16:10
Figure out What Values Are Not in the Domain of x
16:24
Determine if the Numerator and Denominator Share Common Factors and Cancel
16:45
Find Denominator Roots
17:33
Note if Asymptote Approaches Negative or Positive Infinity
18:06
Example 1
18:57
Example 2
21:26
Example 3
23:04
Example 4
30:01
Horizontal Asymptotes

34m 16s

Intro
0:00
Introduction
0:05
Investigating a Fundamental Function
0:53
What Happens as x Grows Large
1:00
Different View
1:12
Idea of a Horizontal Asymptote
1:36
What's Going On?
2:24
What Happens as x Grows to a Large Negative Number
2:49
What Happens as x Grows to a Large Number
3:30
Dividing by Very Large Numbers Results in Very Small Numbers
3:52
Example Function
4:41
Definition of a Vertical Asymptote
8:09
Expanding the Idea
9:03
What's Going On?
9:48
What Happens to the Function in the Long Run?
9:51
Rewriting the Function
10:13
Definition of a Slant Asymptote
12:09
Symbolical Definition
12:30
Informal Definition
12:45
Beyond Slant Asymptotes
13:03
Not Going Beyond Slant Asymptotes
14:39
Horizontal/Slant Asymptotes and Graphs
15:43
How to Find Horizontal and Slant Asymptotes
16:52
How to Find Horizontal Asymptotes
17:12
Expand the Given Polynomials
17:18
Compare the Degrees of the Numerator and Denominator
17:40
How to Find Slant Asymptotes
20:05
Slant Asymptotes Exist When n+m=1
20:08
Use Polynomial Division
20:24
Example 1
24:32
Example 2
25:53
Example 3
26:55
Example 4
29:22
Graphing Asymptotes in a Nutshell

49m 7s

Intro
0:00
Introduction
0:05
A Process for Graphing
1:22
1. Factor Numerator and Denominator
1:50
2. Find Domain
2:53
3. Simplifying the Function
3:59
4. Find Vertical Asymptotes
4:59
5. Find Horizontal/Slant Asymptotes
5:24
6. Find Intercepts
7:35
7. Draw Graph (Find Points as Necessary)
9:21
Draw Graph Example
11:21
Vertical Asymptote
11:41
Horizontal Asymptote
11:50
Other Graphing
12:16
Test Intervals
15:08
Example 1
17:57
Example 2
23:01
Example 3
29:02
Example 4
33:37
Partial Fractions

44m 56s

Intro
0:00
Introduction: Idea
0:04
Introduction: Prerequisites and Uses
1:57
Proper vs. Improper Polynomial Fractions
3:11
Possible Things in the Denominator
4:38
Linear Factors
6:16
Example of Linear Factors
7:03
Multiple Linear Factors
7:48
Irreducible Quadratic Factors
8:25
Example of Quadratic Factors
9:26
Multiple Quadratic Factors
9:49
Mixing Factor Types
10:28
Figuring Out the Numerator
11:10
How to Solve for the Constants
11:30
Quick Example
11:40
Example 1
14:29
Example 2
18:35
Example 3
20:33
Example 4
28:51
V. Exponential & Logarithmic Functions
Understanding Exponents

35m 17s

Intro
0:00
Introduction
0:05
Fundamental Idea
1:46
Expanding the Idea
2:28
Multiplication of the Same Base
2:40
Exponents acting on Exponents
3:45
Different Bases with the Same Exponent
4:31
To the Zero
5:35
To the First
5:45
Fundamental Rule with the Zero Power
6:35
To the Negative
7:45
Any Number to a Negative Power
8:14
A Fraction to a Negative Power
9:58
Division with Exponential Terms
10:41
To the Fraction
11:33
Square Root
11:58
Any Root
12:59
Summary of Rules
14:38
To the Irrational
17:21
Example 1
20:34
Example 2
23:42
Example 3
27:44
Example 4
31:44
Example 5
33:15
Exponential Functions

47m 4s

Intro
0:00
Introduction
0:05
Definition of an Exponential Function
0:48
Definition of the Base
1:02
Restrictions on the Base
1:16
Computing Exponential Functions
2:29
Harder Computations
3:10
When to Use a Calculator
3:21
Graphing Exponential Functions: a>1
6:02
Three Examples
6:13
What to Notice on the Graph
7:44
A Story
8:27
Story Diagram
9:15
Increasing Exponentials
11:29
Story Morals
14:40
Application: Compound Interest
15:15
Compounding Year after Year
16:01
Function for Compounding Interest
16:51
A Special Number: e
20:55
Expression for e
21:28
Where e stabilizes
21:55
Application: Continuously Compounded Interest
24:07
Equation for Continuous Compounding
24:22
Exponential Decay 0<a<1
25:50
Three Examples
26:11
Why they 'lose' value
26:54
Example 1
27:47
Example 2
33:11
Example 3
36:34
Example 4
41:28
Introduction to Logarithms

40m 31s

Intro
0:00
Introduction
0:04
Definition of a Logarithm, Base 2
0:51
Log 2 Defined
0:55
Examples
2:28
Definition of a Logarithm, General
3:23
Examples of Logarithms
5:15
Problems with Unusual Bases
7:38
Shorthand Notation: ln and log
9:44
base e as ln
10:01
base 10 as log
10:34
Calculating Logarithms
11:01
using a calculator
11:34
issues with other bases
11:58
Graphs of Logarithms
13:21
Three Examples
13:29
Slow Growth
15:19
Logarithms as Inverse of Exponentiation
16:02
Using Base 2
16:05
General Case
17:10
Looking More Closely at Logarithm Graphs
19:16
The Domain of Logarithms
20:41
Thinking about Logs like Inverses
21:08
The Alternate
24:00
Example 1
25:59
Example 2
30:03
Example 3
32:49
Example 4
37:34
Properties of Logarithms

42m 33s

Intro
0:00
Introduction
0:04
Basic Properties
1:12
Inverse--log(exp)
1:43
A Key Idea
2:44
What We Get through Exponentiation
3:18
B Always Exists
4:50
Inverse--exp(log)
5:53
Logarithm of a Power
7:44
Logarithm of a Product
10:07
Logarithm of a Quotient
13:48
Caution! There Is No Rule for loga(M+N)
16:12
Summary of Properties
17:42
Change of Base--Motivation
20:17
No Calculator Button
20:59
A Specific Example
21:45
Simplifying
23:45
Change of Base--Formula
24:14
Example 1
25:47
Example 2
29:08
Example 3
31:14
Example 4
34:13
Solving Exponential and Logarithmic Equations

34m 10s

Intro
0:00
Introduction
0:05
One to One Property
1:09
Exponential
1:26
Logarithmic
1:44
Specific Considerations
2:02
One-to-One Property
3:30
Solving by One-to-One
4:11
Inverse Property
6:09
Solving by Inverses
7:25
Dealing with Equations
7:50
Example of Taking an Exponent or Logarithm of an Equation
9:07
A Useful Property
11:57
Bring Down Exponents
12:01
Try to Simplify
13:20
Extraneous Solutions
13:45
Example 1
16:37
Example 2
19:39
Example 3
21:37
Example 4
26:45
Example 5
29:37
Application of Exponential and Logarithmic Functions

48m 46s

Intro
0:00
Introduction
0:06
Applications of Exponential Functions
1:07
A Secret!
2:17
Natural Exponential Growth Model
3:07
Figure out r
3:34
A Secret!--Why Does It Work?
4:44
e to the r Morphs
4:57
Example
5:06
Applications of Logarithmic Functions
8:32
Examples
8:43
What Logarithms are Useful For
9:53
Example 1
11:29
Example 2
15:30
Example 3
26:22
Example 4
32:05
Example 5
39:19
VI. Trigonometric Functions
Angles

39m 5s

Intro
0:00
Degrees
0:22
Circle is 360 Degrees
0:48
Splitting a Circle
1:13
Radians
2:08
Circle is 2 Pi Radians
2:31
One Radian
2:52
Half-Circle and Right Angle
4:00
Converting Between Degrees and Radians
6:24
Formulas for Degrees and Radians
6:52
Coterminal, Complementary, Supplementary Angles
7:23
Coterminal Angles
7:30
Complementary Angles
9:40
Supplementary Angles
10:08
Example 1: Dividing a Circle
10:38
Example 2: Converting Between Degrees and Radians
11:56
Example 3: Quadrants and Coterminal Angles
14:18
Extra Example 1: Common Angle Conversions
-1
Extra Example 2: Quadrants and Coterminal Angles
-2
Sine and Cosine Functions

43m 16s

Intro
0:00
Sine and Cosine
0:15
Unit Circle
0:22
Coordinates on Unit Circle
1:03
Right Triangles
1:52
Adjacent, Opposite, Hypotenuse
2:25
Master Right Triangle Formula: SOHCAHTOA
2:48
Odd Functions, Even Functions
4:40
Example: Odd Function
4:56
Example: Even Function
7:30
Example 1: Sine and Cosine
10:27
Example 2: Graphing Sine and Cosine Functions
14:39
Example 3: Right Triangle
21:40
Example 4: Odd, Even, or Neither
26:01
Extra Example 1: Right Triangle
-1
Extra Example 2: Graphing Sine and Cosine Functions
-2
Sine and Cosine Values of Special Angles

33m 5s

Intro
0:00
45-45-90 Triangle and 30-60-90 Triangle
0:08
45-45-90 Triangle
0:21
30-60-90 Triangle
2:06
Mnemonic: All Students Take Calculus (ASTC)
5:21
Using the Unit Circle
5:59
New Angles
6:21
Other Quadrants
9:43
Mnemonic: All Students Take Calculus
10:13
Example 1: Convert, Quadrant, Sine/Cosine
13:11
Example 2: Convert, Quadrant, Sine/Cosine
16:48
Example 3: All Angles and Quadrants
20:21
Extra Example 1: Convert, Quadrant, Sine/Cosine
-1
Extra Example 2: All Angles and Quadrants
-2
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D

52m 3s

Intro
0:00
Amplitude and Period of a Sine Wave
0:38
Sine Wave Graph
0:58
Amplitude: Distance from Middle to Peak
1:18
Peak: Distance from Peak to Peak
2:41
Phase Shift and Vertical Shift
4:13
Phase Shift: Distance Shifted Horizontally
4:16
Vertical Shift: Distance Shifted Vertically
6:48
Example 1: Amplitude/Period/Phase and Vertical Shift
8:04
Example 2: Amplitude/Period/Phase and Vertical Shift
17:39
Example 3: Find Sine Wave Given Attributes
25:23
Extra Example 1: Amplitude/Period/Phase and Vertical Shift
-1
Extra Example 2: Find Cosine Wave Given Attributes
-2
Tangent and Cotangent Functions

36m 4s

Intro
0:00
Tangent and Cotangent Definitions
0:21
Tangent Definition
0:25
Cotangent Definition
0:47
Master Formula: SOHCAHTOA
1:01
Mnemonic
1:16
Tangent and Cotangent Values
2:29
Remember Common Values of Sine and Cosine
2:46
90 Degrees Undefined
4:36
Slope and Menmonic: ASTC
5:47
Uses of Tangent
5:54
Example: Tangent of Angle is Slope
6:09
Sign of Tangent in Quadrants
7:49
Example 1: Graph Tangent and Cotangent Functions
10:42
Example 2: Tangent and Cotangent of Angles
16:09
Example 3: Odd, Even, or Neither
18:56
Extra Example 1: Tangent and Cotangent of Angles
-1
Extra Example 2: Tangent and Cotangent of Angles
-2
Secant and Cosecant Functions

27m 18s

Intro
0:00
Secant and Cosecant Definitions
0:17
Secant Definition
0:18
Cosecant Definition
0:33
Example 1: Graph Secant Function
0:48
Example 2: Values of Secant and Cosecant
6:49
Example 3: Odd, Even, or Neither
12:49
Extra Example 1: Graph of Cosecant Function
-1
Extra Example 2: Values of Secant and Cosecant
-2
Inverse Trigonometric Functions

32m 58s

Intro
0:00
Arcsine Function
0:24
Restrictions between -1 and 1
0:43
Arcsine Notation
1:26
Arccosine Function
3:07
Restrictions between -1 and 1
3:36
Cosine Notation
3:53
Arctangent Function
4:30
Between -Pi/2 and Pi/2
4:44
Tangent Notation
5:02
Example 1: Domain/Range/Graph of Arcsine
5:45
Example 2: Arcsin/Arccos/Arctan Values
10:46
Example 3: Domain/Range/Graph of Arctangent
17:14
Extra Example 1: Domain/Range/Graph of Arccosine
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
Computations of Inverse Trigonometric Functions

31m 8s

Intro
0:00
Inverse Trigonometric Function Domains and Ranges
0:31
Arcsine
0:41
Arccosine
1:14
Arctangent
1:41
Example 1: Arcsines of Common Values
2:44
Example 2: Odd, Even, or Neither
5:57
Example 3: Arccosines of Common Values
12:24
Extra Example 1: Arctangents of Common Values
-1
Extra Example 2: Arcsin/Arccos/Arctan Values
-2
VII. Trigonometric Identities
Pythagorean Identity

19m 11s

Intro
0:00
Pythagorean Identity
0:17
Pythagorean Triangle
0:27
Pythagorean Identity
0:45
Example 1: Use Pythagorean Theorem to Prove Pythagorean Identity
1:14
Example 2: Find Angle Given Cosine and Quadrant
4:18
Example 3: Verify Trigonometric Identity
8:00
Extra Example 1: Use Pythagorean Identity to Prove Pythagorean Theorem
-1
Extra Example 2: Find Angle Given Cosine and Quadrant
-2
Identity Tan(squared)x+1=Sec(squared)x

23m 16s

Intro
0:00
Main Formulas
0:19
Companion to Pythagorean Identity
0:27
For Cotangents and Cosecants
0:52
How to Remember
0:58
Example 1: Prove the Identity
1:40
Example 2: Given Tan Find Sec
3:42
Example 3: Prove the Identity
7:45
Extra Example 1: Prove the Identity
-1
Extra Example 2: Given Sec Find Tan
-2
Addition and Subtraction Formulas

52m 52s

Intro
0:00
Addition and Subtraction Formulas
0:09
How to Remember
0:48
Cofunction Identities
1:31
How to Remember Graphically
1:44
Where to Use Cofunction Identities
2:52
Example 1: Derive the Formula for cos(A-B)
3:08
Example 2: Use Addition and Subtraction Formulas
16:03
Example 3: Use Addition and Subtraction Formulas to Prove Identity
25:11
Extra Example 1: Use cos(A-B) and Cofunction Identities
-1
Extra Example 2: Convert to Radians and use Formulas
-2
Double Angle Formulas

29m 5s

Intro
0:00
Main Formula
0:07
How to Remember from Addition Formula
0:18
Two Other Forms
1:35
Example 1: Find Sine and Cosine of Angle using Double Angle
3:16
Example 2: Prove Trigonometric Identity using Double Angle
9:37
Example 3: Use Addition and Subtraction Formulas
12:38
Extra Example 1: Find Sine and Cosine of Angle using Double Angle
-1
Extra Example 2: Prove Trigonometric Identity using Double Angle
-2
Half-Angle Formulas

43m 55s

Intro
0:00
Main Formulas
0:09
Confusing Part
0:34
Example 1: Find Sine and Cosine of Angle using Half-Angle
0:54
Example 2: Prove Trigonometric Identity using Half-Angle
11:51
Example 3: Prove the Half-Angle Formula for Tangents
18:39
Extra Example 1: Find Sine and Cosine of Angle using Half-Angle
-1
Extra Example 2: Prove Trigonometric Identity using Half-Angle
-2
VIII. Applications of Trigonometry
Trigonometry in Right Angles

25m 43s

Intro
0:00
Master Formula for Right Angles
0:11
SOHCAHTOA
0:15
Only for Right Triangles
1:26
Example 1: Find All Angles in a Triangle
2:19
Example 2: Find Lengths of All Sides of Triangle
7:39
Example 3: Find All Angles in a Triangle
11:00
Extra Example 1: Find All Angles in a Triangle
-1
Extra Example 2: Find Lengths of All Sides of Triangle
-2
Law of Sines

56m 40s

Intro
0:00
Law of Sines Formula
0:18
SOHCAHTOA
0:27
Any Triangle
0:59
Graphical Representation
1:25
Solving Triangle Completely
2:37
When to Use Law of Sines
2:55
ASA, SAA, SSA, AAA
2:59
SAS, SSS for Law of Cosines
7:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
8:44
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:30
Example 3: How Many Triangles Satisfy Conditions, Solve Completely
28:32
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: How Many Triangles Satisfy Conditions, Solve Completely
-2
Law of Cosines

49m 5s

Intro
0:00
Law of Cosines Formula
0:23
Graphical Representation
0:34
Relates Sides to Angles
1:00
Any Triangle
1:20
Generalization of Pythagorean Theorem
1:32
When to Use Law of Cosines
2:26
SAS, SSS
2:30
Heron's Formula
4:49
Semiperimeter S
5:11
Example 1: How Many Triangles Satisfy Conditions, Solve Completely
5:53
Example 2: How Many Triangles Satisfy Conditions, Solve Completely
15:19
Example 3: Find Area of a Triangle Given All Side Lengths
26:33
Extra Example 1: How Many Triangles Satisfy Conditions, Solve Completely
-1
Extra Example 2: Length of Third Side and Area of Triangle
-2
Finding the Area of a Triangle

27m 37s

Intro
0:00
Master Right Triangle Formula and Law of Cosines
0:19
SOHCAHTOA
0:27
Law of Cosines
1:23
Heron's Formula
2:22
Semiperimeter S
2:37
Example 1: Area of Triangle with Two Sides and One Angle
3:12
Example 2: Area of Triangle with Three Sides
6:11
Example 3: Area of Triangle with Three Sides, No Heron's Formula
8:50
Extra Example 1: Area of Triangle with Two Sides and One Angle
-1
Extra Example 2: Area of Triangle with Two Sides and One Angle
-2
Word Problems and Applications of Trigonometry

34m 25s

Intro
0:00
Formulas to Remember
0:11
SOHCAHTOA
0:15
Law of Sines
0:55
Law of Cosines
1:48
Heron's Formula
2:46
Example 1: Telephone Pole Height
4:01
Example 2: Bridge Length
7:48
Example 3: Area of Triangular Field
14:20
Extra Example 1: Kite Height
-1
Extra Example 2: Roads to a Town
-2
IX. Systems of Equations and Inequalities
Systems of Linear Equations

55m 40s

Intro
0:00
Introduction
0:04
Graphs as Location of 'True'
1:49
All Locations that Make the Function True
2:25
Understand the Relationship Between Solutions and the Graph
3:43
Systems as Graphs
4:07
Equations as Lines
4:20
Intersection Point
5:19
Three Possibilities for Solutions
6:17
Independent
6:24
Inconsistent
6:36
Dependent
7:06
Solving by Substitution
8:37
Solve for One Variable
9:07
Substitute into the Second Equation
9:34
Solve for Both Variables
10:12
What If a System is Inconsistent or Dependent?
11:08
No Solutions
11:25
Infinite Solutions
12:30
Solving by Elimination
13:56
Example
14:22
Determining the Number of Solutions
16:30
Why Elimination Makes Sense
17:25
Solving by Graphing Calculator
19:59
Systems with More than Two Variables
23:22
Example 1
25:49
Example 2
30:22
Example 3
34:11
Example 4
38:55
Example 5
46:01
(Non-) Example 6
53:37
Systems of Linear Inequalities

1h 13s

Intro
0:00
Introduction
0:04
Inequality Refresher-Solutions
0:46
Equation Solutions vs. Inequality Solutions
1:02
Essentially a Wide Variety of Answers
1:35
Refresher--Negative Multiplication Flips
1:43
Refresher--Negative Flips: Why?
3:19
Multiplication by a Negative
3:43
The Relationship Flips
3:55
Refresher--Stick to Basic Operations
4:34
Linear Equations in Two Variables
6:50
Graphing Linear Inequalities
8:28
Why It Includes a Whole Section
8:43
How to Show The Difference Between Strict and Not Strict Inequalities
10:08
Dashed Line--Not Solutions
11:10
Solid Line--Are Solutions
11:24
Test Points for Shading
11:42
Example of Using a Point
12:41
Drawing Shading from the Point
13:14
Graphing a System
14:53
Set of Solutions is the Overlap
15:17
Example
15:22
Solutions are Best Found Through Graphing
18:05
Linear Programming-Idea
19:52
Use a Linear Objective Function
20:15
Variables in Objective Function have Constraints
21:24
Linear Programming-Method
22:09
Rearrange Equations
22:21
Graph
22:49
Critical Solution is at the Vertex of the Overlap
23:40
Try Each Vertice
24:35
Example 1
24:58
Example 2
28:57
Example 3
33:48
Example 4
43:10
Nonlinear Systems

41m 1s

Intro
0:00
Introduction
0:06
Substitution
1:12
Example
1:22
Elimination
3:46
Example
3:56
Elimination is Less Useful for Nonlinear Systems
4:56
Graphing
5:56
Using a Graphing Calculator
6:44
Number of Solutions
8:44
Systems of Nonlinear Inequalities
10:02
Graph Each Inequality
10:06
Dashed and/or Solid
10:18
Shade Appropriately
11:14
Example 1
13:24
Example 2
15:50
Example 3
22:02
Example 4
29:06
Example 4, cont.
33:40
X. Vectors and Matrices
Vectors

1h 9m 31s

Intro
0:00
Introduction
0:10
Magnitude of the Force
0:22
Direction of the Force
0:48
Vector
0:52
Idea of a Vector
1:30
How Vectors are Denoted
2:00
Component Form
3:20
Angle Brackets and Parentheses
3:50
Magnitude/Length
4:26
Denoting the Magnitude of a Vector
5:16
Direction/Angle
7:52
Always Draw a Picture
8:50
Component Form from Magnitude & Angle
10:10
Scaling by Scalars
14:06
Unit Vectors
16:26
Combining Vectors - Algebraically
18:10
Combining Vectors - Geometrically
19:54
Resultant Vector
20:46
Alternate Component Form: i, j
21:16
The Zero Vector
23:18
Properties of Vectors
24:20
No Multiplication (Between Vectors)
28:30
Dot Product
29:40
Motion in a Medium
30:10
Fish in an Aquarium Example
31:38
More Than Two Dimensions
33:12
More Than Two Dimensions - Magnitude
34:18
Example 1
35:26
Example 2
38:10
Example 3
45:48
Example 4
50:40
Example 4, cont.
56:07
Example 5
1:01:32
Dot Product & Cross Product

35m 20s

Intro
0:00
Introduction
0:08
Dot Product - Definition
0:42
Dot Product Results in a Scalar, Not a Vector
2:10
Example in Two Dimensions
2:34
Angle and the Dot Product
2:58
The Dot Product of Two Vectors is Deeply Related to the Angle Between the Two Vectors
2:59
Proof of Dot Product Formula
4:14
Won't Directly Help Us Better Understand Vectors
4:18
Dot Product - Geometric Interpretation
4:58
We Can Interpret the Dot Product as a Measure of How Long and How Parallel Two Vectors Are
7:26
Dot Product - Perpendicular Vectors
8:24
If the Dot Product of Two Vectors is 0, We Know They are Perpendicular to Each Other
8:54
Cross Product - Definition
11:08
Cross Product Only Works in Three Dimensions
11:09
Cross Product - A Mnemonic
12:16
The Determinant of a 3 x 3 Matrix and Standard Unit Vectors
12:17
Cross Product - Geometric Interpretations
14:30
The Right-Hand Rule
15:17
Cross Product - Geometric Interpretations Cont.
17:00
Example 1
18:40
Example 2
22:50
Example 3
24:04
Example 4
26:20
Bonus Round
29:18
Proof: Dot Product Formula
29:24
Proof: Dot Product Formula, cont.
30:38
Matrices

54m 7s

Intro
0:00
Introduction
0:08
Definition of a Matrix
3:02
Size or Dimension
3:58
Square Matrix
4:42
Denoted by Capital Letters
4:56
When are Two Matrices Equal?
5:04
Examples of Matrices
6:44
Rows x Columns
6:46
Talking About Specific Entries
7:48
We Use Capitals to Denote a Matrix and Lower Case to Denotes Its Entries
8:32
Using Entries to Talk About Matrices
10:08
Scalar Multiplication
11:26
Scalar = Real Number
11:34
Example
12:36
Matrix Addition
13:08
Example
14:22
Matrix Multiplication
15:00
Example
18:52
Matrix Multiplication, cont.
19:58
Matrix Multiplication and Order (Size)
25:26
Make Sure Their Orders are Compatible
25:27
Matrix Multiplication is NOT Commutative
28:20
Example
30:08
Special Matrices - Zero Matrix (0)
32:48
Zero Matrix Has 0 for All of its Entries
32:49
Special Matrices - Identity Matrix (I)
34:14
Identity Matrix is a Square Matrix That Has 1 for All Its Entries on the Main Diagonal and 0 for All Other Entries
34:15
Example 1
36:16
Example 2
40:00
Example 3
44:54
Example 4
50:08
Determinants & Inverses of Matrices

47m 12s

Intro
0:00
Introduction
0:06
Not All Matrices Are Invertible
1:30
What Must a Matrix Have to Be Invertible?
2:08
Determinant
2:32
The Determinant is a Real Number Associated With a Square Matrix
2:38
If the Determinant of a Matrix is Nonzero, the Matrix is Invertible
3:40
Determinant of a 2 x 2 Matrix
4:34
Think in Terms of Diagonals
5:12
Minors and Cofactors - Minors
6:24
Example
6:46
Minors and Cofactors - Cofactors
8:00
Cofactor is Closely Based on the Minor
8:01
Alternating Sign Pattern
9:04
Determinant of Larger Matrices
10:56
Example
13:00
Alternative Method for 3x3 Matrices
16:46
Not Recommended
16:48
Inverse of a 2 x 2 Matrix
19:02
Inverse of Larger Matrices
20:00
Using Inverse Matrices
21:06
When Multiplied Together, They Create the Identity Matrix
21:24
Example 1
23:45
Example 2
27:21
Example 3
32:49
Example 4
36:27
Finding the Inverse of Larger Matrices
41:59
General Inverse Method - Step 1
43:25
General Inverse Method - Step 2
43:27
General Inverse Method - Step 2, cont.
43:27
General Inverse Method - Step 3
45:15
Using Matrices to Solve Systems of Linear Equations

58m 34s

Intro
0:00
Introduction
0:12
Augmented Matrix
1:44
We Can Represent the Entire Linear System With an Augmented Matrix
1:50
Row Operations
3:22
Interchange the Locations of Two Rows
3:50
Multiply (or Divide) a Row by a Nonzero Number
3:58
Add (or Subtract) a Multiple of One Row to Another
4:12
Row Operations - Keep Notes!
5:50
Suggested Symbols
7:08
Gauss-Jordan Elimination - Idea
8:04
Gauss-Jordan Elimination - Idea, cont.
9:16
Reduced Row-Echelon Form
9:18
Gauss-Jordan Elimination - Method
11:36
Begin by Writing the System As An Augmented Matrix
11:38
Gauss-Jordan Elimination - Method, cont.
13:48
Cramer's Rule - 2 x 2 Matrices
17:08
Cramer's Rule - n x n Matrices
19:24
Solving with Inverse Matrices
21:10
Solving Inverse Matrices, cont.
25:28
The Mighty (Graphing) Calculator
26:38
Example 1
29:56
Example 2
33:56
Example 3
37:00
Example 3, cont.
45:04
Example 4
51:28
XI. Alternate Ways to Graph
Parametric Equations

53m 33s

Intro
0:00
Introduction
0:06
Definition
1:10
Plane Curve
1:24
The Key Idea
2:00
Graphing with Parametric Equations
2:52
Same Graph, Different Equations
5:04
How Is That Possible?
5:36
Same Graph, Different Equations, cont.
5:42
Here's Another to Consider
7:56
Same Plane Curve, But Still Different
8:10
A Metaphor for Parametric Equations
9:36
Think of Parametric Equations As a Way to Describe the Motion of An Object
9:38
Graph Shows Where It Went, But Not Speed
10:32
Eliminating Parameters
12:14
Rectangular Equation
12:16
Caution
13:52
Creating Parametric Equations
14:30
Interesting Graphs
16:38
Graphing Calculators, Yay!
19:18
Example 1
22:36
Example 2
28:26
Example 3
37:36
Example 4
41:00
Projectile Motion
44:26
Example 5
47:00
Polar Coordinates

48m 7s

Intro
0:00
Introduction
0:04
Polar Coordinates Give Us a Way To Describe the Location of a Point
0:26
Polar Equations and Functions
0:50
Plotting Points with Polar Coordinates
1:06
The Distance of the Point from the Origin
1:09
The Angle of the Point
1:33
Give Points as the Ordered Pair (r,θ)
2:03
Visualizing Plotting in Polar Coordinates
2:32
First Way We Can Plot
2:39
Second Way We Can Plot
2:50
First, We'll Look at Visualizing r, Then θ
3:09
Rotate the Length Counter-Clockwise by θ
3:38
Alternatively, We Can Visualize θ, Then r
4:06
'Polar Graph Paper'
6:17
Horizontal and Vertical Tick Marks Are Not Useful for Polar
6:42
Use Concentric Circles to Helps Up See Distance From the Pole
7:08
Can Use Arc Sectors to See Angles
7:57
Multiple Ways to Name a Point
9:17
Examples
9:30
For Any Angle θ, We Can Make an Equivalent Angle
10:44
Negative Values for r
11:58
If r Is Negative, We Go In The Direction Opposite the One That The Angle θ Points Out
12:22
Another Way to Name the Same Point: Add π to θ and Make r Negative
13:44
Converting Between Rectangular and Polar
14:37
Rectangular Way to Name
14:43
Polar Way to Name
14:52
The Rectangular System Must Have a Right Angle Because It's Based on a Rectangle
15:08
Connect Both Systems Through Basic Trigonometry
15:38
Equation to Convert From Polar to Rectangular Coordinate Systems
16:55
Equation to Convert From Rectangular to Polar Coordinate Systems
17:13
Converting to Rectangular is Easy
17:20
Converting to Polar is a Bit Trickier
17:21
Draw Pictures
18:55
Example 1
19:50
Example 2
25:17
Example 3
31:05
Example 4
35:56
Example 5
41:49
Polar Equations & Functions

38m 16s

Intro
0:00
Introduction
0:04
Equations and Functions
1:16
Independent Variable
1:21
Dependent Variable
1:30
Examples
1:46
Always Assume That θ Is In Radians
2:44
Graphing in Polar Coordinates
3:29
Graph is the Same Way We Graph 'Normal' Stuff
3:32
Example
3:52
Graphing in Polar - Example, Cont.
6:45
Tips for Graphing
9:23
Notice Patterns
10:19
Repetition
13:39
Graphing Equations of One Variable
14:39
Converting Coordinate Types
16:16
Use the Same Conversion Formulas From the Previous Lesson
16:23
Interesting Graphs
17:48
Example 1
18:03
Example 2
18:34
Graphing Calculators, Yay!
19:07
Plot Random Things, Alter Equations You Understand, Get a Sense for How Polar Stuff Works
19:11
Check Out the Appendix
19:26
Example 1
21:36
Example 2
28:13
Example 3
34:24
Example 4
35:52
XII. Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers

40m 43s

Intro
0:00
Polar Coordinates
0:49
Rectangular Form
0:52
Polar Form
1:25
R and Theta
1:51
Polar Form Conversion
2:27
R and Theta
2:35
Optimal Values
4:05
Euler's Formula
4:25
Multiplying Two Complex Numbers in Polar Form
6:10
Multiply r's Together and Add Exponents
6:32
Example 1: Convert Rectangular to Polar Form
7:17
Example 2: Convert Polar to Rectangular Form
13:49
Example 3: Multiply Two Complex Numbers
17:28
Extra Example 1: Convert Between Rectangular and Polar Forms
-1
Extra Example 2: Simplify Expression to Polar Form
-2
DeMoivre's Theorem

57m 37s

Intro
0:00
Introduction to DeMoivre's Theorem
0:10
n nth Roots
3:06
DeMoivre's Theorem: Finding nth Roots
3:52
Relation to Unit Circle
6:29
One nth Root for Each Value of k
7:11
Example 1: Convert to Polar Form and Use DeMoivre's Theorem
8:24
Example 2: Find Complex Eighth Roots
15:27
Example 3: Find Complex Roots
27:49
Extra Example 1: Convert to Polar Form and Use DeMoivre's Theorem
-1
Extra Example 2: Find Complex Fourth Roots
-2
XIII. Counting & Probability
Counting

31m 36s

Intro
0:00
Introduction
0:08
Combinatorics
0:56
Definition: Event
1:24
Example
1:50
Visualizing an Event
3:02
Branching line diagram
3:06
Addition Principle
3:40
Example
4:18
Multiplication Principle
5:42
Example
6:24
Pigeonhole Principle
8:06
Example
10:26
Draw Pictures
11:06
Example 1
12:02
Example 2
14:16
Example 3
17:34
Example 4
21:26
Example 5
25:14
Permutations & Combinations

44m 3s

Intro
0:00
Introduction
0:08
Permutation
0:42
Combination
1:10
Towards a Permutation Formula
2:38
How Many Ways Can We Arrange the Letters A, B, C, D, and E?
3:02
Towards a Permutation Formula, cont.
3:34
Factorial Notation
6:56
Symbol Is '!'
6:58
Examples
7:32
Permutation of n Objects
8:44
Permutation of r Objects out of n
9:04
What If We Have More Objects Than We Have Slots to Fit Them Into?
9:46
Permutation of r Objects Out of n, cont.
10:28
Distinguishable Permutations
14:46
What If Not All Of the Objects We're Permuting Are Distinguishable From Each Other?
14:48
Distinguishable Permutations, cont.
17:04
Combinations
19:04
Combinations, cont.
20:56
Example 1
23:10
Example 2
26:16
Example 3
28:28
Example 4
31:52
Example 5
33:58
Example 6
36:34
Probability

36m 58s

Intro
0:00
Introduction
0:06
Definition: Sample Space
1:18
Event = Something Happening
1:20
Sample Space
1:36
Probability of an Event
2:12
Let E Be An Event and S Be The Corresponding Sample Space
2:14
'Equally Likely' Is Important
3:52
Fair and Random
5:26
Interpreting Probability
6:34
How Can We Interpret This Value?
7:24
We Can Represent Probability As a Fraction, a Decimal, Or a Percentage
8:04
One of Multiple Events Occurring
9:52
Mutually Exclusive Events
10:38
What If The Events Are Not Mutually Exclusive?
12:20
Taking the Possibility of Overlap Into Account
13:24
An Event Not Occurring
17:14
Complement of E
17:22
Independent Events
19:36
Independent
19:48
Conditional Events
21:28
What Is The Events Are Not Independent Though?
21:30
Conditional Probability
22:16
Conditional Events, cont.
23:51
Example 1
25:27
Example 2
27:09
Example 3
28:57
Example 4
30:51
Example 5
34:15
XIV. Conic Sections
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
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

Intro
0:00
Introduction
0:06
Definition: Sequence
0:28
Infinite Sequence
2:08
Finite Sequence
2:22
Length
2:58
Formula for the nth Term
3:22
Defining a Sequence Recursively
5:54
Initial Term
7:58
Sequences and Patterns
10:40
First, Identify a Pattern
12:52
How to Get From One Term to the Next
17:38
Tips for Finding Patterns
19:52
More Tips for Finding Patterns
24:14
Even More Tips
26:50
Example 1
30:32
Example 2
34:54
Fibonacci Sequence
34:55
Example 3
38:40
Example 4
45:02
Example 5
49:26
Example 6
51:54
Introduction to Series

40m 27s

Intro
0:00
Introduction
0:06
Definition: Series
1:20
Why We Need Notation
2:48
Simga Notation (AKA Summation Notation)
4:44
Thing Being Summed
5:42
Index of Summation
6:21
Lower Limit of Summation
7:09
Upper Limit of Summation
7:23
Sigma Notation, Example
7:36
Sigma Notation for Infinite Series
9:08
How to Reindex
10:58
How to Reindex, Expanding
12:56
How to Reindex, Substitution
16:46
Properties of Sums
19:42
Example 1
23:46
Example 2
25:34
Example 3
27:12
Example 4
29:54
Example 5
32:06
Example 6
37:16
Arithmetic Sequences & Series

31m 36s

Intro
0:00
Introduction
0:05
Definition: Arithmetic Sequence
0:47
Common Difference
1:13
Two Examples
1:19
Form for the nth Term
2:14
Recursive Relation
2:33
Towards an Arithmetic Series Formula
5:12
Creating a General Formula
10:09
General Formula for Arithmetic Series
14:23
Example 1
15:46
Example 2
17:37
Example 3
22:21
Example 4
24:09
Example 5
27:14
Geometric Sequences & Series

39m 27s

Intro
0:00
Introduction
0:06
Definition
0:48
Form for the nth Term
2:42
Formula for Geometric Series
5:16
Infinite Geometric Series
11:48
Diverges
13:04
Converges
14:48
Formula for Infinite Geometric Series
16:32
Example 1
20:32
Example 2
22:02
Example 3
26:00
Example 4
30:48
Example 5
34:28
Mathematical Induction

49m 53s

Intro
0:00
Introduction
0:06
Belief Vs. Proof
1:22
A Metaphor for Induction
6:14
The Principle of Mathematical Induction
11:38
Base Case
13:24
Inductive Step
13:30
Inductive Hypothesis
13:52
A Remark on Statements
14:18
Using Mathematical Induction
16:58
Working Example
19:58
Finding Patterns
28:46
Example 1
30:17
Example 2
37:50
Example 3
42:38
The Binomial Theorem

1h 13m 13s

Intro
0:00
Introduction
0:06
We've Learned That a Binomial Is An Expression That Has Two Terms
0:07
Understanding Binomial Coefficients
1:20
Things We Notice
2:24
What Goes In the Blanks?
5:52
Each Blank is Called a Binomial Coefficient
6:18
The Binomial Theorem
6:38
Example
8:10
The Binomial Theorem, cont.
10:46
We Can Also Write This Expression Compactly Using Sigma Notation
12:06
Proof of the Binomial Theorem
13:22
Proving the Binomial Theorem Is Within Our Reach
13:24
Pascal's Triangle
15:12
Pascal's Triangle, cont.
16:12
Diagonal Addition of Terms
16:24
Zeroth Row
18:04
First Row
18:12
Why Do We Care About Pascal's Triangle?
18:50
Pascal's Triangle, Example
19:26
Example 1
21:26
Example 2
24:34
Example 3
28:34
Example 4
32:28
Example 5
37:12
Time for the Fireworks!
43:38
Proof of the Binomial Theorem
43:44
We'll Prove This By Induction
44:04
Proof (By Induction)
46:36
Proof, Base Case
47:00
Proof, Inductive Step - Notation Discussion
49:22
Induction Step
49:24
Proof, Inductive Step - Setting Up
52:26
Induction Hypothesis
52:34
What We What To Show
52:44
Proof, Inductive Step - Start
54:18
Proof, Inductive Step - Middle
55:38
Expand Sigma Notations
55:48
Proof, Inductive Step - Middle, cont.
58:40
Proof, Inductive Step - Checking In
1:01:08
Let's Check In With Our Original Goal
1:01:12
Want to Show
1:01:18
Lemma - A Mini Theorem
1:02:18
Proof, Inductive Step - Lemma
1:02:52
Proof of Lemma: Let's Investigate the Left Side
1:03:08
Proof, Inductive Step - Nearly There
1:07:54
Proof, Inductive Step - End!
1:09:18
Proof, Inductive Step - End!, cont.
1:11:01
XVI. Preview of Calculus
Idea of a Limit

40m 22s

Intro
0:00
Introduction
0:05
Motivating Example
1:26
Fuzzy Notion of a Limit
3:38
Limit is the Vertical Location a Function is Headed Towards
3:44
Limit is What the Function Output is Going to Be
4:15
Limit Notation
4:33
Exploring Limits - 'Ordinary' Function
5:26
Test Out
5:27
Graphing, We See The Answer Is What We Would Expect
5:44
Exploring Limits - Piecewise Function
6:45
If We Modify the Function a Bit
6:49
Exploring Limits - A Visual Conception
10:08
Definition of a Limit
12:07
If f(x) Becomes Arbitrarily Close to Some Number L as x Approaches Some Number c, Then the Limit of f(x) As a Approaches c is L.
12:09
We Are Not Concerned with f(x) at x=c
12:49
We Are Considering x Approaching From All Directions, Not Just One Side
13:10
Limits Do Not Always Exist
15:47
Finding Limits
19:49
Graphs
19:52
Tables
21:48
Precise Methods
24:53
Example 1
26:06
Example 2
27:39
Example 3
30:51
Example 4
33:11
Example 5
37:07
Formal Definition of a Limit

57m 11s

Intro
0:00
Introduction
0:06
New Greek Letters
2:42
Delta
3:14
Epsilon
3:46
Sometimes Called the Epsilon-Delta Definition of a Limit
3:56
Formal Definition of a Limit
4:22
What does it MEAN!?!?
5:00
The Groundwork
5:38
Set Up the Limit
5:39
The Function is Defined Over Some Portion of the Reals
5:58
The Horizontal Location is the Value the Limit Will Approach
6:28
The Vertical Location L is Where the Limit Goes To
7:00
The Epsilon-Delta Part
7:26
The Hard Part is the Second Part of the Definition
7:30
Second Half of Definition
10:04
Restrictions on the Allowed x Values
10:28
The Epsilon-Delta Part, cont.
13:34
Sherlock Holmes and Dr. Watson
15:08
The Adventure of the Delta-Epsilon Limit
15:16
Setting
15:18
We Begin By Setting Up the Game As Follows
15:52
The Adventure of the Delta-Epsilon, cont.
17:24
This Game is About Limits
17:46
What If I Try Larger?
19:39
Technically, You Haven't Proven the Limit
20:53
Here is the Method
21:18
What We Should Concern Ourselves With
22:20
Investigate the Left Sides of the Expressions
25:24
We Can Create the Following Inequalities
28:08
Finally…
28:50
Nothing Like a Good Proof to Develop the Appetite
30:42
Example 1
31:02
Example 1, cont.
36:26
Example 2
41:46
Example 2, cont.
47:50
Finding Limits

32m 40s

Intro
0:00
Introduction
0:08
Method - 'Normal' Functions
2:04
The Easiest Limits to Find
2:06
It Does Not 'Break'
2:18
It Is Not Piecewise
2:26
Method - 'Normal' Functions, Example
3:38
Method - 'Normal' Functions, cont.
4:54
The Functions We're Used to Working With Go Where We Expect Them To Go
5:22
A Limit is About Figuring Out Where a Function is 'Headed'
5:42
Method - Canceling Factors
7:18
One Weird Thing That Often Happens is Dividing By 0
7:26
Method - Canceling Factors, cont.
8:16
Notice That The Two Functions Are Identical With the Exception of x=0
8:20
Method - Canceling Factors, cont.
10:00
Example
10:52
Method - Rationalization
12:04
Rationalizing a Portion of Some Fraction
12:05
Conjugate
12:26
Method - Rationalization, cont.
13:14
Example
13:50
Method - Piecewise
16:28
The Limits of Piecewise Functions
16:30
Example 1
17:42
Example 2
18:44
Example 3
20:20
Example 4
22:24
Example 5
24:24
Example 6
27:12
Continuity & One-Sided Limits

32m 43s

Intro
0:00
Introduction
0:06
Motivating Example
0:56
Continuity - Idea
2:14
Continuous Function
2:18
All Parts of Function Are Connected
2:28
Function's Graph Can Be Drawn Without Lifting Pencil
2:36
There Are No Breaks or Holes in Graph
2:56
Continuity - Idea, cont.
3:38
We Can Interpret the Break in the Continuity of f(x) as an Issue With the Function 'Jumping'
3:52
Continuity - Definition
5:16
A Break in Continuity is Caused By the Limit Not Matching Up With What the Function Does
5:18
Discontinuous
6:02
Discontinuity
6:10
Continuity and 'Normal' Functions
6:48
Return of the Motivating Example
8:14
One-Sided Limit
8:48
One-Sided Limit - Definition
9:16
Only Considers One Side
9:20
Be Careful to Keep Track of Which Symbol Goes With Which Side
10:06
One-Sided Limit - Example
10:50
There Does Not Necessarily Need to Be a Connection Between Left or Right Side Limits
11:16
Normal Limits and One-Sided Limits
12:08
Limits of Piecewise Functions
14:12
'Breakover' Points
14:22
We Find the Limit of a Piecewise Function By Checking If the Left and Right Side Limits Agree With Each Other
15:34
Example 1
16:40
Example 2
18:54
Example 3
22:00
Example 4
26:36
Limits at Infinity & Limits of Sequences

32m 49s

Intro
0:00
Introduction
0:06
Definition: Limit of a Function at Infinity
1:44
A Limit at Infinity Works Very Similarly to How a Normal Limit Works
2:38
Evaluating Limits at Infinity
4:08
Rational Functions
4:17
Examples
4:30
For a Rational Function, the Question Boils Down to Comparing the Long Term Growth Rates of the Numerator and Denominator
5:22
There are Three Possibilities
6:36
Evaluating Limits at Infinity, cont.
8:08
Does the Function Grow Without Bound? Will It 'Settle Down' Over Time?
10:06
Two Good Ways to Think About This
10:26
Limit of a Sequence
12:20
What Value Does the Sequence Tend to Do in the Long-Run?
12:41
The Limit of a Sequence is Very Similar to the Limit of a Function at Infinity
12:52
Numerical Evaluation
14:16
Numerically: Plug in Numbers and See What Comes Out
14:24
Example 1
16:42
Example 2
21:00
Example 3
22:08
Example 4
26:14
Example 5
28:10
Example 6
31:06
Instantaneous Slope & Tangents (Derivatives)

51m 13s

Intro
0:00
Introduction
0:08
The Derivative of a Function Gives Us a Way to Talk About 'How Fast' the Function If Changing
0:16
Instantaneous Slop
0:22
Instantaneous Rate of Change
0:28
Slope
1:24
The Vertical Change Divided by the Horizontal
1:40
Idea of Instantaneous Slope
2:10
What If We Wanted to Apply the Idea of Slope to a Non-Line?
2:14
Tangent to a Circle
3:52
What is the Tangent Line for a Circle?
4:42
Tangent to a Curve
5:20
Towards a Derivative - Average Slope
6:36
Towards a Derivative - Average Slope, cont.
8:20
An Approximation
11:24
Towards a Derivative - General Form
13:18
Towards a Derivative - General Form, cont.
16:46
An h Grows Smaller, Our Slope Approximation Becomes Better
18:44
Towards a Derivative - Limits!
20:04
Towards a Derivative - Limits!, cont.
22:08
We Want to Show the Slope at x=1
22:34
Towards a Derivative - Checking Our Slope
23:12
Definition of the Derivative
23:54
Derivative: A Way to Find the Instantaneous Slope of a Function at Any Point
23:58
Differentiation
24:54
Notation for the Derivative
25:58
The Derivative is a Very Important Idea In Calculus
26:04
The Important Idea
27:34
Why Did We Learn the Formal Definition to Find a Derivative?
28:18
Example 1
30:50
Example 2
36:06
Example 3
40:24
The Power Rule
44:16
Makes It Easier to Find the Derivative of a Function
44:24
Examples
45:04
n Is Any Constant Number
45:46
Example 4
46:26
Area Under a Curve (Integrals)

45m 26s

Intro
0:00
Introduction
0:06
Integral
0:12
Idea of Area Under a Curve
1:18
Approximation by Rectangles
2:12
The Easiest Way to Find Area is With a Rectangle
2:18
Various Methods for Choosing Rectangles
4:30
Rectangle Method - Left-Most Point
5:12
The Left-Most Point
5:16
Rectangle Method - Right-Most Point
5:58
The Right-Most Point
6:00
Rectangle Method - Mid-Point
6:42
Horizontal Mid-Point
6:48
Rectangle Method - Maximum (Upper Sum)
7:34
Maximum Height
7:40
Rectangle Method - Minimum
8:54
Minimum Height
9:02
Evaluating the Area Approximation
10:08
Split the Interval Into n Sub-Intervals
10:30
More Rectangles, Better Approximation
12:14
The More We Us , the Better Our Approximation Becomes
12:16
Our Approximation Becomes More Accurate as the Number of Rectangles n Goes Off to Infinity
12:44
Finding Area with a Limit
13:08
If This Limit Exists, It Is Called the Integral From a to b
14:08
The Process of Finding Integrals is Called Integration
14:22
The Big Reveal
14:40
The Integral is Based on the Antiderivative
14:46
The Big Reveal - Wait, Why?
16:28
The Rate of Change for the Area is Based on the Height of the Function
16:50
Height is the Derivative of Area, So Area is Based on the Antiderivative of Height
17:50
Example 1
19:06
Example 2
22:48
Example 3
29:06
Example 3, cont.
35:14
Example 4
40:14
XVII. Appendix: Graphing Calculators
Buying a Graphing Calculator

10m 41s

Intro
0:00
Should You Buy?
0:06
Should I Get a Graphing Utility?
0:20
Free Graphing Utilities - Web Based
0:38
Personal Favorite: Desmos
0:58
Free Graphing Utilities - Offline Programs
1:18
GeoGebra
1:31
Microsoft Mathematics
1:50
Grapher
2:18
Other Graphing Utilities - Tablet/Phone
2:48
Should You Buy a Graphing Calculator?
3:22
The Only Real Downside
4:10
Deciding on Buying
4:20
If You Plan on Continuing in Math and/or Science
4:26
If Money is Not Particularly Tight for You
4:32
If You Don't Plan to Continue in Math and Science
5:02
If You Do Plan to Continue and Money Is Tight
5:28
Which to Buy
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
Do Your Research
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
Tips for Purchasing
9:17
Buy Online
9:19
Buy Used
9:35
Ask Around
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
Read the Manual
0:06
Skim It
0:20
Play Around and Experiment
0:34
Syntax
0:40
Definition of Syntax in English and Math
0:46
Pay Careful Attention to Your Syntax When Working With a Calculator
2:08
Make Sure You Use Parentheses to Indicate the Proper Order of Operations
2:16
Think About the Results
3:54
Settings
4:58
You'll Almost Never Need to Change the Settings on Your Calculator
5:00
Tell Calculator In Settings Whether the Angles Are In Radians or Degrees
5:26
Graphing Mode
6:32
Error Messages
7:10
Don't Panic
7:11
Internet Search
7:32
So Many Things
8:14
More Powerful Than You Realize
8:18
Other Things Your Graphing Calculator Can Do
8:24
Playing Around
9:16
Graphing Functions, Window Settings, & Table of Values

10m 38s

Intro
0:00
Graphing Functions
0:18
Graphing Calculator Expects the Variable to Be x
0:28
Syntax
0:58
The Syntax We Choose Will Affect How the Function Graphs
1:00
Use Parentheses
1:26
The Viewing Window
2:00
One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
2:01
For Example
2:30
The Viewing Window, cont.
2:36
Window Settings
3:24
Manually Choose Window Settings
4:20
x Min
4:40
x Max
4:42
y Min
4:44
y Max
4:46
Changing the x Scale or y Scale
5:08
Window Settings, cont.
5:44
Table of Values
7:38
Allows You to Quickly Churn Out Values for Various Inputs
7:42
For example
7:44
Changing the Independent Variable From 'Automatic' to 'Ask'
8:50
Finding Points of Interest

9m 45s

Intro
0:00
Points of Interest
0:06
Interesting Points on the Graph
0:11
Roots/Zeros (Zero)
0:18
Relative Minimums (Min)
0:26
Relative Maximums (Max)
0:32
Intersections (Intersection)
0:38
Finding Points of Interest - Process
1:48
Graph the Function
1:49
Adjust Viewing Window
2:12
Choose Point of Interest Type
2:54
Identify Where Search Should Occur
3:04
Give a Guess
3:36
Get Result
4:06
Advanced Technique: Arbitrary Solving
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
Advanced Technique: Calculus
7:18
Derivative
7:22
Integral
7:30
But How Do You Show Work?
8:20
Parametric & Polar Graphs

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (2)

1 answer

Last reply by: Professor Selhorst-Jones
Mon Sep 30, 2013 9:05 AM

Post by Mike Olyer on September 29, 2013

Why is 1 not a vertical asymptote for the function in example one? Why does canceling out allow this?

Rational Functions and Vertical Asymptotes

  • A rational function is the quotient of two polynomials:
    f(x) =N(x)

    D(x)


     
    ,
    where N(x) and D(x) are polynomials and D(x) ≠ 0.
  • Since a rational function f(x) = [N(x)/D(x)] is inherently built out of the operation of division, we must watch out for the possibility of dividing by zero. The domain of a rational function is all real numbers except the zeros of D(x).
  • As the denominator of a rational function goes to 0 (and assuming the numerator is not also 0), the fraction becomes very large. While it can't actually divide by 0, as it gets extremely close to 0, the function "blows out" to very large values. We call this location a vertical asymptote. A vertical asymptote is a vertical line x = a where as x gets close to a, |f(x)| becomes arbitrarily large. Symbolically, we show this as
    x → a     ⇒     f(x) → ∞   or   f(x) → − ∞.
  • On a graph, we show the location of a vertical asymptote with a dashed line. This aids us in drawing the graph and in understanding the graph later.
  • To find the vertical asymptotes of a rational function, we need to find the x-values where the denominator becomes 0 (the roots of the denominator function). However, not all of these zeros will give asymptotes. It's possible for the numerator to go to 0 at the same time, which will cause the function to just have a hole at that x-value, but not "blow out" to infinity.
  • We can find the vertical asymptotes of a rational function by following these steps:

      1. Begin by figuring out what x-values are not in the domain of f: these are all the zeros of D(x).
      2. Determine if N(x) and D(x) share any common factors. If so, cancel out those factors. [Alternately, this step can be done by checking that the zeros to D(x) are not also zeros to N(x).]
      3. After canceling out common factors, determine any zeros (roots) left in the denominator. These are the vertical asymptotes.
      (4.) If graphing the function, figure out which way the graph goes (+∞ or −∞) on either side of each asymptote. Do this by using test values very close to the asymptote. For example, if the asymptote is at x=2, look at f(1.99) and f(2.01). [You can also do this in your head by thinking in terms of positive vs. negative, which we discuss in the Examples.]

Rational Functions and Vertical Asymptotes

Find the domain of the function below, then find any vertical asymptote(s).
f(x) = 7

x−5
  • The domain of a function is the set of all values that the function can accept. The only thing that could "break" a function like this would be dividing by 0. That means we need to figure out where the denominator can equal 0.
  • Set the denominator equal to 0:
    x−5=0
    We see that the only place where the function "breaks" (is not defined) is at x=5. Thus, x=5 is the only x-value not allowed. Therefore any x value other than 5 is in the domain.
    Domain:    x ≠ 5
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative). This happens when the numerator is non-zero and the denominator is getting very close to 0.
  • To figure out where this occurs, begin by seeing if there are any common factors that can be canceled in the numerator and denominator:
    7

    x−5
    The above cannot be simplified any more, so there are no common factors. [This means we don't have to worry about the numerator also being 0 when the denominator goes to 0.]
  • Once any common factors have been canceled (or there aren't any, such as in this problem), find what zeros (roots) are left in the denominator:
    7

    x−5
    The above denominator still only has a zero at x=5, as we figured out before. Thus, x=5 is a vertical asymptote.
Domain:    x ≠ 5;        Vertical asymptote at x=5
Find the domain of the function below, then find any vertical asymptote(s).
g(x) = 17x

3x+9
  • The domain of a function is the set of all values that the function can accept. The only thing that could "break" a function like this would be dividing by 0. That means we need to figure out where the denominator can equal 0.
  • Set the denominator equal to 0:
    3x+9=0
    We see that the only place where the function "breaks" (is not defined) is at x=−3. Thus, x=−3 is the only x-value not allowed. Therefore any x value other than −3 is in the domain.
    Domain:    x ≠ −3
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative). This happens when the numerator is non-zero and the denominator is getting very close to 0.
  • To figure out where this occurs, begin by seeing if there are any common factors that can be canceled in the numerator and denominator:
    17x

    3x+9
    The above cannot be simplified any more, so there are no common factors. [This means we don't have to worry about the numerator also being 0 when the denominator goes to 0.]
  • Once any common factors have been canceled (or there aren't any, such as in this problem), find what zeros (roots) are left in the denominator:
    17x

    3x+9
    The above denominator still only has a zero at x=−3, as we figured out before. Thus, x=−3 is a vertical asymptote.
Domain:    x ≠ −3;        Vertical asymptote at x=−3
Find the domain of the function below, then find any vertical asymptote(s).
h(x) = 3

x2−2x−24
  • The domain of a function is the set of all values that the function can accept. The only thing that could "break" a function like this would be dividing by 0. That means we need to figure out where the denominator can equal 0.
  • Set the denominator equal to 0:
    x2−2x−24=0
    We can find the solutions to this equation by factoring.
  • (x+4)(x−6)=0, so the function "breaks" (is not defined) at x=−4 and x=6. These are the only two x-values not allowed. Therefore any x value other than them is in the domain.
    Domain:    x ≠ −4,  6
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative). This happens when the numerator is non-zero and the denominator is getting very close to 0.
  • To figure out where this occurs, begin by seeing if there are any common factors that can be canceled in the numerator and denominator:
    3

    x2−2x−24
    = 3

    (x+4)(x−6)
    We see there are no common factors, so we can continue. [This means we don't have to worry about the numerator also being 0 when the denominator goes to 0.]
  • Once any common factors have been canceled (or there aren't any, such as in this problem), find what zeros (roots) are left in the denominator:
    3

    (x+4)(x−6)
    The above denominator still has zeros at x=−4,  6, as we figured out before. Thus, x=−4 and x=6 are the vertical asymptotes.
Domain:    x ≠ −4,  6;        Vertical asymptotes at x=−4 and x=6
Find the domain of the function below, then find any vertical asymptote(s).
v(x) = 2x+14

x3+5x2−14x
  • The domain of a function is the set of all values that the function can accept. The only thing that could "break" a function like this would be dividing by 0. That means we need to figure out where the denominator can equal 0.
  • Set the denominator equal to 0:
    x3+5x2−14x=0
    We can find the solutions to this equation by factoring.
  • x(x−2)(x+7)=0, so the function "breaks" (is not defined) at x=0, x=2, and x=−7. These are all the x-values not allowed. Therefore any x value other than them is in the domain.
    Domain:    x ≠ −7,  0,   2
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative). This happens when the numerator is non-zero and the denominator is getting very close to 0.
  • To figure out where this occurs, begin by seeing if there are any common factors that can be canceled in the numerator and denominator:
    2x+14

    x3+5x2−14x
    = 2(x+7)

    x(x−2)(x+7)
    We see that there is a common factor: (x+7), so we cancel this factor out before trying to find the vertical asymptote(s).
    2

    x(x−2)
    [Note that the (x+7) factor still affects the domain, as we figured out before. It just cannot contribute to the creation of vertical asymptotes because both the top and bottom of the fraction produce 0 at the x-location of −7.]
  • Once any common factors have been canceled, find what zeros (roots) are left in the denominator:
    2

    x(x−2)
    The above denominator only has zeros at x=0 and x=2. These are the vertical asymptotes.
Domain:    x ≠ −7,  0,   2;        Vertical asymptotes at x=0 and x=2
From the graph below, identify the location of its vertical asymptote.
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative).
  • Look on the graph for a horizontal location where the graph shoots toward ±∞ vertically.
  • We see that happen at x=2, so that is the vertical asymptote.
x=2
From the graph below, identify the location of its vertical asymptotes.
  • A vertical asymptote is some specific x-value that, as x approaches it, the function grows very large (positive or negative).
  • Look on the graph for the horizontal locations where the graph shoots toward ±∞ vertically.
  • We see that happen at x=−4 and x=6, so those are the vertical asymptotes.
x=−4 and x=6
Draw the graph of the rational function below.
f(x) = 1

x+3
  • Just like we did in the previous questions, begin by identifying the function's domain and the vertical asymptotes. They'll help when we need to draw the graph.
  • Because the fraction is as simplified as possible, the "holes" in the domain are the same as the location of the vertical asymptote:
    x+3 = 0
    The function has a vertical asymptote of x=−3. [It also has a domain of x ≠ −3, but that won't really affect the graph we draw because we already have a vertical asymptote there. This is because the vertical asymptote can't have a point on it (otherwise the function could not have flown off to ±∞ on either side).]
  • Draw graph axes and draw a dashed vertical line at x=−3 to indicate the asymptote is there.
  • At this point, it would help to have more points to get a sense of how the function curves. Make a table of values to help plot some points. Notice that the most "interesting" thing for this function is the asymptote at x=−3 so we want to focus our table around that.
    x
    f(x)
    −9
    −0.166
    −5
    −0.5
    −4
    −1
    −3.5
    −2
    −2.5
    2
    −2
    1
    −1
    0.5
    3
    0.166
    Plot these points on the graph. [As you become more comfortable with graphing functions like this, you will need fewer points because you'll have a better idea of the shape of the function. For now though, plot however many you need to be comfortable with graphing it.]
  • As x → −3, the graph will fly off to ±∞. We need to figure out which direction it will fly off, though. From our table of points, it seems that on the left side, it flies off to − ∞, while on the right side it goes to + ∞. We might want to be sure, though, so we could also figure out where f(−3.01) and f(−2.99) would go. Alternatively, we could also just think about whether or not the function will go up or down based on if the x-value is slightly more or less than x=−3. Once you know which direction it goes, draw a curve that approaches the dashed line, but does not actually cross it. As it gets closer, it gets lower/higher.
  • Finally, think about what happens to the function as x goes very far to the right or left. Because the numerator never changes, a very large x (+ or −) will cause the function output to become very small. Thus, as x goes very far to the right or left, it will get pulled to a very small height. Combine all of these ideas, and draw in all the curves based off your plotted points.
Draw the graph of the rational function below.
g(x) = 15

24−6x
  • Just like we did in the previous questions, begin by identifying the function's domain and the vertical asymptotes. They'll help when we need to draw the graph.
  • First, figure out the function's domain by setting the denominator to 0:
    24−6x = 0
    The function will "break" at x=4, so the domain is x ≠ 4.
  • Next find the vertical asymptotes. Begin by canceling any common terms on the top and bottom:
    15

    24−6x
        =     3·5

    3(8−2x)
        =     5

    8−2x
    Using the new, simplified version, find the location of any vertical asymptotes by setting the denominator to 0:
    8 − 2x = 0
    The function has a vertical asymptote at x=4. [Notice that this is basically what we figured out when we were looking for the domain. While we canceled out a common factor, nothing changed because the common factor was just a number. Still, we want to make sure to check again after canceling factors because it can change what we find for the asymptotes.]
  • Draw graph axes and draw a dashed vertical line at x=4 to indicate the asymptote is there.
  • At this point, it would help to have more points to get a sense of how the function curves. Make a table of values to help plot some points. Notice that the most "interesting" thing for this function is the asymptote at x=4 so we want to focus our table around that.
    x
    f(x)
    −1
    0.5
    2
    1.25
    3
    2.5
    3.5
    5
    4.5
    −5
    5
    −2.5
    6
    −1.25
    9
    −0.5
    Plot these points on the graph. [As you become more comfortable with graphing functions like this, you will need fewer points because you'll have a better idea of the shape of the function. For now though, plot however many you need to be comfortable with graphing it.]
  • As x → 4, the graph will fly off to ±∞. We need to figure out which direction it will fly off, though. From our table of points, it seems that on the left side, it flies off to + ∞, while on the right side it goes to − ∞. We might want to be sure, though, so we could also figure out where f(3.99) and f(4.01) would go. Alternatively, we could also just think about whether or not the function will go up or down based on if the x-value is slightly more or less than x=4. Once you know which direction it goes, draw a curve that approaches the dashed line, but does not actually cross it. As it gets closer, it gets lower/higher.
  • Finally, think about what happens to the function as x goes very far to the right or left. Because the numerator never changes, a very large x (+ or −) will cause the function output to become very small. Thus, as x goes very far to the right or left, it will get pulled to a very small height. Combine all of these ideas, and draw in all the curves based off your plotted points.
Draw the graph of the rational function below.
h(x) =−2x

x3−x2−6x
  • Just like we did in the previous questions, begin by identifying the function's domain and the vertical asymptotes. They'll help when we need to draw the graph.
  • First, figure out the function's domain by setting the denominator to 0:
    x3−x2−6x = 0
    We solve by factoring and get
    x(x+2)(x−3) = 0,
    thus it has "holes" at x=−2,  0,   3. Therefore its domain is x ≠ −2,  0,  3.
  • Next find the vertical asymptotes. Begin by canceling any common terms on the top and bottom:
    −2x

    x3−x2−6x
        =    −2x

    x(x+2)(x−3)
        =    −2

    (x+2)(x−3)
    Using the new, simplified version, find the location of any vertical asymptotes by setting the denominator to 0:
    (x+2)(x−3) = 0
    The function has vertical asymptotes at x=−2,   3. [Notice that this is different from the holes in our domain. At the very end, we'll deal with the fact that there is a hole in the domain at x=0 that doesn't show up in our asymptotes.]
  • Draw graph axes and draw dashed vertical lines at both x=−2 and x=3 to indicate the asymptotes.
  • At this point, it would help to have more points to get a sense of how the function curves. Make a table of values to help plot some points. Notice that the most "interesting" things for this function are the asymptotes at x=−2,  3 so we want to focus our table around them. [Also, we can use the simplified version of the function ([(−2)/((x+2)(x−3))]) to make it easier to compute points. With one exception, it works exactly the same. That exception is at x=0. From earlier, we found that the function does not exist there. Still, we can plot the point in the simplified version to know the best place to put the "hole" and to help us draw the function.]
    x
    f(x)
    −7
    −0.04
    −4
    −0.14
    −3
    −0.33
    −2.5
    −0.73
    −1.5
    0.88
    −1
    0.5
    0
    0.33 / DNE
    1
    0.33
    2
    0.5
    2.5
    0.88
    3.5
    −0.73
    4
    −0.33
    5
    −0.14
    8
    −0.04
    Plot these points on the graph. [As you become more comfortable with graphing functions like this, you will need fewer points because you'll have a better idea of the shape of the function. For now though, plot however many you need to be comfortable with graphing it.]
  • As x → −2 and x→ 3, the graph will fly off to ±∞. We need to figure out which direction it will fly off, though. From our table of points, it seems that to the left of x=−2 it goes to −∞, then on the right of x=−2 it goes to +∞. For x=3, on the left it goes to +∞, while on the right it goes to −∞. We might want to be sure, though, so we could also figure out where f(−2.01), f(−1.99), f(2.99), and f(3.01) would go. Alternatively, we could also just think about whether or not the function will go up or down based on if the x-value is slightly more or less at each of those asymptote locations. Once you know which direction it goes, draw a curve that approaches the dashed line, but does not actually cross it. As it gets closer, it gets lower/higher.
  • To help plot far to the right and left, think about what happens to the function as x goes very far to the right or left. Because the numerator grows so much less than the denominator, a very large x (+ or −) will cause the function output to become very small. Thus, as x goes very far to the right or left, it will get pulled to a very small height.
  • And there's one last thing to do: remember the "hole" in the graph. From the very beginning, we know that the function does not exist at x=0. We indicate by drawing a little empty circle at where the function "would have been" if it had existed.
Explain why f(x) = [(x+7)/(x+7)] does not have a vertical asymptote.
  • To have a vertical asymptote, there needs to be some x value that causes the denominator to become very, very small as the numerator remains comparatively large. In other words, where the denominator goes to 0, the numerator cannot also go to 0.
  • For this function, the only x-value that will make the denominator go to 0 is at x=−7. However, that location will also cause the numerator to go to 0. Therefore this function can not have a vertical asymptote.
  • An alternative way to look at this function is by canceling common factors. For the most part,
    x+7

    x+7
    = 1,
    so f(x) is just a constant function. A constant function can't fly off to ±∞, like a vertical asymptote requires, so f(x) has no vertical asymptote. The only place where f(x) is different than the simplified version is at x=−7 (because that breaks f, [0/0]), but that doesn't affect whether or not it has a vertical asymptote.
The only place where the denominator approaches 0 is at x=−7, but that causes the numerator to also approach 0, so no vertical asymptote is possible.

*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

Rational Functions and Vertical Asymptotes

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
  • Introduction 0:05
  • Definition of a Rational Function 1:20
    • Examples of Rational Functions
    • Why They are Called 'Rational'
  • Domain of a Rational Function 3:15
    • Undefined at Denominator Zeros
    • Otherwise all Reals
  • Investigating a Fundamental Function 4:50
    • The Domain of the Function
    • What Occurs at the Zeroes of the Denominator
  • Idea of a Vertical Asymptote 6:23
  • What's Going On? 6:58
    • Approaching x=0 from the left
    • Approaching x=0 from the right
    • Dividing by Very Small Numbers Results in Very Large Numbers
  • Definition of a Vertical Asymptote 10:05
  • Vertical Asymptotes and Graphs 11:15
    • Drawing Asymptotes by Using a Dashed Line
    • The Graph Can Never Touch Its Undefined Point
  • Not All Zeros Give Asymptotes 13:02
    • Special Cases: When Numerator and Denominator Go to Zero at the Same Time
    • Cancel out Common Factors
  • How to Find Vertical Asymptotes 16:10
    • Figure out What Values Are Not in the Domain of x
    • Determine if the Numerator and Denominator Share Common Factors and Cancel
    • Find Denominator Roots
    • Note if Asymptote Approaches Negative or Positive Infinity
  • Example 1 18:57
  • Example 2 21:26
  • Example 3 23:04
  • Example 4 30:01

Transcription: Rational Functions and Vertical Asymptotes

Hi--welcome back to Educator.com.0000

Today, we are going to talk about rational functions and vertical asymptotes.0002

The next few lessons are going to be about rational functions and asymptotes; and they are pronounced aa-sim-tohts.0006

They are spelled kind of funny--asymptotes--but we want to know how to pronounce them; they are pronounced aa-sim-tohts, asymptotes.0012

In exploring these ideas, we will see some very interesting behavior; and we will learn why it occurs.0020

But before you start these lessons, it is important that you have a reasonable understanding of polynomials.0024

Polynomials are going to be central to this, as we are about to see in the definition of a rational function.0028

You will need to know their basics; you will need to know how to factor them to find zeroes and roots.0032

And you will also need to know polynomial division.0036

If you haven't watched the previous section of lessons on polynomials, you might find it helpful to watch0038

the three lessons Introduction to Polynomials, Roots/Zeroes of Polynomials, and Polynomial Division;0042

or just the ones that you need specific help with, because those ideas are going to definitely come up as we explore this stuff.0048

Also, it should be mentioned that we would be working only with the real numbers; so we are back to working just with the reals.0054

And while we briefly worked with the complex numbers in the last couple of lessons, it is back to the reals.0059

We are not going to see anything other than real numbers anymore.0063

If we would have solutions in the complex, too bad; we are not going to really care about them right now.0067

We are back to focusing just on the reals; they are still pretty interesting--there is lots of stuff to explore in the reals.0071

You can leave complex numbers to a future, later course in math.0076

All right, first let's define a rational function: a rational function is the quotient of two polynomials.0079

So, f(x) is our rational function; it is equal to n(x) divided by d(x), where n(x) and d(x) are both polynomials, and d(x) is not equal to 0.0086

And by "not equal to 0," I mean that d(x) isn't 0, like straight "this is zero, all the time, forever."0097

So, d(x) = 0, the constant function, just 0, all the time, forever--that is not allowed.0104

But d(x) can have roots; we can say that d(x) can have roots.0115

For example, d(x) could be x2 - 1, where d would be 0 at positive 1 and -1; that is OK.0120

d(x) can have roots, but it is not going to be the constant function of 0 all the time.0130

That is what is not allowed, because then our function would be broken, completely, everywhere.0137

But we are allowed to have slight breaks occasionally, when we end up having roots appear in the denominator.0141

So, it is just a normal polynomial, but not just 0; that would be bad.0147

All right, some examples: g(x) = (2x + 1)/(x2 - 4) or 1/x3 or (x4 - 3x2)/(x + 2).0152

In all of these cases, we have a polynomial divided by a polynomial--as simple as that.0161

You might wonder why they are called rational functions--what is so rational about them?0166

Remember: many lessons back, when we have talked about the idea of sets, we called fractions made up of integers,0170

things like 3/5 or -47/2, the rational numbers, because they seem to be made in a fairly rational, sensible way.0175

Thus, we are using a similar name, because the rational functions are built similarly.0184

Rational numbers are built out of division, and rational functions are built out of division.0188

So, we are using a similar name; cool.0193

Since a rational function f(x) = n(x)/d(x) is inherently built out of the operation of division, we have to watch out for dividing by 0.0197

That is going to be the Achilles' heel of rational functions; and also, it is going to be what makes them so interesting to look at.0205

Dividing by 0 is not defined--it is never defined; so the zeroes, the roots, the places where it becomes 0,0211

of our denominator polynomial, d(x), will break the rational function.0218

The zeroes of d(x) are not in the domain of our function; so a rational function, right here--0222

wherever it ends up being 0, it is not defined there; that is not in the domain.0231

It is not in the set of numbers that we are allowed to use, the domain, because if we plug in a number0236

that makes us divide by 0, we don't know what to do; we just blew up the world, so it is no good.0241

So, we are not going to be allowed to plug in numbers--we are not going to have those in our domain--0246

when d(x) = 0, when we are looking at the zeroes, the roots, of our denominator polynomial.0250

Now, other than these zeroes in the denominator, a rational function is defined everywhere else,0256

because polynomials have all of the real numbers as their domain.0262

Since you can plug any real number into a polynomial, and get something out of it,0265

the only place we will have any issues is where we are accidentally dividing by 0.0269

So, everything other than these locations where we divide by 0--they are all good.0272

The domain of a rational function is all real numbers, except the zeroes of d(x)--all real numbers, with the exceptions of these zeroes in our denominator polynomial.0276

Everything will be allowed, with a very few exceptions for that denominator polynomial's zeroes.0285

All right, to help us understand rational functions better, let's consider this fundamental rational function, 1/x.0290

So, immediately we see that f is not going to be defined at x = 0.0299

So, our domain for this will be everything except x = 0; domain is everything where x is not 0--so most numbers.0303

We see that we can't divide by 0; but everything else would give us an actual thing.0316

So, notice how it behaves near x = 0; it isn't actually going to be defined at x = 0--we will never see that.0320

But as it approaches it, f(x) grows very, very large--see how it is shooting up?0328

We have it going up and down--it is very, very large.0333

What is going on here? Let's look at a different viewing window to get a sense for just how large f(x) manages to become.0338

If we look at a very small x width, only going from -0.5 to +0.5, we manage to see a very large amount that we move vertically.0344

We can go all the way...as we come from the left, we manage to go from near 0 all the way to -100 on this window.0353

And we can go from near 0 all the way up to positive 100 on this window.0363

And in fact, it will keep going; it just keeps shooting up.0368

It shoots all the way out to...well, not out to infinity, because we can't actually hit infinity...but the idea of shooting out to something arbitrarily large.0371

It is going to get as big as we want to talk about it getting out to; so in a way, we say it goes out to infinity.0378

We see this behavior of going out to infinity in many rational functions.0384

If we look at 1/(x + 2), we get it going out to infinity at -2.0389

Notice that, when we plug in -2 into the denominator, we have a 0 showing up; so we see this behavior around zeroes.0395

If we look at 5/(x2 - 9), we see it at -3 and +3.0401

We have these places where it goes out to infinity.0406

In a few moments, we will formally define this behavior, and we will name it a vertical asymptote.0410

But first, let's understand why this strange behavior is occurring.0415

To understand that, let's look, once again, at f(x) = 1/x, our fundamental rational function, with what makes the most basic form of it.0419

Notice that, while x = 0 is not defined (that would blow up the world, because we are dividing by 0--we are not allowed to do that--0428

it just doesn't make sense), everywhere else is going to be defined.0436

So, let's see what happens as x goes to 0: this little arrow--we say that x is going to 0.0441

We are looking as x gets close to 0--not actually at 0; but as x sort of marches towards 0.0446

Let's look at what happens as we go from the negative side.0452

As we plug in -2, we get -0.5, 1/-2; if we plug in -1, we get -1.0455

But as the x becomes very, very small, we see f(x) become very, very large.0462

-0.5 gets us -2; -0.1 gets us -10; -0.01 gets us -100; -0.001 gets us -1000.0467

As we divide a number by a very small number, 1 divided by 1/10 is equal to positive 10, because the fraction will flip up.0477

We are going to end up having to look at its reciprocal, since it is in the denominator again.0489

We are going to see this behavior: as we get really, really small--as we get to these really small numbers that we are dividing by--0493

we will end up having the whole thing become very, very large.0500

We can see this on the positive side, as well, if we go in from the positive side.0504

The negative side is us starting from the negative side and us moving towards the 0.0508

And the positive one is us starting from the positive side, and us moving towards the 0.0514

It is moving from the right, into the left, versus moving from the left into the right.0519

Positive: we look at 2; we get one-half, 1/2; we plug in 1; we get 1/1, so 1.0524

As we get smaller and smaller, we see it begin to get larger and larger.0529

We plug in 0.5, and we will get 2; we plug in 0.1, and we will get 10; we plug in 0.01, and we will get 100; and so on and so forth.0532

As we get in smaller and smaller and smaller numbers, we are going to get larger and larger things.0540

1/(1 divided by 1 million) is a very small number, but it is one of the things that is defined, because it is not x = 0; it is just very close to x = 0.0545

It will cause it to flip up to the top, and we will get 1 million.0558

We will get to any size number that we want.0562

We can't actually divide by 0; but dividing by very small numbers gives large results.0565

This is what is happening: as we get near 0, we go out to infinity, because we are dividing by very, very small numbers.0576

And when you divide by something very, very small, you get a very big thing.0583

How many times does something very, very small go into a reasonable-sized number?0587

It goes in many times; the smaller the chunk that we are trying to see how many times it fits in, the more times it is going to be able to fit in.0592

So, the smaller the thing that we are dividing by, the larger the number we will get out in response.0598

And that is what is going to cause this behavior of "blowing out."0603

With this idea in mind, we will define a vertical asymptote.0607

A vertical asymptote is a vertical line x = a, where as x gets close to a, the size of f,0610

the absolute value of f(x), becomes arbitrarily large--becomes very big.0619

Symbolically, we show this as x → a will cause f(x) to go to infinity or f(x) to go to negative infinity.0624

Informally, we can see a vertical asymptote as a horizontal location, a, where the function blows out to infinity,0633

either positive infinity or negative infinity, as it gets near a.0641

So, it is going to be some location a that, as we get close to it, we go out to infinity.0645

We manage to blow out to infinity as we get close to it; we get very, very large numbers.0656

So, a vertical asymptote is some horizontal location, some vertical line defined by x = a,0661

where, as we get close to it, we blow out to infinity.0668

Maybe we blow up; maybe we blow down; but we are going to blow out to infinity, one way or the other.0671

Vertical asymptotes and graphs; we show the location of a vertical asymptote with a dashed line, as we have been doing previously.0677

This aids in drawing the graph and in understanding the graph later.0683

So, notice: we just use a dashed line here to show us that this is where the vertical asymptote is.0686

So, for the graph of f(x) = 1/(x - 1), we see, right here at x = +1 (because if we plug in x = +1, we would be dividing by 0 there)--0691

so near that location--that we will be dividing by very small numbers, which will cause us to blow out to infinity around that location.0702

We will see a vertical asymptote there; so as our function approaches that vertical asymptote,0708

it ends up having to get very, very large, because it is now dividing by very, very small numbers.0714

Notice that the graph gets very close to the asymptote; but it won't be able to touch the asymptote.0721

So, it will become arbitrarily close to the line, but it never touches the line.0726

Why can't it touch the line? Because it is not defined there.0731

We have 1 here; so if we plug in x = 1, then we have 1/(1 - 1), which is 1/0.0735

We are not allowed to divide by 0, so we are not allowed to do this, which means we are not allowed to plug in x = 1.0743

Our domain is everything where x is not equal to 1; so we are not defined at that thing.0748

Our function is not actually defined on the asymptote; it will become very large--it will become very close to this asymptote.0754

But it won't actually be allowed to get onto the asymptote, because to be on that point,0761

to be on that horizontal location, would require it to be defined there.0765

And it is not defined there, because that would require us to divide by 0, which is not allowed.0768

So, we are not allowed to do that, which means that we are not actually going to be able to get on it.0773

We will never touch the vertical asymptote, because it is not defined.0777

Now, not all 0's give asymptotes; so far, all of the vertical asymptotes we have seen have happened at the location of a 0 in the denominator.0783

And that is going to be the case: all vertical asymptotes will require there to be a 0 in the denominator.0789

But a 0 in the denominator does not always imply an asymptote.0795

We are not going to always see an asymptote.0801

Consider f(x) = x/x; now, normally, x/x is a very boring function; the x's cancel, and we are left with this effectively being just 1,0803

except at one special place: when we look at x = 0, we are given 0/0.0814

Now, 0/0 is definitely not defined, so we are not allowed to do anything there.0820

We have a pretty good sense that it would be going to 1.0825

But when we actually look at that place, it is not defined, because we are looking at 0/0.0828

0/0 is definitely not defined; so there is going to be a hole in the graph.0834

f is a rational function, x/x, a polynomial divided by a polynomial; they are simple linear polynomials, but they are both polynomials.0839

So, f is a rational function; and it is not defined at x = 0; but it still has no asymptote there,0848

because normally, we just have x/x; it is just really a constant function, 1.0855

We see it here; it is going to be 1 all the time, forever, always, except when we look at x = 0,0861

at which point it blips out of existence, because it is not allowed to actually have anything at 0/0, because that is not defined.0868

So, formally, we can't give it a location; so instead, we have to use this hole.0876

We show that there is a missing place there with this empty circle.0880

As opposed to a filled-in circle to show that there is something there, we use an empty circle to show that we are actually missing a location there.0885

It has a hole at that height there, because it would be going there, but it doesn't actually show up there.0891

There is no asymptote there; why does it happen--why do we lack this asymptote?0896

It is because the numerator and the denominator go to 0 at the same time, so the asymptote never happens.0901

Because they are both going to 0, there is nothing to divide out to blow up to a very large number--to blow down to a very large negative number.0906

The asymptote can't happen, because normally they just cancel out to 1/1...well, not 1/1; they cancel out to something over something.0916

But since it is the same something, it becomes just 1.0923

So, if we have .1 divided by .1, well, they cancel out, and we just get 1.0926

If we have .0001 divided by .0001, well, once again, they cancel out, and we have just 1.0930

So, we don't end up seeing this asymptote behavior, because we have to have something divided by very small numbers.0936

But if we have a very small number divided by a very small number, they end up going at the same rate; and so we don't get this asymptote.0942

To find an asymptote, then, that means we first need to cancel out common factors.0949

We need to be able to say, "Oh, x/x means I am going to cancel these out; and I will get it be equivalent to being just 1."0955

We will have to remember that we are forbidden to actually plug in x = 0; but other than that, it is just going to be 1 all the time, forever.0963

So, with our newfound understanding of all these different things that make up rational functions,0971

we can create a step-by-step guide for finding vertical asymptotes.0975

So, given a rational function in the form n(x)/d(x), where n and d are both polynomials,0978

the first thing we want to do is figure out all of the x-values that we aren't going to be allowed to have in our domain--0984

all of our forbidden x-values that would break our function.0990

These are the zeroes of d(x), because they would cause us to divide by 0.0993

Since dividing by 0 is not allowed, then our domain is everything where we won't divide by 0.0997

So, the zeroes of d(x) are not in our domain.1002

Next, we want to determine if n(x) and d(x) share any common factors.1006

So, to do this, you might have to factor the two polynomials.1010

If they share common factors, cancel out those factors.1013

Alternatively, this step can be done by checking that the zeroes to d(x) are not also zeroes to n(x),1016

because for them to have a common factor, where they would both be going to 0 at the same time,1025

then a common factor means that they have the same root.1029

So, if they are both going to be zeroes at the location, that means we will see this effect.1032

So, we can either cancel out the factors, or we can just and make sure that the roots to d(x) are not also roots to n(x).1036

But the easiest way to find roots in the first place is usually to factor.1043

So, you will probably want to factor, for the most part.1046

But occasionally, it will be easier to just see, "If we plug in this number, does it come out to be 0?"1048

The next step: once you cancel out the common factors, you determine any zeroes that are left in the denominator.1053

These are the vertical asymptotes, because once we have canceled out common factors,1059

we don't have to worry about both the top and the bottom going to 0 at the same time.1063

So, any roots that are left are going to have to make very small numbers to blow everything out to infinity.1067

So, anything that is left after canceling out (it is crucial that it is after you cancel out common factors)--1073

whatever is left as zeroes in the denominator after canceling out will give you vertical asymptotes.1080

Finally, an optional step: if we are graphing the function, you want to figure out which way the graph goes:1086

positive infinity or negative infinity on either side of each asymptote.1091

You can do this by using test values that are very close to the asymptote.1095

For example, if we had an asymptote at x = 2, we might want to check out f at 1.99 and f at 2.01.1098

They are very close to the asymptote, so we will have an idea of where it is going.1105

Alternatively, you can also do this in your head by thinking in terms of positive versus negative,1108

and just thinking, "If I was very slightly more or very slightly less," and we will discuss this in the examples;1114

you will see it specifically in Example 3; we will see this idea of "If I was just a little bit over;1118

if I was just a little bit under," and it will make a little bit more sense when we actually do it in practice.1123

But you can always just test a number, use a calculator, and figure out, "Oh, that is what that value would be";1127

"I am clearly going to be positive on this side; I am clearly going to be negative on this side."1132

All right, let's look at some examples.1136

f(x) = (x - 1)/(2x2 + 8x - 10); what is the domain of f?1139

To figure out the domain of f, we need to figure out where our zeroes on the bottom thing are.1145

So, our very first step, where we do factoring: 2x2 + 8x - 10--how are we going to factor this?1155

Well, let's start out by pulling out that 2 that is in the front; it is just kind of getting in our way.1161

So, x2 +...8x, pulling out a 2, becomes 4x; -10, pulling out a 2, becomes -5.1165

We can check this if we multiply back out; we see that we have the same thing.1171

So, 2(x2 + 4x - 5); could we factor this further? Yes, we can.1174

We see that, if we have +5 and -1, that would factor; so 2(x + 5)(x - 1)...sure enough, that checks out.1178

x2 - x + 5x...that changes to 4x...plus 5 times -1...-5...great; that checks out.1186

So, that means we can rewrite (x - 1)/(2x2 + 8x - 10).1193

It is equivalent to saying (x - 1) over the factored version of the denominator, 2(x + 5)(x - 1).1201

So, all of the places to figure out the domain of f, all of the places where our denominator will go to 0, are going to be x at -5 and x at +1.1210

Now, we don't want to allow those; so when x is not -5, and x is not 1, then we are in the domain.1222

We are allowed those values; so our domain is everything where x is not -5 and not positive 1.1228

Great; the next question is where the vertical asymptotes are.1235

To find the vertical asymptotes, we need to cancel any common factors,1241

so that the numerator and the denominator don't go to 0 at the same time.1244

We notice that we have x - 1 here and x - 1 here; so we can cancel these out, and we get 1/[2(x + 5)].1248

So, now we are looking for where the denominator goes to 0.1258

That is going to happen at...when is x + 5 equal to 0? We get x = -5.1263

So, we have a vertical asymptote when the denominator in our new form with the common factors canceled out.1269

So, we will have a vertical asymptote after we have canceled out common factors,1275

when we can figure out that the denominator still goes to 0, at x = -5; great.1280

The second example: Given that the graph below is of f, our function, find a and b if our function f is 3x/[(x - a)(x - b)].1287

So, we notice, first, that we have a vertical asymptote here, and a vertical asymptote here, off of this graph.1297

So, we see that one of our vertical asymptotes happens at -3; the other one happens at positive 2.1302

So, x = -3; x = +2; we have vertical asymptotes.1308

OK, so if x = -3 and x = 2 mean vertical asymptotes, then that means we need to have a factor that will cause a 0 in the denominator at those locations.1315

So, we want a 0 in the denominator at -3; so x + 3 would give us a 0 when we plug in -3.1324

And x - 2 would give us a 0 when we plug in positive 2.1333

Plus 3...we plug in -3 + 3; that gives us a 0 in our denominator.1338

We plug in +2; 2 - 2...that gives us a 0 in our denominator; great.1342

So, that means that (x + 3)(x - 2)...we need those factors in our denominator to be able to have vertical asymptotes.1346

We don't have to worry about it interfering with our numerator, because 3x doesn't have any common factors with (x + 3)(x - 2).1352

They are very different linear factors.1358

At this point, we are ready...we want -a to be + 3, so that means that a must be equal to -3, because it comes out to be positive 3.1361

There is our a; and b...minus b is -2, so it must be that b is equal to positive 2, so it is still actually subtracting.1371

And there are our answers.1380

All right, the third example: Draw the graph of f(x) = 1/(3 - x).1383

It is a pretty simple one, so we don't have to worry about factoring; it has already been factored.1389

We don't have to worry about canceling any common factors; we see that there are very clearly no common factors.1392

So, we can just get right to figuring this out.1398

We have a vertical asymptote; first, we notice that we have a vertical asymptote at x = 3,1401

because when we plug in 3, 3 minus x...we have 3 minus 3, so we would have a 0, so we would be dividing by very small numbers;1409

so we would be dividing 1 by very small numbers at x = 3.1416

That means we will blow out when we are near x = 3; so we have a vertical asymptote there.1421

Let's draw in our graph axes: let's put this at...we will need more to the right, because the interesting thing happens out here.1426

So, here is 1, 2, 3, 4, 5; I will extend that a little bit further: 6, 1, 2, 3, 4, 5, 6.1436

I will go for roughly the same scale; OK.1450

And 1, 2, 3, -1, -2, -3, -1, -2; great, we have drawn out our graph.1459

We know that we have a vertical asymptote at 3; so let's draw in a vertical line.1467

OK, we see our asymptote on our graph; and now we need to draw it in.1475

So, at this point, we realize that we actually probably should have some points; so let's try some points.1481

If we try x =...oh, let's say 4, just one unit to the right of our asymptote; when we plug that in, we will get 1/(3 - 4), which gets us 1/-1.1486

1/-1 is -1; so at 4, we are at -1; so we plot in that point.1500

Let's try one unit to the left, x = 2: we will get 1/(3 - 2), so we will get 1 over positive 1 (3 - 2 is positive 1).1505

We will be at positive 1 when we are at 2.1515

We might still want a little bit more information, so maybe we will also try x = 0.1518

At x = 0, we will get 1/(3 - 0), so we will be at 1/3; so at 0, we are just 1/3 of the way up to our 1.1525

So, we will put in that point there at x = 6; let's try that one.1533

We will be at 1/(3 - 6), so 1/-3; so we will be at -1/3, so once again, we will be just a little bit...one-third of the way out there.1540

All right, now we are starting to get some sense; what happens is that we actually get near that asymptote.1550

Are we going to go positive on the left side, or are we going to go negative on the left side?1556

Are we going to go positive on the right side, or are we going to go negative on the right side?1560

What we can think about is: we can consider just a hair to the left of 3: consider 2.99999.1564

Consider this number; now, we could actually plug it into a calculator, and we could get a value.1575

But we can also just think about this in our head, and say, "1/(3 - 2.99999)...if we have a bunch of 9's,1578

3 - 2.99999, well, is it going to be a positive number, or is it going to be a negative number?"1591

Well, 3 - 2.99999...2.99999 is still smaller than 3, so it is going to come out as a positive thing.1595

We have 1, a positive number, divided by a positive number; we can just think of it as dividing by a small positive number.1601

That means that the whole thing will come out to be a positive thing, when we are on the left side,1608

because we are subtracting by something, but it ends up being just under; so we stay positive.1613

So, we can think in terms of "Are we being positive, or are we being negative?"1620

We are being positive; so we know we are going to go up on this side.1623

If we consider a very small number above 3, 3.00001--if we consider that, it will be 1/(3 - 3.00001).1627

Well, 3 minus something that is just a little bit above it is going to end up being negative.1642

It is going to be a very small number, but it is going to be negative, and that is key.1647

1 over a negative number on the bottom: since it is a positive divided by a negative, the whole thing will stay negative.1649

And so, we will get very large negative numbers; so we will be going down on the right side.1655

So, at 3.00001, we see that we end up being still negative; so we will have to be negative when we are on the right side of our asymptote.1659

On the left side, 2.99999, just to the left of our asymptote at x = 3...we see that we are going to end up being positive.1668

And so, that guides us on how our asymptote should grow.1675

So, at this point, we can draw in this curve; it is going to be growing and growing and growing.1679

And as it gets to the asymptote, it is going to get larger and larger and larger and larger and larger.1684

It will never actually touch the asymptote; it will end up just getting larger and larger and larger and going up and up and up.1689

It never becomes perfectly vertical; it never touches the asymptote; but it is going to blow out to infinity, slowly but surely.1694

It won't ever actually touch infinity, because we can't touch infinity--infinity is just an idea of ever-larger numbers.1700

But it will go out to ever-larger numbers.1706

On the other side, we see a very similar thing; sorry, that should be curved just a little bit more--not quite straight.1708

And it is going to get very, very large, and once again, going to curve out.1717

It never quite touches that vertical asymptote, but we will get very, very, very close.1721

My right side is a little bit closer to the vertical asymptote than it probably should be.1725

But we have a good idea, and that would be acceptable when turning that in on a test or homework.1729

What happens as we go very, very far to the left?1734

Well, let's think about as we plug in some really big negative number, like, say, -100.1736

Well, x = -100...we are going to end up having 1/(3 - -100); those cancel out, and we get a positive.1741

So, we get 1/103, which is a very small number.1749

So, since it is 3 - x, as we plug in very large negative numbers, we are going to effectively get very large positive numbers in our denominator.1752

And so, we will be dividing by very large things; so we will sort of crush down to nothing.1760

As we get larger and larger and larger, we are going slowly approach 0.1765

We will never quite hit 0 (we will talk about this more when we talk about horizontal asymptotes in the next lesson),1769

but we will get very, very close to the 0 height.1773

The same thing happens as we go out to the right: but we will be coming from the negative side,1776

because if we look at a very, very large positive number, like 1/(3 - 100), a positive 100 plugged in for x,1781

we will get 1/-97, so we will be a very small number,1789

but we will be a very small negative number, because we have that negative on the bottom.1793

And so, we will be on the bottom side; we will be below the x-axis.1797

Great; the final example, Example 4: Give a rational function where the graph goes up on both sides of an asymptote.1800

Also give one where it goes down on both sides.1807

So, so far, we have always seen our asymptotes where on one side it goes up, and on the other side it goes down (or maybe the other way).1809

But is it possible to create something where it is going to go up on both sides or going to go down on both sides?1815

Let's look at our first one that we think of when we think of a rational function.1821

We immediately think, "Oh, the fundamental, basic thing that makes up rational functions is 1/x."1825

This is the simplest form we can think of that really shows these ideas we are looking for.1830

1/x ends up giving us a graph that gets very, very large, that gets to the right side, but goes down on the right.1834

And then, the same thing blows down.1845

The issue here with 1/x is: on one side, it is negative; on the left side, we were dividing by negative numbers, so we have negative.1849

On the other side, it is positive; that is why we go up and down.1865

That is what causes up and down effect--well, down and up, if we go negative to positive.1872

But that is why we are seeing this split, where we are going in two opposite ways,1879

because on one side of the vertical asymptote, we are dividing by a negative number.1882

On the other side of the vertical asymptote, we are dividing by a positive number.1885

They are both very small numbers, but that negative versus positive causes this incredible difference in whether you go to -∞ or +∞.1888

So, what we want is positive on both sides, if we are going to go up on both sides.1894

So, what could we create that would be similar to 1/x, but always putting out positive things on the bottom?1903

Let's keep it as 1 over; but what is always positive?1909

Well, instead of dividing by x, let's divide by x2.1912

We can force it to always be positive: 1/x2...x2 is still a polynomial, so it is still a rational function.1916

And if we draw that one in, we are going to see it behaving very similarly on the right side.1922

It will end up being here and here and here as we get to small numbers, as we get below 1.1927

We are going to be getting to dividing by very small numbers, so it is going to blow out to infinity.1932

And we will get crushed down to 0 as it goes to the right.1939

On the left side, though, when we square a negative number, it becomes positive; so we are going to see a mirror image on the left side, as well.1941

So, we are going to see it going up on both sides.1950

If we want to get one that is going to go down on both sides, if we want it to be down on both sides--1953

well, we see 1/x2; we just want it to go down; so let's make it as easy as just flipping it down.1958

We flip it down by making everything negative, so it is -1/x2.1964

And we will see a graph that looks like this; we have managed to make it go down on both sides.1969

So, a simple way to make both sides go in the same direction is by having it be squared, or by having it be negative and then squared...1978

not negative and then squared, because that would be positive; but a negative square,1986

because then it is a square number, times a negative; cool.1989

All right, we will see you at Educator.com later.1992

Next time, we will talk about horizontal asymptotes and have an understanding of why it is getting crushed to 0,1994

and see that it can, in fact, be crushed to value other than 0--pretty interesting; goodbye!1998