Home » Mathematics » Pre Calculus » Graphing Asymptotes in a Nutshell

Vincent Selhorst-Jones

Vincent Selhorst-Jones

Graphing Asymptotes in a Nutshell

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Table of Contents

Section 1: 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
Section 2: 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
Section 3: 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
Section 4: 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
Section 5: 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
Section 6: 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
Section 7: 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
Section 8: 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
Section 9: 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
Section 10: 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
Section 11: 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
Section 12: 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
Section 13: 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
Section 14: 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
Section 15: 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
Section 16: 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
Section 17: 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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Graphing Asymptotes in a Nutshell

  • This lesson is all about learning how to graph rational functions. It's strongly recommended that you watch the previous lessons beforehand, because we'll be pulling from that work. Also, while we won't be going over it in the lesson, using a graphing utility (calculator or program) can be a great way to understand how rational functions work. Playing with function graphs can quickly build your intuition.
  • Here is a step-by-step process to create an accurate graph for any rational function:
    1. Begin by factoring the numerator polynomial and the denominator polynomial of the rational function you're working with.
    2. Find the domain of the function by looking for where the denominator equals 0. Each of these "forbidden" locations will become one of two things: a vertical asymptote (if the zero does not occur in the numerator) or a hole in the graph (if the zero occurs in the numerator).
    3. Now that we've found the function's domain, simplify the function by canceling out any factors that are in both the numerator and the denominator. [It's important that we don't do this in step #1 (factoring), otherwise we won't be able to find all the "forbidden" x-values in step #2 (domain).]
    4. Once the function is simplified, we can find the vertical asymptotes. The vertical asymptotes occur at all the x-values that still cause the denominator (after simplifying) to become 0.
    5. Find the horizontal/slant asymptotes by looking at the degree of the numerator, n, and the degree of the denominator, m. There are a total of four possible cases:
    • n < m  ⇒ horizontal asymptote at y=0.
    • n = m  ⇒ horizontal asymptote at a height given by ratio of leading coefficients in numerator and denominator.
    • n=m+1  ⇒ slant asymptote, which can be found by using polynomial division.
    • n > m+1  ⇒ no horizontal or slant asymptote.

    6. Find the x- and y-intercepts so we have some useful points that we can graph from the start. [It's possible for a rational function to be missing one type or both.]
    7. Finally, using all this information, draw the graph. Place the asymptotes (drawn as dashed lines) and intercepts. You will probably need some more points, so plot more points as necessary until you see how to draw in the appropriate curves.
  • A useful concept for graphing is the idea of test intervals. Because a rational function is continuous between vertical asymptotes, we know that the function can only change signs at x-intercepts or vertical asymptotes. This means we can put our x-intercepts and vertical asymptote locations in order and break the x-axis into intervals. In each of these intervals, we can test just one point to find if the function is + or − in the interval.

Graphing Asymptotes in a Nutshell

From the graph below, name the type and location of all the asymptotes.
  • A vertical asymptote is a horizontal location that, as approached, causes the function to fly off vertically to ±∞.
  • We see that as the graph nears x=4, the function flies off vertically. Thus, x=4 is a vertical asymptote.
  • A horizontal asymptote is the vertical height that the function tends towards as x goes goes very far to the right or left. We can think of it as a height that the function is "pulled" towards over the long-run.
  • We see from the graph that as x becomes very large (far to the right or left), the graph approaches the vertical height of y=0.
Vertical Asymptote: x=4,       Horizontal Asymptote: y=0
From the graph below, name the type and location of all the asymptotes.
  • A vertical asymptote is a horizontal location that, as approached, causes the function to fly off vertically to ±∞.
  • We see that the graph flies off vertically in two locations: as the graph nears x=−2 and as it nears x=2. Thus, there are two vertical asymptotes: x=−2,  2.
  • A horizontal asymptote is the vertical height that the function tends towards as x goes goes very far to the right or left. We can think of it as a height that the function is "pulled" towards over the long-run.
  • We see from the graph that as x becomes very large (far to the right or left), the graph approaches the vertical height of y=−4.
Vertical Asymptotes: x=−2,  2,       Horizontal Asymptote: y=−4
From the graph below, name the type and location of all the asymptotes. [Note: a dashed line has been put in for one of the asymptotes to assist in identifying it.]
  • A vertical asymptote is a horizontal location that, as approached, causes the function to fly off vertically to ±∞.
  • We see that as the graph nears x=0, the function flies off vertically. Thus, x=0 is a vertical asymptote.
  • A horizontal asymptote is the vertical height that the function tends towards as x goes goes very far to the right or left. We can think of it as a height that the function is "pulled" towards over the long-run.
  • We see from the graph that no such height exists. As the graph goes far to the right or left, it does not "pull" to a specific height. This graph continues to go up and down forever. However, we do notice that it "pulls" along a very specific line. This is a slant asymptote.
  • Looking at the dashed line (indicating the slant asymptote), we can figure out an equation for it. For any interval, the amount it goes right equals the amount it goes down, so the line has a slope of m=−1. It has a vertical intercept of b=0. Putting these observations in the slope-intercept line equation y=mx+b, we find that the slant asymptote is y=−x.
Vertical Asymptote: x=0,       Slant Asymptote: y=−x
Write out a rational function that has the following qualities. Vertical asymptote: x=7,    Horizontal asymptote: y=−8
  • Remember, a rational function is something in the form of a fraction where the top and bottom are both polynomials. Let's figure out what each asymptote requires of the function.
  • To have a vertical asymptote at x=7, the denominator needs to go to 0 at x=7. Thus, we must have a factor of (x−7) in the denominator [and we need to make sure that factor doesn't get canceled out by something in the numerator].
  • To have a horizontal asymptote of y=−8, the numerator and denominator polynomials must have the same degree. Once they have the same degree, the ratio of their coefficients must be −[8/1].
  • So far, we've figured out the function as
    f(x) = ?
    x−7
    Thus, we will make the numerator have a degree of 1 as well (because the denominator already has degree 1). The easiest thing would be to just use x. However, we must also have a coefficient in front of it to get the horizontal asymptote of y=−8. Since the denominator has a coefficient of 1, we can use −8 for the numerator's coefficient. Thus, we'll have −8x as the numerator.
  • Putting these together, we get
    f(x) =−8x
    x−7
    		 .
    [Note: there are other ways to write out a rational function with the given two qualities. This is just one of the easiest ways to write it. Here are some other possible rational functions that would all have the given vertical and horizontal asymptotes:
    −16x
    2x−14
    		      −8x(x3+1)
    (x−7)(x3+1)
    		      −8x3+5x2−18
    (x−7)(x+2)(x−15)
    (Some of the above have additional asymptotes or "holes" in them or both, but they all have the asymptotes required by the question and so could be considered correct as well.)]
f(x) = [(−8x)/(x−7)] [Note: There are other possible answers for this question, the above is just one of the easiest answers to write out. Look at the final step for some other possible answers if you're curious.]
Write out a rational function that has the following qualities. Vertical asymptotes: x=−4, −3, 4,    Horizontal asymptote: y=[1/2]
  • Remember, a rational function is something in the form of a fraction where the top and bottom are both polynomials. Let's figure out what each asymptote requires of the function.
  • To have a vertical asymptote, the denominator needs to go to 0 at the given horizontal location. Since we have three vertical asymptotes (x=−4, −3, 4), we need three factors that will each go to 0 at each respective location. Thus, we must have factors of (x+4), (x+3), and (x−4) in the denominator [and we need to make sure that those factors are not canceled out by something in the numerator].
  • To have a horizontal asymptote of y=[1/2], the numerator and denominator polynomials must have the same degree. Once they have the same degree, the ratio of their coefficients must be [1/2].
  • So far, we've figured out the function as
    f(x) = ?
    (x+4)(x+3)(x−4)
    Thus, we will make the numerator have a degree of 3 as well (because the denominator already has degree 3 [if you don't see it, expand the bottom]). The easiest thing would be to just use x3. However, we must also have a coefficient in front of it to get the horizontal asymptote of y=[1/2]. Since the denominator has a coefficient of 1, we can use [1/2] for the numerator's coefficient. Thus, we'll have [1/2]x3 as the numerator.
  • Putting these together, we get
    f(x) =
    1
    2
    		 x3
    (x+4)(x+3)(x−4)
    		 .
    [Note: there are other ways to write out a rational function with the given asymptotes. This is just one of the easiest ways to write it. Here are some other possible rational functions that would all have the given vertical and horizontal asymptotes:
    x3
    2(x+4)(x+3)(x−4)
    		       4x4
    8(x+4)(x+3)(x−4)(x+47)
    		      
    1
    2
    		 x3(x−4)
    (x+4)(x+3)(x−4)2
    (Some of the above have additional asymptotes or "holes" in them or both, but they all have the asymptotes required by the question and so could be considered correct as well.)]
f(x) = [([1/2]x3)/((x+4)(x+3)(x−4))]    [equivalently expanded: [([1/2]x3)/(x3+3x2−16x−48)]  ] [Note: There are other possible answers for this question, the above is just one of the easiest answers to write out. Look at the final step for other possible answers if you're curious.]
Graph the rational function below.
f(x) = 2
x−3
  • Begin by factoring the numerator and denominator [but don't simplify yet]. In this case, it's already done, so we can move right along to the next step.
  • Find the domain. The function will "break" when you divide by 0, so that occurs at the zeros of our denominator.
    x−3=0
    Thus, we have a "forbidden" location of x=3, so the domain is x ≠ 3.
  • Simplify the function. Once again, quite easy because there's nothing to cancel out.
  • The vertical asymptotes are wherever the denominator still has zeros (after simplifying). Since we couldn't simplify it anymore, the vertical asymptote is the same as the "forbidden" location: x=3.
  • Find the horizontal asymptote. Compare the degree of the numerator to the degree of the denominator. In this case, the numerator has lower degree, so the horizontal asymptote is y=0.
  • Find points to plot on the graph. It can be useful to find any x-intercepts and the y-intercept in addition to finding other points as needed to draw the graph.
    x
    f(x)
    0
    −0.67
    2
    −2
    2.5
    −4
    3.5
    4
    4
    2
    6
    0.67
  • Set up the graph axes, draw in the asymptotes with dashed lines, plot points, and connect with curves. Remember, as the graph gets close to an asymptote, it "pulls" alongside.
Graph the rational function below.
g(x) = x+5
x2+3x−10
  • Begin by factoring the numerator and denominator [but don't simplify yet]. The numerator is already factored, and we see that we can factor the denominator into (x+5)(x−2).
  • Find the domain. The function will "break" when you divide by 0, so that occurs at the zeros of our denominator.
    (x+5)(x−2)=0
    Thus, we have "forbidden" locations at x=−5, 2, so the domain is x ≠ −5, 2.
  • Simplify the function.
    g(x) = x+5
    x2+3x−10
    		    =     x+5
    (x+5)(x−2)
    		    ⇒     1
    x−2
  • The vertical asymptotes are wherever the denominator still has zeros (after simplifying). Since our new, simplified denominator is just x−2, we only have a vertical asymptote at x=2. [Notice that we still have to care about the other "forbidden" value of x=−5 by making a "hole" in the graph at the end.]
  • Find the horizontal asymptote. Compare the degree of the numerator to the degree of the denominator. In this case, the numerator has lower degree, so the horizontal asymptote is y=0.
  • Find points to plot on the graph. It can be useful to find any x-intercepts and the y-intercept in addition to finding other points as needed to draw the graph. [Also, we can use the simplified version to find where the "hole" at x=−5 should go. It doesn't actually exist there, but we can graph it as if it did, then just put a "hole" in that spot.]
    x
    g(x)
    −5
    −0.14  /  DNE
    0
    −0.5
    1
    −1
    1.5
    −2
    2.5
    2
    3
    1
    4
    0.5
    5
    0.67
  • Set up the graph axes, draw in the asymptotes with dashed lines, plot points, and connect with curves. Remember, as the graph gets close to an asymptote, it "pulls" alongside. Also, don't forget to put a "hole" at x=−5 because the rational function technically does not exist there. We do this with an empty circle.
Graph the rational function below.
h(x) = 3x2+3x−36
x2−x−6
  • Begin by factoring the numerator and denominator [but don't simplify yet]. We can factor as follows:
    Numerator: 3x2+3x−36    =     3(x2 + x − 12)     =     3(x+4)(x−3)

    Denominator: x2 −x −6     =     (x+2)(x−3)
  • Find the domain. The function will "break" when you divide by 0, so that occurs at the zeros of our denominator.
    (x+2)(x−3)=0
    Thus, we have "forbidden" locations at x=−2, 3, so the domain is x ≠ −2, 3.
  • Simplify the function.
    h(x) = 3x2+3x−36
    x2−x−6
    		    =     3(x+4)(x−3)
    (x+2)(x−3)
    		    ⇒     3(x+4)
    x+2
  • The vertical asymptotes are wherever the denominator still has zeros (after simplifying). Since our new, simplified denominator is just x+2, we only have a vertical asymptote at x=−2. [Notice that we still have to care about the other "forbidden" value of x=3 by making a "hole" in the graph at the end.]
  • Find the horizontal asymptote. Compare the degree of the numerator to the degree of the denominator. In this case, the numerator and denominator have equal degrees, so we make a fraction from their leading coefficients. This gives a horizontal asymptote of y=[3/1] = 3.
  • Find points to plot on the graph. It can be useful to find any x-intercepts and the y-intercept in addition to finding other points as needed to draw the graph. [Also, we can use the simplified version to find where the "hole" at x=3 should go. It doesn't actually exist there, but we can graph it as if it did, then just put a "hole" in that spot.]
    x
    h(x)
    −6
    1.5
    −4
    0
    −3
    −3
    −2.5
    −9
    −1.5
    15
    −1
    9
    0
    6
    3
    4.2  /  DNE
  • Set up the graph axes, draw in the asymptotes with dashed lines, plot points, and connect with curves. Remember, as the graph gets close to an asymptote, it "pulls" alongside. Also, don't forget to put a "hole" at x=3 because the rational function technically does not exist there. We do this with an empty circle.
Graph the rational function below.
f(x) =−2x3−2x
x3−3x2−4x
  • Begin by factoring the numerator and denominator [but don't simplify yet]. We can factor as follows:
    Numerator: −2x3−2x    =     −2x(x2 + 1)

    Denominator: x3−3x2−4x     =     x(x2−3x−4)    =     x(x+1)(x−4)
  • Find the domain. The function will "break" when you divide by 0, so that occurs at the zeros of our denominator.
    x(x+1)(x−4)=0
    Thus, we have "forbidden" locations at x=−1, 0, 4, so the domain is x ≠ −1, 0, 4.
  • Simplify the function.
    f(x) =−2x3−2x
    x3−3x2−4x
    		    =    −2x(x2 + 1)
    x(x+1)(x−4)
    		    ⇒    −2(x2+1)
    (x+1)(x−4)
  • The vertical asymptotes are wherever the denominator still has zeros (after simplifying). Since our new, simplified denominator is (x+1)(x−4), we have vertical asymptotes at x=−1 and x=4. [Notice that we still have to care about the other "forbidden" value of x=0 by making a "hole" in the graph at the end.]
  • Find the horizontal asymptote. Compare the degree of the numerator to the degree of the denominator. In this case, the numerator and denominator have equal degrees, so we make a fraction from their leading coefficients. This gives a horizontal asymptote of y=[(−2)/1] = −2.
  • Find points to plot on the graph. It can be useful to find any x-intercepts and the y-intercept in addition to finding other points as needed to draw the graph. [Also, we can use the simplified version to find where the "hole" at x=0 should go. It doesn't actually exist there, but we can graph it as if it did, then just put a "hole" in that spot.]
    x
    f(x)
    −7
    −1.52
    −4
    −1.42
    −3
    −1.43
    −2
    −1.67
    −1.5
    −2.36
    −0.5
    1.11
    0
    0.5  /  DNE
    1
    0.67
    2
    1.67
    3
    5
    3.5
    11.78
    4.5
    −15.45
    5
    −8.67
    8
    −3.61
  • Set up the graph axes, draw in the asymptotes with dashed lines, plot points, and connect with curves. Remember, as the graph gets close to an asymptote, it "pulls" alongside. Also, don't forget to put a "hole" at x=0 because the rational function technically does not exist there. We do this with an empty circle.
Graph the rational function below.
g(x) = 6x2−12x+7
3x−3
  • Normally we would begin by factoring the numerator and denominator. However, for this problem, it would be difficult to factor the numerator (give it a try if you don't believe me). It's alright though: the only reason we care about factoring the numerator is to see if it has any common factors with the denominator. We see that the denominator breaks down to 3(x−1). Furthermore, while it's difficult to factor the numerator, we can pretty easily see that (x−1) can't be a factor in 6x2−12x+7 (the numerator), so we're safe from having to worry about canceling factors later on. Thus, we can just leave the numerator unfactored (which will wind up being useful later on).
  • Find the domain. The function will "break" when you divide by 0, so that occurs at the zeros of our denominator.
    3(x−1)=0
    Thus, we have a "forbidden" location at x=1, so the domain is x ≠ 1.
  • Because we can't cancel out any factors between the numerator and denominator, the "forbidden" location of x=1 is also a vertical asymptote.
  • Notice that the function does not have a horizontal asymptote. This is because the numerator's degree (2) is higher than denominator's degree (1). Thus, no horizontal asymptote. However, because it is only one number higher, it will have a slant asymptote. We find the slant asymptote using polynomial long division:
    3x
    −3
    ⎞
    ⎠
    6x2
    −12x
    +7

  • 2x
    −2
    R:  1
    3x
    −3
    ⎞
    ⎠
    6x2
    −12x
    +7
    6x2
    −6x
    −6x
    +7
    −6x
    +6
    1
    Thus, through long division, we have shown that
    g(x) = 6x2−12x+7
    3x−3
    		    =     2x −2 + 1
    3x−3
    		 ,
    which means we have a slant asymptote of y = 2x−2.
  • Find points to plot on the graph. It can be useful to find any x-intercepts and the y-intercept in addition to finding other points as needed to draw the graph.
    x
    g(x)
    −4
    −10.07
    −2
    −6.11
    −1
    −4.17
    −0.5
    −3.22
    0
    −2.33
    0.5
    −1.67
    1.5
    1.67
    2
    2.33
    2.5
    3.22
    3
    4.17
    5
    8.08
  • Set up the graph axes, draw in the asymptotes with dashed lines, plot points, and connect with curves. Remember, as the graph gets close to an asymptote, it "pulls" alongside.

*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

Graphing Asymptotes in a Nutshell

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
  • 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
    • Horizontal Asymptote
    • Other Graphing
  • Test Intervals 15:08
  • Example 1 17:57
  • Example 2 23:01
  • Example 3 29:02
  • Example 4 33:37

Transcription: Graphing Asymptotes in a Nutshell

Hi--welcome back to Educator.com.0000

Today, we are going to talk about graphing and asymptotes in summary.0002

In the past two lessons, we learned how rational functions work.0006

We have studied and come to understand their behavior, along with vertical asymptotes and horizontal/slant asymptotes.0009

In this lesson, we use this knowledge to learn how to graph rational functions.0015

It is strongly recommended that you watch the previous lessons beforehand, because we will be pulling things from that work.0019

Also, while we won't be going over it in this lesson, using a graphing utility, whether it is a calculator or a program0024

or something even on a smartphone, can be an absolutely great way to understand how rational functions work.0029

Playing with function graphs can quickly build your intuition.0035

Just like it can build your intuition for any function, being able to play around and change how things are working0038

and what your denominator is and what your numerator is will build a really great, intuitive understanding0042

of how this stuff works much faster than trying to do it all by hand.0047

You will be able to get a really good grasp of how this stuff works, and be able to see it in your mind's eye0051

in a way that you just can't develop by trying to do ten problems in hand, because you can just do 100 of them0055

in a matter of a few minutes if you are just playing around on a calculator.0061

So, I highly recommend that you take the chance and use a graphing utility and play with something.0065

If you haven't already checked it out, there is an appendix to this course all about graphing utilities,0069

graphing calculators...all of that stuff... that can give you some idea of how to start in that sort of thing,0073

because they can be really useful for helping you in a course like this and in future math courses.0078

All right, let's get started: the majority of this lesson is going to be about a process for graphing rational functions.0082

By following the steps of this process, you will obtain what you need to graph.0088

You will get all of the information that you need to make a good graph.0091

This process also gives you a way to analyze rational functions in general, though.0094

So, it is not something that you can only use when you want to graph a function--just if you want to look at a rational function0098

and get a good idea of how it works, you might find this process useful.0104

So, you might use it, even if you don't need to graph anything; let's go!0106

The first step: a rational function starts in the form n(x)/d(x), where n(x) and d(x) are both polynomials--this division.0110

It is useful as a very first step to begin by factoring n and d.0119

While this won't directly tell us anything--factoring the numerator and the denominator doesn't immediately tell us anything directly--0124

it is very useful in the coming steps to have the numerator and denominator broken into smallest factors.0130

A lot of our steps are going to revolve around having these things already factored.0135

So, it is something to get out of the way, right from the beginning.0138

So, we are going to have a running example as we go through this.0141

If we have f(x) = (x2 - 3x + 2)/(2x2 - 2x - 4), we could factor this,0143

and we would get (x - 1)(x - 2) for the numerator, divided by 2(x + 1)(x - 2) for the denominator.0149

So, we started by factoring it, by just breaking these into their factors.0156

By seeing what these things become, we are able to get a good sense for later steps.0161

It will help us out in the later steps; so we just re-format it with the numerator factored and the denominator factored; and that is all we do for right now.0167

The second step: find the domain of the function.0174

Remember: we can't have division by 0--it is one of the critical ideas here.0177

So, we want to find all of the x-values where the denominator is 0, because those are going to be forbidden.0181

We do this by finding the zeroes, the roots, of d(x); so find the zeroes of our denominator.0186

And since we just factored the denominator function, this is going to be pretty easy.0191

We just factored it, in our example, into (x + 1) and (x - 2), so we see at this point that we wouldn't be allowed -1 or +2,0196

because they would cause a 0 to pop up.0203

So, each of the zeroes of our denominator is not going to be allowed in the domain.0206

They will not be allowed in the domain if they are a zero of our denominator polynomial.0210

All other real numbers are in the domain, because everything else is fine for a polynomial.0214

The only problem is when we are accidentally dividing by 0; so we just have to clip those out.0219

Those are the forbidden locations that we can't take in.0223

Now, each of these forbidden locations will become one of two things: they will become a vertical asymptote,0226

if the zero does not occur in the numerator; or they will become a hole in the graph, if the zero does occur in the numerator.0232

The next step: simplify the function--once we have found the function's domain, we can simplify the function0240

by canceling out any factors that are in both n(x) and d(x).0245

So, we noticed that we had (x - 2) on the top and (x - 2) on the bottom; we have common linear factors.0249

So, we knock them both out, and we are left with (x - 1)/2(x + 1).0254

It is important to note that we couldn't do this in our very first step, factoring,0260

because if we did that, we wouldn't be able to find all of the forbidden x-values.0264

There is a forbidden x-value at x = 2; so that is something that we are not allowed to have from this.0268

So, the only way to find that is if we haven't already gotten rid of it.0275

If we start by canceling it, we will never realize that x is not equal to 2, because x = 2, that horizontal location of x = 2, is forbidden.0278

If we cancel out that factor before we notice that it is a forbidden location, we will never be able to figure that out.0287

So, we have to get that information before we cancel it; and that is why we do factoring, and then check the domain, then simplify.0292

The next step: find vertical asymptotes--once the function is simplified, we can find the vertical asymptotes.0300

The vertical asymptotes will occur at all the x-values0305

that cause the denominator, after we have simplified the denominator, to become 0--0307

so, all of the zeroes of our now-newly-simplified denominator.0312

So, if we have 2(x + 1) in our denominator, we are going to get a vertical asymptote at x = -1,0316

because that will cause our denominator to turn into a 0.0321

The next step: find the horizontal or slant asymptotes, or figure out if it has absolutely none of those.0325

Let n be the degree of the numerator; and m is the degree of our denominator.0331

Then, there are a total of four possible cases: n is less than m--the degree of the numerator is less than the degree of the denominator.0339

Our denominator grows faster than our numerator--that means that we will eventually be crushed down to nothing.0350

The whole fraction will be crushed down to nothing; and we have a horizontal asymptote, y = 0.0356

Another possibility: if n equals m--the degree of the numerator is equal to the degree of the denominator--0360

then there will be a horizontal asymptote at a height given by a ratio of the leading coefficients,0367

because they are both growing at the same class of speed, so we need to compare how their fronts go.0373

If we have effectively 5x5 divided by 2x5, the other stuff has some effect.0379

But in the long run, it won't be as important, and it will turn into 5/2, because the x5's cancel out.0385

That is one way of looking at it: n = m means that we get a horizontal asymptote based on the ratio of leading coefficients.0391

Next, n = m + 1; that is a slant asymptote--we will have a slant asymptote, and we find that by using polynomial division.0399

You can also use polynomial division to find the horizontal asymptotes,0408

but it is pretty easy and fast to find it just by comparing the leading coefficients, so we don't worry about that as much.0411

And then finally, if we have n > m + 1, there is no horizontal or slant asymptote whatsoever,0418

because the numerator is growing so much faster than the denominator0425

that it is not going to be a horizontal thing that it goes to; it is not even going to be a slant that it goes to.0429

It is just going to blow into something even larger and more interesting.0433

But we are not going to worry about that in this course.0436

f(x) = (x - 1)/2(x + 1): we realize that this is a degree of 1; this is a degree of 1.0438

So now, we go and we compare our leading coefficients: we have a 1 on the top and a 2 on the bottom.0445

So, we get a horizontal asymptote of y = 1/2 in our running example.0451

The sixth step: find the intercepts--by finding the x- and y-intercepts, we have a couple points from in the graph that we just have to start with.0456

Now, it is possible for a rational function to be missing one type or both.0464

So, it is possible that these things won't be there, because the location of our y-intercept, x = 0, could be a forbidden location.0468

And all of the places where we cross over--it could either not cross the x-axis at all,0477

or the locations where it would cross the x-axis are actually disappeared holes in our graph, so they aren't technically intercepts.0481

So, it is possible to be missing these things.0488

But if we have them there, they are nice, and they are not that hard to find.0490

The x-intercepts occur wherever the function has an output value of 0, because that means we have a height of 0;0494

so, we are on the x-axis; we are an x-intercept with a y height of 0.0499

Thus, all of our zeroes mean that our numerator is at 0.0505

This is all of the zeroes of our simplified numerator: x - 1...what are all the places where that gives us a 0?0509

That gives us a 0 at x = 1, so at x = 1, we get a 0 out of it; so if we plug in 1, we get a zero out of that, because the numerator is now at 0.0515

The y-intercept is where the function's input is 0, because that will put us on the y-axis.0528

So, we plug in the x-value...0533

I'm sorry; I might have said the wrong thing there; I am not quite sure what I said.0535

The y-intercept occurs where the function's input, its x-value, the x that we are plugging in, is 0,0537

because that will put us right on the y-axis.0542

So, just evaluate our function with a 0 plugged into it, f(0): to find it, we plug in 0: (0 - 1)/2(0 + 1) from our simplified function.0544

That simplifies to -1/2, so we get (0,-1/2); so we have some points to start with when we are plotting.0555

The final thing: draw the graph--now that we have all of this information on our function, we are ready to graph it.0563

Begin by drawing in the asymptotes on the graph, and plot the intercept points.0567

If we need more points to graph the function (and we are probably going to need more points,0571

since the intercepts--really, there are only a couple of things that come out of the intercepts), evaluate the function0575

at a few more points, as you need, and plot those points, as well.0579

Plot those extra points; and then, at that point, you can start drawing in curves.0583

It is useful to plot at least one point between and beyond each vertical asymptote.0587

So, if we have vertical asymptotes, like here and here, you probably want to make sure you plot at least one thing in each of these locations.0594

And there is a good chance that you will need a couple more than that, to be able to really get a sense0601

for how the thing is going to come together as a picture--that is something to think about.0605

Then, draw in the graph, connecting the points with smooth curves, and pulling the graph along the asymptotes.0610

And of course, if you are not quite sure where it goes and how it works together,0616

you can just plot in even more points, and you will get a better sense of how the picture comes together.0619

Make sure you know which direction, positive or negative, the graph will go in either side of your vertical asymptotes.0624

And one last thing: don't forget that the forbidden values, our forbidden x locations, will create holes in the graph.0629

We are going to be having these locations that aren't really there, which we will denote with an open circle0644

to say, "Well, we would be going here; but it actually is missing that location,0651

because it is a forbidden thing, because it will cause us to divide by 0."0655

So, we are either going to have vertical asymptotes--which we would never get to anyway--0658

or we are going to have actual holes in our graph, which we denote with a hole,0661

of a just round circle that has nothing inside of it.0665

So, don't forget that you have to remember about the holes when you are actually drawing it in the graph.0668

Not every graph will have holes; but if yours does have forbidden x-values0674

that aren't just vertical asymptotes, you will need to do that, as we are about to see.0678

f(x) = (x2 - 3x + 2)/(2x2 - 2x - 4); we simplify that into (x - 1)/2(x + 1).0681

We figured out that our domain had forbidden values at -1 and 2, because they caused our original denominator,0690

not just our simplified denominator, to turn into a 0; we are not allowed to divide by 0.0696

To figure out our vertical asymptote, we looked: when does our simplified denominator go to 0? That happens at x = -1.0701

To find our horizontal asymptote, we noticed that we have a degree of 1 on the top and the bottom; so 1/2 gave us y = 1/2 as a horizontal asymptote.0710

So, we plot our horizontal asymptote; we plot our vertical asymptote.0719

And we also figured out our intercepts; 0 comes out as -1/2, so we plot that point right there.0725

and 1 comes out as 0, so we plot that point right there; those are our intercepts.0731

Now, that is not quite enough information (for me, at least) to figure out how this is going to graph.0736

So, we decide, "Let's plot a few more points."0740

We try out -2 and -3, because -2 is one step to the left of our asymptote.0743

So, we plot in this point; we plot in -3; that comes out there; so we have -2 at 3/2 and -3 at positive 1.0747

And then also, we can say, "Well, we are not allowed to put in 2; but we are still curious to know where that would be, if it was there."0755

So, let's see where 2 would go if it wasn't a forbidden location.0764

Remember, it is forbidden here, because it caused us to divide by 0.0769

But in this one, it doesn't actually cause anything weird to happen in our simplified version, because we have canceled out the 0/0, effectively.0772

Since we have canceled it out, it doesn't cause anything weird to happen.0779

So, we can see where it would go by looking at our simplified version.0781

So, if we plug 2 into our simplified version, we get (2 - 1), 1, over 2(2 + 1), 2(3), 6; so we get 1/6.0784

So, we plug in, not just a point, but a hole, to tell us, "Look, there is where you would go;0793

you will run through that location, but you are not actually going to occupy that."0800

So, at this point, it seems like we are starting to get enough information.0804

One last thing we might want to figure out is which way we are going on each of these asymptotes.0807

We probably can get a sense of this at this point.0812

So, if we have plugged in, say, -1.0001, if we were just to the left side of our vertical asymptote,0814

then over here we would have 2 and -1.0001 plus 1; and up here, we would have -1.0001.0822

On the top we would have a negative...minus a negative, so it is a negative; and then divided by 2 times negative +1,0834

-1.0001, so just a little bit more negative; so it is going to be a negative number down there; 2 doesn't change it from being negative.0841

So, we have a negative over a negative, which cancels out to a positive.0848

It is going to come out being positive on this side; so we are going to be going up with our vertical asymptote on this side.0852

If we do the opposite, -0.999, like this, then if we plugged in -0.999 - 1, we see that that is going to end up being a negative.0858

And then, divided by 2 times -0.999 plus 1...we see -0.999 + 1...that stays positive, just barely.0871

It is a very small positive, but it is positive; so we have a positive on the bottom.0880

A negative divided by a positive--that remains being negative; so on the right side of our vertical asymptote,0884

over here, we are going to be going down, because we are going to be having negative values coming out of it.0890

Now, we know which way it should be going; and we know, on our horizontal asymptote, it is just going to stay on the same side and run with it.0896

We draw all of these things in; and sure enough, that is what happens when we draw in our picture; great.0903

Test intervals: this is a useful idea that can occasionally be used.0909

We didn't use it in our initial seven steps; but it is something that we kind of vaguely...I vaguely used it without explaining it,0913

on the last thing, where I said we knew that we were going to have to stay0919

under the horizontal, below the horizontal, in these various locations.0922

Let's see what it is: a useful concept for graphing is the idea of the test interval.0925

Because a rational function is continuous (remember, there are no jumps in a rational function,0929

because it is built out of polynomials, so there can't be any jumps between vertical asymptotes),0934

we know, by the intermediate value theorem, since we aren't allowed jumps,0939

that a function can only change signs at x-intercepts where it crosses from positive to negative.0942

It can only change signs when it hits 0 to be able to go from positive to negative--0947

or vertical asymptotes, because after a vertical asymptote, it does jump.0951

This means we can put our x-intercepts and our vertical asymptote locations in order, and then break the x-axis into intervals.0955

In each of these intervals, we can test just one point, because we know we can't jump0961

until we hit an x-intercept or we hit a vertical asymptote--we can't flip signs.0966

So, we can test just one point to figure out if it is going to be positive in that interval, or it is going to be negative in that interval.0970

So, in each of these intervals, we only have to test one point to figure out positive- or negative-ness in there.0976

If we have f(x) = (x - 1)/2(x + 1), our example that we have been working this whole time,0980

we have a vertical asymptote at x = -1, and we have an x-intercept at x = +1.0985

So, that means we can go from negative infinity, right here, up until -1, our first interval location stopping.0990

And then, from -1 up until 1 is our next thing--from -1 here to positive 1 here; and then 1 out into infinity, because we don't have any others.0998

There are three intervals, and the function is going to maintain its sign within that interval.1006

So, let's test any point: notice, -2 is inside of that interval, so let's try out -2.1012

We plug in -2, and we get -3 divided by -2, which becomes +3/2, so it is positive; everywhere between negative infinity and -1, it is positive.1016

From -1 to 1...let's look at 0: 0 is definitely in that interval.1029

We plug that in; we get -1 divided by 2, so we get -1/2; that means it is going to be negative everywhere inside of that interval.1032

And 1 to infinity--let's try out 3; 3 is in there--we plug in 3; we get 2/2(4), so we get 1/4, because we had 2/8, so 1/4.1040

But it is a positive, more importantly, so we know we are going to be positive everywhere from 1 out until positive infinity.1052

If you go back just a little bit to the previous graph, where we saw the thing actually get graphed in,1058

and pause it there, you will actually be able to see that it ends up being always positive between negative infinity and -1,1063

always negative from -1 to positive 1, and always positive from 1 out to infinity.1069

You will be able to see that in the graph; that is the idea of test intervals.1074

All right, let's look at some examples: f(x) = 3/(x + 2).1078

Our first step: let's figure out what is forbidden in our domain.1084

So, our domain can't cause anything to go to 0; so when is x + 2 equal to 0?1088

That happens at x = -2, so our domain will not allow -2, because we do not want a denominator to allow a 0.1094

Next, what are our vertical asymptotes?1102

That is going to happen when...are we simplified?...yes, 3/(x + 2) is already simplified,1108

so that is going to be whenever the denominator is 0, so we have a vertical asymptote at x =...1113

oh, sorry, not 0, but x = -2, where the denominator becomes 0.1118

So, our domain location that is not allowed is our vertical asymptote here.1123

And finally, a horizontal asymptote--what will this go to in the long run?1128

In the long run, we have a numerator's degree that is 0, and our denominator's degree is 1.1137

So, in the long run, the denominator will grow and eventually crush our numerator to effectively nothing.1143

So, in the long run, our fraction goes to 0; so it has y = 0 as its horizontal asymptote.1148

Now, it would probably be a good idea to find some intercept locations, so we can have a better idea of what is going on here.1155

Let's see, at 0, where are we? At 0, we plug in 0; 3/(0 + 2) gets us 3/2.1160

Do we have a y-intercept?...yes, we have a y-intercept--that is what we just figured out.1169

Do we have any x-intercepts?...no, because the numerator is always 3.1173

The numerator never goes to 0, so there are not going to be any x-intercepts.1177

Let's draw in our graph, and then we will fill it out.1182

So...oops, that got kind of curvy; I will erase that really quickly.1186

We know that -2 (that is our vertical asymptote) is the most interesting location; so we will give a little extra space on our left side.1191

1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5; OK.1201

We know about the -2...and I hope you don't mind; I am just not going to mark it down.1219

I think we can see pretty easily, just counting out what those are; we just know those types of marks mean 1.1223

So, we have a vertical asymptote at -2, and we have a horizontal asymptote at y = 0, which is just our x-axis.1228

Great; all right, so how can we draw this in?1238

We have (0,3/2); we are right here; now, we probably want some more points.1242

That is not quite enough information, so let's try some points; let's go just one to the left of our vertical asymptote--let's try out -3.1248

We plug in -3; we get 3/(-3 + 2), so -1...3/-1 gets us -3.1258

Let's try -4, as well: we plug in -4; 3/(-4 + 2) is -2, so that is 3/-2, or -3/2.1266

We plug in -5, and we will get 3/(-5 + 2), so -3, so we will get -1.1275

We can plot the stuff on the left side now: at -3, we are at a height of 1, 2, 3, -3.1282

At -4, we are at 3/2; and at -5, we are at -1.1290

Let's go on the other side; let's look at...we already have this point here, so let's try -1.1296

At -1, 3/(-1 + 2) becomes positive 1, so we are at positive 3, because 3 divided by 1 is 3.1302

At 0, we already figured out we are at 3/2; and at positive 1, we are going to be at 3/(1 + 2), so 3, so we will be at 1 there.1310

Positive 1...we are at 1; at -1, we are at 3; great; at this point, we have a pretty good idea.1323

We can see that we are going up here; we can see that we are going down here.1330

We could plug in -2.0001, and we would see that that would be 3 divided by a negative, so we are going negative.1333

And -1.99999 would be 3 divided by a positive, so we are going to go positive.1342

But we can also just see, from enough points that we have at this point--see where the curve is going.1347

So, we draw this in; it curves; as it approaches the asymptote, it curves more and more and more and more and more.1351

The other way, it is going to curve and approach its horizontal asymptote; and it approaches it, but it never quite touches it; and it goes out that way.1359

A similar thing is going on over here; it curves; it approaches the vertical asymptote; it won't quite touch it.1368

And here, it curves; it approaches the horizontal asymptote in the long run, but it doesn't quite touch it, either; great.1373

And there we are--the first one done.1380

The next one: Graph f(x) = (x + 1)/(x3 + x2 + x + 1).1382

The first thing we want to do is factor this; the top is still just going to be (x + 1); what does the bottom become?1388

Well, we know that there is going to be an (x + 1) factor in there; we can figure that out by trying plugging in -1.1394

We see that it would work; but we can also eventually notice that we will eventually factor it into (x + 1)(x2 + 1).1402

So, what is not allowed by our domain? Our domain does not allow (x + 1);1411

so x is not allowed to be -1, because -1 + 1 would give us a 0.1419

The domain is not allowed to be -1; does x2 + 1 provide anything?1424

No, x2 + 1 is an irreducible quadratic; there are no roots there, because x2 + 1 = 0 is always going to remain positive.1428

You can't solve x2 + 1 if you are using real numbers, so we don't have to worry about a hole appearing there.1435

So, our only issue is that = -1; so we disallow -1: -1 is a forbidden location.1440

Now, at this point, we can simplify: we see (x + 1) and (x + 1); we cancel those out; and so, we get 1/(x2 + 1).1446

All right, so at this point, we can figure out what is our vertical asymptote; do we have any vertical asymptotes?1455

We don't have any vertical asymptotes, because in our simplified form,1461

1/(x2 + 1), there is no x we can plug in to get 0 to show up in the bottom.1464

x2 + 1 has no solutions; it never intersects the x-axis, if we think about it.1469

So, a vertical asymptote...we have no vertical asymptote here.1474

What about a horizontal asymptote? If we plug in for a horizontal asymptote, we see--look, the denominator has a higher degree than the numerator.1478

So, the numerator is going to eventually get crushed by that large denominator, so we have it eventually going to y = 0.1491

Let's look for some intercepts: are there any intercepts?1499

Can we plug in anything to get an intercept on the top?1505

Well, in our simplified form, 1/(x2 + 1), there are no intercepts that we can get1509

that will be x-axis intercepts, because there is nothing we can plug in to get a 0 to show up in our numerator.1514

But we will be able to plug in 0 for our x, so we can see what the y-axis intercept is.1520

We plug that in, and we get 1/(0 + 1), so we get just (0,1).1526

All right, at this point, we can probably draw in our graph.1532

We don't have a vertical asymptote; we know that -1 is interesting, and we have an intercept; so let's just make it even down the middle.1535

I am not quite sure how much will end up showing up on either side, so let's just draw something to begin with.1545

1, 2, 3, 1, 2, 3, 1, 2, 1; great; once again, every tick mark just means a distance of one--we won't worry about writing in all of those numbers.1554

So, at this point, we probably want a few more points.1574

(0,1)...well, let's see; our horizontal asymptote is just our x-axis.1578

The vertical asymptote...we have no vertical asymptote.1585

We can plot in our intercept at (0,1), so here is a point; and now, let's start looking at some other points.1588

If we plug in positive 1, 1/(12 + 1) gets us 1/2; OK, (1,1/2); here we are.1594

We plug in positive 2: 1 over 22, 4, plus 1...1/5, one-fifth.1605

Plug in 3; with 2, we are at 1/5; we are getting pretty low here; plug in 3: 1 over 32, 9, plus 1, so 1/10; we get one-tenth.1612

Now, we are getting really low, so we get crushed down pretty quickly there.1626

What about on the left side? Well, look: it is x2, and we don't have any other things there.1630

So, it is going to do the exact same thing on the other side, because -1 is going to square to effectively what it would have been if it had been positive 1.1635

-2 will go to positive 3, and -3 to positive 3; so -3 will also be 1/10; -2 will also be 1/10; and -1...wait a second!1641

Domain...x is not allowed to be -1; so this is actually a hole, so -1 is going to be a location where we can see where it would have gone;1653

but we are going to have to denote it with a hole, because it is not actually allowed to show up there.1664

It is allowed to be very close on either side, but at -1 precisely, it is technically forbidden.1668

-1...when we plug that in, we would also get 1/2; but these are both going to be a hole.1673

So, we can plot that point as if it was there; but we will have to plot it with a hole,1678

because there, it is not actually there; it is at that point where it disappears briefly.1682

Momentarily, the function breaks, and it just sort of isn't there.1688

Now, let's plug in the other -2...not 1/10; sorry about that; it should have been 1/5; I wasn't paying attention...there, and then 1/10 here.1692

Great; at this point, we see that we are going to see some sort of curve, like this,1704

where, over time, it gets closer and closer, but then it flattens out; so we will never quite touch that horizontal asymptote.1710

But we will get very, very close...this way, as well...it gets there; it blips out of existence,1717

very, very briefly, for that single location of -1; it blips out of existence, and then it pops right back into existence.1723

And then, once again, it goes back to getting very, very close to that horizontal asymptote, but never quite touching it.1730

So, that is rough; my drawing is not quite perfect--sorry; but that is a pretty good sense of what that would look like.1735

All right, the next one: f(x) = (x2 - 1)/(x2 + 1).1743

Let's factor the top into its factors: we would get (x + 1)(x - 1); and it is divided by (x2 + 1); great.1748

So, our domain...is there anything forbidden in the domain?1760

No, everything is allowed, because with x2 + 1,1764

we can never get any zeroes to show up there; so all of the real numbers are allowed in here.1767

So, there is nothing to cancel out; we don't have to worry about canceling out any factors,1774

because we have (x + 1)(x - 1); (x2 + 1) is irreducible,1777

so it can't break into any linear factors to cancel with those linear factors up top.1780

So, at this point, let's see: are there any vertical asymptotes?1784

Vertical asymptotes would be where the denominator is equal to 0; there are no vertical asymptotes,1788

because x2 + 1...once again, there is nothing we can plug in to make it turn into a 0.1794

What about horizontal asymptotes? There are two...no, sorry, there can only be one horizontal asymptote; my apologies.1798

For the horizontal asymptote, we compare the degrees: squared here, squared here; both a quadratic on the top and a quadratic on the bottom.1807

That means that we are going to see a horizontal asymptote that isn't just the x-axis.1814

So, now we compare, and we see--what are the leading coefficients?1818

We have a 1 here and a 1 here, so we are going to see a horizontal asymptote of 1/1.1821

And since 1/1 just simplifies to 1, we have a horizontal asymptote of y = 1.1827

A height of 1 is what it will eventually, slowly move towards.1833

Let's look at what our intercepts are: can we get intercepts for our y-intercept?1838

Yes, we have no problem plugging in a 0; nothing is forbidden, so we plug in 0.1847

We get 1(-1)...it is probably easier to actually figure it out from this equation up here, so it is -1 divided by +1, or just -1.1851

We have two intercepts when the numerator becomes 0; so at -1 it becomes 0; at positive 1, it becomes 0.1860

And we will probably want a few more points; but let's draw this down so we get a sense of what is going on first.1868

So, once again, we don't have any vertical asymptotes; we have nothing special that we really want to look for.1873

So, let's just center our graph nicely; 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 1, 2; great.1878

So, we can plot in our intercepts; we have intercepts...well, let's first plot our horizontal asymptote, y = 1.1901

We have a dashed line at the height of y = 1; OK.1910

There are no vertical asymptotes; we don't have to worry about that.1918

Intercepts--where are our intercepts? 0 is at -1; -1 is at 0; positive 1 is also at 0.1920

We probably want a few more points; let's see what happens at positive 2, negative 2, and things like that.1929

Let's try those out: at positive 2...we plug that in, and we get 22 - 1,1933

so that is going to be 4 - 1, or 3, divided by 22 + 1; that is going to be 5; so 3/5.1938

Great; let's try 3--plug that in; we get 9 - 1 is 8, divided by 9 + 1 is 10; so 8/10 simplifies to 4/5.1946

And what about if we tried -2 and -3? Well, notice: we have x2 up here.1956

We have x2 down here; so if we plug in a negative, it is going to behave just the same as if we had plugged in a positive.1960

So, we see that -2 will also be at 3/5; -3 will also be at 4/5; great.1966

Let's plot those points: -2 is at 3/5, so 3/5 of the way up...positive 3 (I should have said positive 2, as well, back there)...1976

positive 3 is at 4/5; -2, now, is at 3/5; -3 is at 4/5; and we will see a curve like this,1985

where it gets really, really close, but it will never quite touch that horizontal asymptote.2000

It goes through our points--a nice, smooth curve; it is really close to the horizontal asymptote, but manages to never quite touch it.2007

It is symmetric left and right; great.2014

On to our final example: this one is going to be a little bit complicated.2016

All right, (x3 + 1)/(x2 - 4); and notice, there is also this negative sign--we don't want to forget about that negative sign.2020

So, -x3 + 1, divided by x2 - 4...the first thing: we want to break this into factors, because that is normally our first step.2027

How can we factor the top? Well, x3 + 1...how can we break that down?2037

Well, we notice that if we plug in -1, that is going to turn to 0; so we know that x + 1 has to be a factor.2041

x + 1...now, there is going to be some amount of x2, some amount of x, and some amount of constant.2048

Constant...some x...some x2...we have x3, so it must be that there is only one x2.2056

We have 1 at the end, so it must be a constant of 1 at the end, as well.2062

x times x2 is the only x3 that we will get, so we want only one x3.2066

1 times 1 is the constant that we will get, so we want to make sure that they are both 1.2070

So now, we need something in the middle that will cause the x2's and the x to cancel out.2075

So, if we have x2 here, then we need to have it be -1 here,2079

so that when x times x is here, it will come out as -x2, because we have positive 1x2.2086

So, those two cancel out; so we see that x3 + 1 is the same thing as (x + 1)(x2 - x + 1).2091

That is the same thing on our top; how can we factor our bottom for our rational function?2104

That is going to break into (x - 2)(x + 2).2109

Remember, we had a negative sign out front in the beginning, so we have a negative here still.2113

So, f(x) = -(x + 1)(x2 - x + 1)/(x - 2)(x + 2).2118

Great; that is helpful; but there is something else that we have to do.2126

If we want to get to what the horizontal asymptote or slant asymptote is...let's figure out which one it is.2129

Cubed here; squared here...oh, so our numerator is one degree higher than our denominator; that means we have a slant asymptote.2134

How do we figure out a slant asymptote?--through polynomial division.2141

So, we want to do polynomial division on this: x2 - 4 divides into...now, here is where things get a little bit tricky.2145

We have this negative here once again, so we can't divide into x3 + 1, because then we would have to remember to deal with that negative.2153

So, we can make things a little bit easier on ourselves, and we can distribute that negative.2159

And we will get -x3 - 1--that is the same thing as what was initially there--divided by x2 - 4.2163

So now, we can be safe by doing that instead.2170

x2 - 4....-x3...how many x2's do we have? 0 x2's. How many x's? 0 x's. Minus 1...2172

How many times does x2 go into -x3?2183

It is going to go in -x; -x times x2 is -x3; -x times -4 becomes positive 4x.2184

We subtract this whole thing, distribute the subtraction...-x3 + x3 becomes 0; 0x - 4x becomes -4x.2193

We bring things down; we get 0x2 - 1; so we have a remainder of -4x - 1; and what came out was -x.2204

So, what we have is that f(x) is also equal to (can be written in the form of) -x plus its remainder of -4x - 1, divided by x2 - 4.2220

Great; so now we have been able to figure out what the slant asymptote is; the slant asymptote is this -x right here.2235

Now, we might want to check this, because it is easy to make a mistake with polynomial division, if...2241

well, it is just easy to make a mistake with polynomial division.2245

So, let's check it; let's make sure that this f is the same as the f that we started with.2248

So, we can put this over a common denominator, -x times x2 - 4, over x2 - 4, plus -4x - 1, over x2 - 4.2252

We distribute up here; we have -x3 - 4x; let's combine our two, since they are over a common denominator:2265

x2 - 4 + -4x - 1...the...that was my mistake; -x times -4 becomes positive 4x, so it does cancel out.2273

Positive 4x and -4x cancel out; and we get -x3 - 1, over x2 - 4,2289

which, as we already talked about, is the same thing as -x3 + 1.2297

We can distribute that negative, and we get -x3 - 1; so it checks out--that is good; great.2301

Let's see both of our two ways of looking at this.2307

There is the factored form that we figured, and then there is also the polynomial division form.2309

So, the polynomial division form is necessary, because it gives us our slant asymptote.2316

And the factored form is necessary, because it tells us our vertical asymptotes, and also helps us figure out some other things.2320

And then also, we just initially started with (just so we can still have it on our paper) (x2 + 1), -(x2 + 1), over (x2 - 4).2327

That might be helpful, since it is not that many numbers; it might be helpful for when we actually have to calculate some extra points.2338

OK, what is allowed in our domain? Our domain forbids when x + 2 or x - 2 becomes 0; so that is going to happen at -2 and +2.2344

x is not allowed to be -2; it is not allowed to be positive 2.2358

Where are our vertical asymptotes? Notice that there are no common factors between the top and the bottom.2362

We have (x + 1) and (x2 - x + 1) on the top, and (x + 2) and (x - 2); none of these things have anything in common.2369

They are not the same factor exactly, so we can't cancel anything out.2375

So, we have vertical asymptotes at where we are forbidden for our domain, -2 and positive 2.2379

What about horizontal asymptotes? We figured out that it wasn't a horizontal asymptote.2384

We figured out that it is a slant asymptote, because the degree was one higher on the top; so we will write it as a slant asymptote.2389

But the idea, in either case, is the same thing: what happens in the long term to this function?2397

That is going to be y = -x; the part that, in the long term...this part in the right, that I just circled--it goes to 0.2401

With very large x's, that thing will eventually get crushed down to 0; so we are left with just this thing in the box, the -x.2411

And that is why it is our slant asymptote.2417

And we might want to know the intercepts, just because they are not too hard to graph.2420

So, we will have intercepts: we plug in 0: -(0 + 1)/(0 - 4), so -1/-4 becomes positive 1/4.2423

Let's plug in some other ones; -1 and 0...we can figure that one out, because if we plug in a -1 here, the whole top goes to 0.2439

So, that is an x-axis intercept.2449

So, let's draw this thing in, because it is going to be hard to work with.2452

OK, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8.2466

OK, I am going to mark in a location of 5...3, 4, 5...just so we have a little bit of...1, 2, 3, 4, 5...reference that we can easily find our way around.2493

1, 2, 3, 4, 1, 2, 3, 4, 5, -5, 1, 2, 3, 4, 5, -5; great.2502

We have a little bit of reference there, on our thing.2510

Now, let's draw in our vertical asymptotes at -2 and positive 2.2513

And we have a slant asymptote at y = -x; so what does that look like?2530

It goes at a 45-degree angle; it has a slope of -1, so for every step to the right it takes, it takes a step down.2536

It cuts through...nice and...oops, that was not quite as nice and even as I want it to be.2543

OK, we see a slant asymptote like that, cutting through the whole thing.2552

All right, we can plot our intercepts: 0 is at 1/4, just above, and -1 is at 0.2557

Now, at this point, we say, "Oh, we need a lot more information."2568

We need to plot a lot of points, so I have brought a bunch of points; I figured them out beforehand with a calculator.2571

We don't have enough room to calculate it all by hand, and honestly, it would be kind of boring.2577

But occasionally, you will have to run through this yourself; so just be aware that, when you need more points, you just work through them.2580

We have -(x3 + 1)/(x2 - 4); we can just plot in points and figure them out.2586

So, let's look at what happens as we get close to this vertical asymptote of -2.2592

We plug in -1.5, and we find out that we are at -1.35; so we can plot that point, -1.5, at -1.35.2597

OK, and I am just going to put down a whole bunch of points, and then we will plot them all at once.2612

1; we are at 0.66; 6 is repeating, so it is just 2/3.2617

What about -3? We are at 5.2; at -4, we are also at 5.2; that is interesting.2625

At -5, we are at 5.9; at positive 3, we are at -5.6; at positive 4, we are at -5.42.2633

At positive 5, we are at -6; OK, that should be about enough for us to figure things out.2646

So, at positive 1, we are at 0.66666; we are at 2/3, more accurately.2653

We also might want to know where we are at 1.5; I didn't do that one--oops.2662

But at 1.5...we could figure that out...1.5 cubed, plus 1, 3/2, cubed...that is a little difficult.2670

We know that is going to end up going up here, though, because we are thinking about a number that is just a little under 2,2677

and just a little bit under 2 is going to cause this bottom part to be negative;2684

and it is going to cause the top part to remain positive; so a negative is in front (remember: we can't forget about this negative in front);2689

negative in front; positive on top; negative on the bottom; this cancels out, and we get positive.2695

So, we are going up this way on this side; and when we are at -1.999, just to the right2699

of our vertical asymptote on the left, we know that we are going to be going down,2706

because we see that, at -1.5, we are going down already.2709

If we consider -1.99999, x2 - 4 is once again going to be negative, because it is smaller.2712

x3 + 1...-1.99999...that is going to make a number that is negative on the top, because negative cubed is larger than positive 1.2718

-1 point anything, cubed, is going to be larger than positive 1, so that will be a negative.2731

So, we have a negative on the top, a negative on the bottom, and a negative in front; it comes out to one negative left, so we are down here.2734

What about a little bit to the left, if we were at 2.0001?2740

Then, the x2 - 4 will end up being a positive, because it will be just large enough to beat out that -4.2744

x3 + 1 is still going to be negative, but we have that negative in front, so it cancels, so we will be going up on this side.2751

And over here at positive 2, a little bit over 2, x2 - 4...if we were at 2.0001, that squared, minus 4,2758

is going to be larger than the -4; so it will be positive; x3 + 1...it will remain positive.2767

We have that negative in front, so it will be going down here.2771

Now, we know the directions of all of our asymptotes; let's plot in the rest of our points, and we will be ready to draw this in.2774

1, 2, 3...5.2...5, and just a hair up; -4 is going to also be at 5.2.2780

My graph is a little bit high on the y = x; my graph is not quite perfect--sorry.2797

-5 will be at 5.9; and if we go this way, we are going to go up here.2802

One thing to notice is that there is something odd going on between -3 and -4.2808

If we were to calculate another point (we might want to try -3.5 or -3.2 to get a sense of where it is lower),2816

we would eventually notice that it actually dips down to its absolute lowest minimum around here.2822

We could figure that out precisely, if we had a calculator and a lot of time.2827

3...we plug in 3; we are at -5.6, so a little below...there we are.2831

At 4, we are at -5.42; once again, there is that interesting thing where it will curve back up.2838

It turns out that the minimum is somewhere between those two.2842

At positive 5, we are at -6; once again, my graph of the green slant asymptote isn't quite perfect.2847

Probably, really, my red axes aren't quite exactly the same scale.2858

That is the problem with drawing it all by hand, without having a ruler.2862

But at this point, we are finally ready to graph this thing.2866

We are going to get on this side of the slant asymptote.2868

In the middle part, where we don't have to worry about the slant asymptote, because it is too close to these verticals, we are going to be like this.2873

When we have the slant asymptotes, we will be pulled off this way; and then, we get pulled along the slant asymptote here.2884

We will get closer and closer over time.2893

It is pulled along the vertical asymptote, and then it dips, and then it gets pulled along the slant asymptote.2895

It wouldn't curve away there; that is just my imperfections; there we go.2902

And then, it gets closer and closer to that slant asymptote the whole time.2908

It is pretty tough to draw something this complex; but this is absolutely as complex as you are going to end up seeing2911

in a class or have any homework or tests that have to do with.2917

So, at the worst case, you just end up having to take a lot of points and plot a bunch of points, and you can figure out how this thing behaves.2920

You can figure out where the asymptotes are; you can figure the slant asymptote, horizontal asymptote, vertical asymptote...2926

all of that business...your domain...you can figure out all of these things.2931

And when it comes time to graph, you just have to punch out a lot of numbers, so you can actually see what the picture looks like.2935

But all in all, it is not that hard, if you just make enough numbers.2940

All right, I hope everything there made sense; and we will see you at Educator.com later--goodbye!2943

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