  Vincent Selhorst-Jones

Horizontal Asymptotes

Slide Duration:

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
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
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
18:50
19:04
19:21
20:04
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
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
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
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
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
13:08
13:21
Form of Factored Binomials
13:38
Factoring Examples
14:40
16:58
Factoring Higher Degree Polynomials
18:19
Factoring a Cubic
18:32
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
9:56
11:38
Derivation
11:43
Final Form
12:23
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
22:00
Example 3: Solve for Zeroes
25:28
Example 4: Using the Quadratic Formula
30:52

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
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
26:49
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
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
19:24
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
6:29
Non-Distinct Roots
6:59
Denoting Multiplicity
7:20
7:41
8:55
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
8:25
9:26
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
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
2:08
2:31
2:52
Half-Circle and Right Angle
4:00
6:24
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
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
9:43
Mnemonic: All Students Take Calculus
10:13
13:11
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
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

52m 52s

Intro
0:00
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
11:42
Example of Using a Point
12:41
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
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
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
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
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
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
11:51
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
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
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
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
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

10m 41s

Intro
0:00
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
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
5:44
Which Graphing Calculator is Best?
5:46
Too Many Factors
5:54
6:12
The Old Standby
7:10
TI-83 (Plus)
7:16
TI-84 (Plus)
7:18
9:17
9:19
9:35
10:09
Graphing Calculator Basics

10m 51s

Intro
0:00
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
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
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
5:10
Find Out What Input Value Causes a Certain Output
5:12
For Example
5:24
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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• ## Related Books 1 answer Last reply by: Professor Selhorst-JonesSat Apr 11, 2020 12:40 PMPost by Chessdongdong on April 6, 2020At the very end you accidentally added 20x^2 to 3x^2 and ended up with 20x^2, when it should be 23x^2. 0 answersPost by Scott Yang on September 2, 2019the lesson contents said vertical asymptote, but it is explaining horizontal asymptotes. at 8:09 1 answer Last reply by: Professor Selhorst-JonesSun Oct 5, 2014 11:32 AMPost by Nico Han on October 1, 2014me too! Math suddenly become so interesting to me. I was getting 85, but now I am getting 98. Thanks a bunch! 1 answer Last reply by: Professor Selhorst-JonesTue Nov 19, 2013 5:41 PMPost by Tim Zhang on November 19, 2013such a great teacher!!, I used to got so confused on the math theory my high school told me, because she never told me the thinking process to get these conclusion. Now, I got 100 on my pre-cal all the time, haha. than you so much!

### Horizontal Asymptotes

• A horizontal asymptote is a way of asking what happens to a rational function in the "long run". Is there a vertical location which the function approaches as the horizontal location "slides to infinity"? Symbolically, we can express a horizontal asymptote as
 x → ±∞       ⇒        f(x) → b.
Informally, a horizontal asymptote is a vertical height that the function is "pulled" towards as it moves very far left or right.
• Not all rational functions have horizontal asymptotes.
• To find a horizontal asymptote, expand the polynomials of the numerator and denominator so you have something in the form
 f(x) = an xn + an−1 xn−1 + …+ a1 x + a0bm xm + bm−1 xm−1 + …+ b1 x + b0 .
Notice that n is the numerator's degree and m is the denominator's degree. There are three possibilities:
• If n < m, then there is a horizontal asymptote at y=0 (the x-axis).
• If n=m, then there is a horizontal asymptote based on the ratio of the leading coefficients:
 y = anbm .
• If n > m, there is no horizontal asymptote.
• A slant asymptote is similar to a horizontal asymptote, but instead of approaching a horizontal line, it approaches some slanted line in the "long run". It doesn't settle down to a single value, but it does get "pulled" along the slanted line as it moves very far left or right.
• A rational function has a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator (n = m+1 from the above). We find the asymptote by polynomial division. Break the function into two parts: a portion with no denominator (the asymptote) and the remainder to the division still over the denominator (which goes to 0 in long term). [This method also works to find horizontal asymptotes.]
• Graphically, we represent horizontal and slant asymptotes the same way we did vertical asymptotes: with a dashed line. However, unlike vertical asymptotes, the graph can cross over a horizontal or slant asymptote. Furthermore, there is only ever one horizontal/slant asymptote.

### Horizontal Asymptotes

Find the horizontal asymptote (if it exists) of the rational function.
 f(x) = 3 2x−5
• A horizontal asymptote is the vertical height that the function is pulled to as x goes very far to the right or left. It is the value that the function approaches for very large x (+ or −).
• To find the horizontal asymptote, compare the degree of the numerator to the degree of the denominator. [Remember, degree is the largest exponent on a variable in a polynomial.]
• For this problem, the numerator has a degree of 0, while the denominator has a degree of 1. Because the denominator's degree is larger than the numerator's, in the "long-run" the denominator will grow much larger than the numerator, thus causing the function to approach a value of 0. Thus, the horizontal asymptote is y=0.
y=0
Find the horizontal asymptote (if it exists) of the rational function.
 g(x) = x2−5x+8 x2+12x+42
• A horizontal asymptote is the vertical height that the function is pulled to as x goes very far to the right or left. It is the value that the function approaches for very large x (+ or −).
• To find the horizontal asymptote, compare the degree of the numerator to the degree of the denominator. [Remember, degree is the largest exponent on a variable in a polynomial.]
• For this problem, the numerator has a degree of 2, while the denominator has a degree of 2. Because the numerator and denominator both have a degree of 2, they grow at the same "rate", so we will have a non-zero horizontal asymptote. Next, we compare the coefficient in front of the numerator's lead variable to the coefficient in front of the denominator's lead variable. They both have a coefficient of 1, so the function will approach [1/1] as x becomes very large. Thus, the horizontal asymptote is y=1.
y=1
Find the horizontal asymptote (if it exists) of the rational function.
 h(x) = −12x3+1 3x3−16x2+3x
• A horizontal asymptote is the vertical height that the function is pulled to as x goes very far to the right or left. It is the value that the function approaches for very large x (+ or −).
• To find the horizontal asymptote, compare the degree of the numerator to the degree of the denominator. [Remember, degree is the largest exponent on a variable in a polynomial.]
• For this problem, the numerator has a degree of 3, while the denominator has a degree of 3. Because the numerator and denominator both have a degree of 3, they grow at the same "rate", so we will have a non-zero horizontal asymptote. Next, we compare the coefficient in front of the numerator's lead variable to the coefficient in front of the denominator's lead variable. The numerator has −12, while the denominator has 3, so the function will approach [(−12)/3] as x grows very large. Thus, the horizontal asymptote is y=−4.
y=−4
Find the horizontal asymptote (if it exists) of the rational function.
 f(x) = x2+5 x
• A horizontal asymptote is the vertical height that the function is pulled to as x goes very far to the right or left. It is the value that the function approaches for very large x (+ or −).
• To find the horizontal asymptote, compare the degree of the numerator to the degree of the denominator. [Remember, degree is the largest exponent on a variable in a polynomial.]
• For this problem, the numerator has a degree of 2, while the denominator has a degree of 1. Because the denominator's degree is smaller the numerator's, in the "long-run" the numerator will grow much larger than the denominator, thus causing the function to go towards ±∞. It will not be pulled towards a single value as x goes very far to the left or right. Thus, the function has no horizontal asymptote.
No horizontal asymptote
Find the slant asymptote of the rational function.
 f(x) = x2+5 x
• A slant asymptote is a (non-horizontal) line that the function is pulled along as x goes very far to the right or left. It is the line that the function approaches for very large x (+ or −).
• To find the slant asymptote, we need to re-write the original rational function in a form that allows us to see the slant asymptote. We do this by breaking the function into a part with no denominator (the asymptote) and a part with a denominator where the numerator is smaller (which goes to 0 in the long term).
• We can always use polynomial long division to do this. However, for this problem in specific, it's even easier. We can just break apart the fraction:
 x2+5 x = x2x + 5 x =     x + 5 x
Thus, in the long-run, the part that doesn't go to 0 is the part not over a denominator. Therefore our slant asymptote is: y=x.
y=x
Find the slant asymptote of the rational function.
 g(x) = 4x2−8x+5 2x+3
• A slant asymptote is a (non-horizontal) line that the function is pulled along as x goes very far to the right or left. It is the line that the function approaches for very large x (+ or −).
• To find the slant asymptote, we need to re-write the original rational function in a form that allows us to see the slant asymptote. We do this by breaking the function into a part with no denominator (the asymptote) and a part with a denominator where the numerator is smaller (which goes to 0 in the long term).
• We do so with polynomial long division. [If you don't know or remember polynomial long division, you'll need to watch the lesson on it.]
 2x
 +3
 ⎞⎠
 4x2
 −8x
 +5

•  2x
 −7
 R:  26
 2x
 +3
 ⎞⎠
 4x2
 −8x
 +5
 4x2
 +6x
 −14x
 +5
 −14x
 −21
 26
Through polynomial long division, we have now shown
 g(x) = 4x2−8x+5 2x+3 =     2x−7 + 26 2x+3
Thus, in the long-run, the fractional part goes to 0, leaving us with a slant asymptote of y=2x−7.
y=2x−7
From the graph below, identify the location of its horizontal asymptote. • A horizontal asymptote is some specific value that the function tends to go towards as x becomes very large.
• Look on the graph for the vertical location that the graph approaches as it goes far to the left/right.
• We see that near the left and right edges, the graph appears to be pulling close to a height of y=4, so that is the horizontal asymptote.
y=4
From the graph below, identify the location of its horizontal asymptote. • A horizontal asymptote is some specific value that the function tends to go towards as x becomes very large.
• Look on the graph for the vertical location that the graph approaches as it goes far to the left/right.
• We see that near the left and right edges, the graph appears to be pulling close to a height of y=0, so that is the horizontal asymptote.
y=0
Graph the function below.
 f(x) = 5x2+2x−5 x2+1
• We want to begin by checking to see if the function has any "holes" in its domain and/or any vertical asymptotes. [See the lesson on vertical asymptotes if you are unfamiliar with this.] We set the denominator equal to 0, and look for any solutions:
 x2+1=0
However, x2 never goes below 0, so when we combine that with the +1, we see that there are no solutions to the above equation. Thus the function has no "holes" and no vertical asymptotes.
• Next, let's find the location of the horizontal asymptote (if it exists). Compare the degree of the top to the degree of the bottom. In this case, they're both 2, so we next create a fraction out of the leading coefficients for the numerator and denominator. Thus, our horizontal asymptote is y = [5/1] = 5
• Draw graph axes and plot the horizontal asymptote (y=5) as a dashed line for future use.
• Create a table of values to plot points.
 x
 f(x)
 −6
 4.41
 −3
 3.4
 −2
 2.2
 −1
 −1
 0
 −5
 1
 1
 2
 3.8
 3
 4.6
 6
 5.05
• If you need more points because you're not quite sure how it will behave between points, plot more points as needed.
• As x goes to very large values, the value of the function will tend towards the horizontal asymptote (y=5). Thus, as the graph goes far to the right or left, it will be "pulled" towards the horizontal asymptote. Graph the function below.
 g(x) = 2x2+2x−1 4x−4
• We want to begin by checking to see if the function has any "holes" in its domain and/or any vertical asymptotes. [See the lesson on vertical asymptotes if you are unfamiliar with this.] We set the denominator equal to 0, and look for any solutions:
 4x−4=0
Thus, the domain is x ≠ 4 and we wind up having a vertical asymptote at x=4. [Technically, we don't know for sure yet because we haven't checked to see if the numerator and denominator have common factors, but they won't.]
• Next, let's find the location of the horizontal asymptote (if it exists). Compare the degree of the top to the degree of the bottom. In this case, the numerator has degree 2 while the denominator has degree 1. Thus the function has no horizontal asymptote. However, because the degree is only one greater in the numerator, there will be a slant asymptote. Set up polynomial long division to find the slant asymptote. [If you don't know or remember polynomial long division, you'll need to watch the lesson on it.]
 4x
 −4
 ⎞⎠
 2x2
 +2x
 −1

•  1 2 x
 +1
 R:  3
 4x
 −4
 ⎞⎠
 2x2
 +2x
 −1
 2x2
 −2x
 4x
 −1
 4x
 −4
 3
Through polynomial long division, we have now shown
 g(x) = 2x2+2x−1 4x−4 = 1 2 x + 1 + 3 4x−4
Thus, in the long-run, the fractional part goes to 0, leaving us with a slant asymptote of y=[1/2]x+1.
• Draw graph axes and plot the slant asymptote (y=[1/2]x+1) as a dashed line for future use.
• Create a table of values to plot points. [Notice that you can use the equivalent version of g(x) that we found through polynomial division, because that will make computation a little easier.] We'll want to pay special attention to the values around x=1 because that's where the vertical asymptote is.
 x
 g(x)
 −4
 −1.15
 −1
 0.13
 0
 0.25
 0.5
 −0.25
 0.9
 −6.05
 1.1
 9.05
 1.5
 3.25
 2
 2.75
 3
 2.88
 6
 4.15
• If you need more points because you're not quite sure how it will behave between points, plot more points as needed.
• Remember that we have two asymptotes-the vertical at x=1 and the slant of y = [1/2]x+1. As x gets close to x=1, the graph will shoot off to ±∞ (on the left, down, on the right, up), while as x gets very large, it will be "pulled" along the slant asymptote. *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.

### Horizontal Asymptotes

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:05
• Investigating a Fundamental Function 0:53
• What Happens as x Grows Large
• Different View
• Idea of a Horizontal Asymptote 1:36
• What's Going On? 2:24
• What Happens as x Grows to a Large Negative Number
• What Happens as x Grows to a Large Number
• Dividing by Very Large Numbers Results in Very Small Numbers
• Example Function
• 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?
• Rewriting the Function
• Definition of a Slant Asymptote 12:09
• Symbolical Definition
• Informal Definition
• 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
• Compare the Degrees of the Numerator and Denominator
• How to Find Slant Asymptotes 20:05
• Slant Asymptotes Exist When n+m=1
• Use Polynomial Division
• Example 1 24:32
• Example 2 25:53
• Example 3 26:55
• Example 4 29:22

### Transcription: Horizontal Asymptotes

Hi--welcome back to Educator.com.0000

Today, we are going to talk about horizontal asymptotes.0002

In the previous lesson, we learned about the idea of a vertical asymptote,0005

a horizontal location where the function blows out to infinity, either up or down.0008

Symbolically, we can express this as a vertical asymptote as x approaches some horizontal location, a;0012

and when that happens, f(x) goes to positive or negative infinity.0020

We can flip this idea to the reverse and discuss the idea of a horizontal asymptote,0025

a vertical location which is approached as the horizontal location slides to infinity.0030

As our x becomes very, very large, what vertical height do we go to?0035

Symbolically, we can express it as x goes to positive or negative infinity (as in, x becomes very, very large,0039

either positively or negatively), and f(x) goes to some b, goes to some specific height y = b.0045

To understand this, let's take a look at our old friend from last time, our fundamental function, 1/x.0054

Notice that, as x grows large, we see f(x) shrink down very small; as we go far out, it becomes very, very small.0060

We can see that we are at 1/10 over here and -1/10 over here.0067

We can expand this to an even larger viewing window, and we can get a sense for just how small f(x) eventually becomes.0072

With our y going only from -0.5 to +0.5, we can see that, by the time we have made it to 100, we are at these tiny numbers.0078

We are 1/100 and -1/100, respectively; so we see that becomes really, really, really small, given enough time.0085

So, the farther we go out, this f(x) is going to sort of crush down to 0.0092

We can see this behavior, being sucked towards a certain height, in many rational functions.0098

In 5x/(x - 2), we see that it has this horizontal asymptote at 5; it sort of gets pulled towards a height of 5 in the long run.0102

Over here, with g(x) = (-3x2 + 6)/(x2 + 1), it gets pulled towards a height of positive 3.0112

In fact, it gets pulled really, really quickly.0120

Just like we had with vertical asymptotes, where it never quite touches the asymptote,0122

with a horizontal asymptote, it will not quite actually get to there.0127

It is going to get very, very close to it; and we will see that as we explore why this is occurring in just a few moments.0131

So, we will formally define this behavior in a little bit; and we will name it a horizontal asymptote.0137

But first, let's understand why it occurs: what is actually making this happen?0141

Let's start by investigating f(x) = 1/x: since x is in the denominator, as it grows really, really, really large,0146

there is a giant denominator that crushes the numerator.0152

The numerator just stays still--it just stays at 1; but the denominator gets big--it has x, and so it is able to march out forever.0156

So, as it gets really, really big, it crushes the numerator down to 0 in the long run.0163

So, if we look at the negative side...over here is negative; we have our vertical asymptote at 0,0168

and we are looking at what happens as it slides to the left.0176

We have -0.5, and it is at -2; -1, then -1; but as the numbers get larger and larger...at -5, we are at -0.2; at -10, we are at -0.1.0179

At -1000, we have made it to -.001; and it is just going to keep getting smaller and smaller and smaller.0188

Negative one billion will be a very, very tiny number.0195

Now, notice that there is no number we could plug in to actually get 0; we are just going to get very, very small numbers.0197

These giant denominators are going to make very, very small numbers that will approach 0.0202

We won't actually make it to 0, but we will get really, really, really close to it.0206

The same thing happens on the positive side, if we look at what happens as we go positive.0211

We start looking from our 0, and we go to the right; at 0.5, we are at 2; at 1, we are at 1.0215

At 5, we are at 0.2, and so on; we get that, at 1000, we are at .001.0222

And we as we get really, really large numbers, we will get crushed smaller and smaller and smaller.0227

Since the denominator grows so much faster than the numerator (the numerator isn't moving at all,0232

so it is not even growing at all), the fraction will eventually shrink to 0, as we get very large denominators crushing our numerator.0237

For any rational function, if the denominator's degree is greater than the numerator's degree--0243

that is, if the denominator is able to grow faster than the numerator is able to grow--the rational function will eventually go to 0.0248

If we have x2/(x3 + 5), that is eventually going to get crushed down to 0,0254

because x2 doesn't grow as fast as x3 + 5.0261

So, the x3 + 5 can effectively outrun x2 in the long run, so it will get much larger than x2 will.0265

And so, it will crush the whole thing down to 0.0271

So, if the degree is greater in the denominator (3 versus the numerator's degree, 2), it will eventually get crushed to 0.0273

How do we get rational functions with a horizontal asymptote that isn't at 0, then?0282

Let's look at one: f(x) = 5x/(x - 2); and of the two graphs that we saw, that was the one on the left, the red one that had a horizontal asymptote at 5.0287

If we plug in 1 (once again, this will be the negative side), what happens as we go more and more negative?0295

If we plug in +1, we will get 5(1)/(1 - 2), so we get 5/-1, which is -5.0302

If we plug in -10, we will get -50/-12, approximately 4.17.0314

-100 gives -500/-102, which is approximately 4.90; -1000 gives -5000/-1002, which is 4.99, approximately.0319

So, notice: as we plug in these things, the 5x here and the positive x here,0328

they end up growing at the same rate, other than this multiplicative factor of 5.0334

So, the top grows 5 times faster, precisely 5 times faster, than the bottom does.0339

So, as they go out one way or the other, they are going to end up approaching...the top is growing0344

5 times faster than the bottom is growing, so it is going to end up approaching 5,0350

because when we have very, very large numbers that we are going to put in (eventually, like 1 million),0354

it will be 5(really big number), divided by (really big number); so it will cancel out to 5.0358

We have this other factor of the -2 here; but as the numbers get much, much larger, like 1 million minus 2...1 million hardly notices the -2.0364

It has a slight effect, but it is not much of an effect.0373

And so, as we get down to farther and farther values out, as we get to larger and larger values,0376

it will have less and less of a relative effect, and we are going to get closer and closer to 5.0381

The exact same thing happens if we look at what happens on the positive side.0386

If we start by plugging in 3, we are at 15/1, so we are very different at 3; we are at 15.0390

But when we plug in 10, we are at 50/8; at 100, 500/98; at 1000, 5000/998.0396

We have this difference that becomes less and less impressive; this -2 becomes less and less meaningful.0402

And eventually, it becomes 5(number)/(number), which goes to 5; look at how close we have managed to make it by the time we are at 1000.0408

And this little pattern will just get closer and closer to 5 as we continue this pattern out; we will just get much closer to 5.0417

Once again, we will never touch 5, because we will always be off by this factor of -2; we will always be slightly imperfectly equal to 5.0424

So, it won't ever actually equal that horizontal asymptote precisely.0432

But it is going to get arbitrarily close to it; it is going to get really, really, really close, until we are dealing with numbers like 5.00001.0435

And so on, and so on...we will eventually be able to get to any arbitrary closeness we want, as long as we look at an x large enough.0442

Once we get far enough from the vertical asymptote at x = 2, we see that the numerator and denominator grow at a constant ratio, 5x and x.0449

So, for any rational function, if the degrees of the numerator and the denominator are equal,0456

we will get a horizontal asymptote that isn't equal to 0; we will get a horizontal asymptote at some height.0461

Notice: the 5x here and the x here have the same degree, a degree of 1.0466

So, since they have the same degree on the top and the bottom, they are going at the same rate, in a way.0471

Other than that multiplicative factor of 5, they are running in the same scale of growth.0476

So, since they have the same scale of growth, they are going to grow around the same rate,0481

which means it is only that multiplicative factor that is going to determine the height that it ends up at.0485

A horizontal asymptote is a horizontal line y = b, where, as x becomes very large (positive or negative), f(x) gets arbitrarily close to b.0490

Symbolically, we show this as "x goes to negative infinity" or "x goes to positive infinity"0505

means that f(x) will go to b; we are going to get to this height; we are going to move toward this height, surely, steadily.0510

It might not get perfectly to b; in fact, it almost certainly won't, as what we were just talking about.0520

But it will get really, really close to b; it will get arbitrarily close to it.0524

Informally, we can think of a horizontal asymptote as a vertical height that the function is pulled towards, as it moves very far left or right.0527

Over the long term, it will start somewhere else, but it gets pulled, in the long term, to a certain height,0535

until it gets really, really close to this horizontal asymptote.0540

We can take this idea and go beyond just having a horizontal asymptote.0544

Consider f(x) = (x3 - 1)/x2; as x gets large, we see f(x) grow very close to the line y = x.0548

We can see that on that dashed orange line going through the y = x line.0557

We see how close it becomes; in fact, it grows so close, so quickly, that we almost can't tell the difference between the two on the far parts in this graph.0561

It will never be perfectly the same, because we have this -1 here; but it will become really, really close to it.0569

This idea is similar to a horizontal asymptote, but it is no longer horizontal.0577

Since it is at a slant, we can't call it a horizontal asymptote; so instead, we call it a slant asymptote.0581

Sometimes, it is also called an oblique asymptote.0586

So, what is going on--why do we see this?0589

Once again, we are trying to consider what happens to the function in the long run.0591

The idea of all this horizontal asymptote/slant asymptote stuff is a question of what happens to this function0594

as we look at very large x--as we go really far right/as we go really far left.0600

In this case, we could plug in large numbers to see what happens; but that will slightly obscure some details for other slant asymptotes.0604

So, instead, what we want to notice is that we can rewrite the function.0612

If we have (x3 - 1)/x2, we could say, "Look, we can divide out the x3,0616

and we can break our fraction apart so that we get x3/x2 - 1/x2."0623

And so, the x3 and x2 cancel down to just x - 1/x2.0627

So, by using division, we can see this function in a new way.0632

In this form, it is clear that, as x goes to positive or negative infinity, this part right here,0636

since it is 1/x2, is just going to sort of get crushed down to 0 by its much larger denominator.0641

But the x here will end up just continuing on; it will just keep moving forward.0646

So, in the long run, as we get to very large x's, this part here goes to 0; but x continues going out.0651

So, effectively, the function will become just x.0658

We can also get this (x3 - 1)/x2...we can break it into this format0662

through the process of polynomial long division, which will be necessary0666

when we have slightly more complicated denominators that we are dealing with.0668

So, you write this as this, minus 1; so in that way, we would have x2.0674

How many times does x2 go into x3?0683

It goes in just x; x times x2 gets us x3.0684

We have nothing else, so we subtract by x3; that gets us 0.0688

We bring everything down; and so we have 0x2 + 0x - 1, which leaves us with just a remainder of -1.0692

We have this remainder of -1; so (x3 - 1)/x2 is equal to x, plus the remainder.0702

The remainder is -1; and then we put it back over our original denominator, the thing that we divided by.0710

We get x plus -1 over x2, which is the exact same thing that we had down here on this line.0717

We can get this through polynomial division, which will be necessary when we are dealing with slightly more complicated denominators.0724

We can also define a slant asymptote similarly to how we defined a horizontal one.0730

It is also called an oblique asymptote sometimes.0734

A slant asymptote is a line y = mx + b (remember, mx + b is just our normal slope-intercept form for a line),0737

where, as x becomes very large (positive or negative), f(x) gets arbitrarily close to that line, y = mx + b.0744

Symbolically, we can show this as: as x approaches negative infinity (very large negative values),0751

or x approaches positive infinity (very large positive values), f(x) will go to mx + b.0755

f(x) will approach just being the same as this line, mx + b.0761

Informally, a slant asymptote is a non-horizontal line that the function is pulled towards as it moves very far to the left or to the right.0765

So, we might start at different heights; but as we get farther and farther to x, we end up getting pulled along this line, this slant asymptote.0774

We can even go beyond the idea of a slant asymptote.0784

Really, the question we have been working on can be phrased as "What does the function look like in the long term?"0786

What is the long-term behavior of this function?0792

So far, we have answered that with horizontal and slant asymptotes.0795

But a function could also tend to a curve; it could tend to anything, really.0798

If we had (x4 + 17x + 20)/(10x2 - 10x - 20), well, notice:0802

we could make that as squared and to the fourth up here.0807

So, the degree of the top is one, two steps larger than the degree on the bottom.0812

That means that, over time, it is going to effectively be the same as x2 coming out of that, x4 divided by x2.0818

We are going to get something that looks kind of like an x2, which is a parabola, which is exactly what we see here.0826

As we get to very large x-values, we see it get pulled along this curve.0832

Notice this curve that we have of a parabola through here.0837

Now, it will behave differently when we are at the vertical asymptotes, because we have vertical asymptotes x - 2 and x + 1.0841

So, there are vertical asymptotes at -1 and vertical asymptotes at +2.0847

We will get pulled into these vertical asymptotes in various ways.0853

But in the long run, as it gets to very large x-values, it gets pulled into this parabolic shape.0857

In fact, if we were to divide this out through dividing the bottom into the top through polynomial division,0863

we would be able to find that it eventually is approaching the parabola 1/10 times x2 plus x plus 3.0870

And that is why we see that parabolic curve right there--pretty cool.0876

That said, we are not going to really see this in this course, or probably in any other course that you are taking right now.0880

While this shows us an interesting idea, don't expect to see this in a normal class.0885

Few textbooks and very few teachers will discuss anything beyond the idea of a slant asymptote.0888

As such, we will not be exploring the idea any further in this class, either.0893

However, it is useful to notice how all of these ideas have been linked.0896

They are about answering: "Where does this go in the long term? What is happening eventually to my function?0899

How will it behave when I look at very large values being plugged in?"0906

We can get a sense of this by thinking about what happens to the function as the numbers get larger and larger and larger.0910

What will happen--how, in what general way, will this function behave when we are plugging in x that is a million, a billion, a trillion--a really, really big x?0915

That is what all of these ideas in this lesson have been about.0924

What happens as x becomes very, very large--how does this thing behave?0927

It could behave in these non-slant asymptote things where it pulls into a parabola or some other polynomial shape.0931

But we are going to restrict ourselves to just horizontal and slant asymptotes, since that is what most other courses look at.0936

And they are also the easiest for us to approach right now.0941

Horizontal/slant asymptotes and graphs: Just like their vertical cousins, it is customary to show horizontal/slant asymptotes with a dashed line.0945

We normally use a dashed line to show, "Look, here is an asymptote."0953

So, if we had (4x4 + 3x3 + 10x2)/(2x4 + 1), we would get this graph over here on the right.0957

And notice: it has a horizontal asymptote at 2, and so, over the long run, our graph gets pulled towards this.0964

Now, notice that, in the middle, it has a behavior that is totally different.0972

It has this interesting behavior in the middle.0976

Unlike vertical asymptotes, the graph can cross the horizontal asymptote.0978

It is allowed to actually cross over that horizontal or slant asymptote.0983

Furthermore, there is only ever one horizontal or slant asymptote.0989

You can't have multiple horizontal/slant asymptotes, in the same way you can't have a vertical asymptote.0993

In any case, over the long run, the graph will be pulled along the asymptote.0998

That is the idea of an asymptote to really get across: that an asymptote is about the function eventually being pulled along it,1002

or, if it is a vertical asymptote, being stretched up along it vertically.1008

How to find horizontal and slant asymptotes: a horizontal or slant asymptote tells us how a function behaves in the long run; that is the idea here.1013

It is fairly easy to determine if a function has a horizontal asymptote, and, if so, what it is.1021

We will see a method for that first.1025

Finding a slant asymptote is a little bit trickier, though, and we will look at its method second; but it is not that difficult.1027

Any rational function is in the form n(x)/d(x), where they are both polynomials.1033

We can expand these polynomials into their normal form, our _xn + _xn - 1 + _xn - 2,1038

all the way until we eventually hit a constant; and the bottom one will be the same thing--some other blank.1047

So, an will be the blank on the top; bn will be the blank on the bottom.1052

And we have two different things; n will be the numerator's degree, and m will illustrate the denominator's degree.1056

There are going to be three possibilities: first, if n is less than m, then there is a horizontal asymptote at y = 0.1067

Why is that the case? Well, if n is less than m, then that means we have a big denominator, compared to the numerator.1075

The numerator will have a smaller degree; so it is going to grow slower,1083

which means that the bottom one will eventually grow large enough to crush the numerator.1087

So, we are going to crush it down to y = 0.1091

So, if we have a numerator degree that is less than the degree of the denominator,1094

the denominator will eventually grow large enough to crush the whole fraction down to 0.1100

If n is equal to m--if they are the same degree--if we have the same degree--then what we are going to see is:1105

we will see a horizontal asymptote that is based on the ratio of the leading coefficients.1114

We will get a horizontal asymptote, but it will no longer be set at y = 0.1118

Since xn and xm are basically the things that, in the long run,1121

are really going to determine how these polynomials are, and n = m,1125

then ultimately it is going to be anxn/bmxm,1129

in the long run, since xn and xm are the same value.1133

Since those things are at the front, they are going to really determine how the polynomial works in the long run.1137

And they will just cancel each other out, because we will have anxn/b...1142

and I will call it n as well, since we have n equal to m...xn.1147

Well, in the long run, what is effectively going to happen is that we will see these two things cancel each other.1151

And we will be left with just the ratio of the leading coefficients.1155

That will determine what the horizontal asymptote is going to go to--this ratio.1161

What is the leading coefficient on the top, divided by the leading coefficient on the bottom?1166

What is our first coefficient here and our first coefficient there?1170

Finally, if n is greater than m, if we have the numerator degree bigger, then there is no horizontal asymptote,1173

because the numerator is able to run faster than the denominator.1185

And so, it is able to escape the clutches of the denominator and actually keep going on to growing forever, and getting less and less.1189

It depends on how it is set up, specifically; but we will be able to have freedom on both the right and the left side,1195

as it manages to have very large values, because it will be able to outrun the denominator, because it has a larger degree.1200

All right, so let's talk about how to find slant asymptotes.1205

A rational function has a slant asymptote if the degree of the numerator is exactly 1 greater than the degree of the denominator.1208

So, in the terms we were using before, where n was the numerator's degree, and m was the denominator's degree, it would be n = m + 1.1215

We can find the asymptote through polynomial division.1222

For example, if we have (3x3 - 2x2 + 7x + 8)/(x2 - 3x), we say,1225

"Oh, look: there is a 2 in the denominator; there is a 3 on the numerator; is 3 = 2 + 1, the same thing as what is going on up here."1230

So, we are exactly 1 greater in the degree of the numerator than we are in the denominator.1240

So, we are going to have a slant asymptote.1247

At that point, we use polynomial division; so let's see how polynomial division would work here.1250

We have x2 - 3x as dividing into...what is in our numerator? 3x3 - 2x2 + 7x + 8.1254

So, how many times does x2 go into 3x3?1268

It is going to go in 3x; 3x times x2 gets us 3x3, so yes, we were right.1271

3x times -3x gets us -9x2; then we subtract this whole thing, so let's distribute that negative:1277

minus 3x2...that becomes + 9x2; so 3x3 - 3x3 is 0.1284

-2x2 + 9x2 is positive 7x2.1291

The next step: bring down the 7x, so + 7x; how many times does x2 go into 7x2? It goes in + 7 times.1295

So, 7x2...that checks out; 7 times -3 is -21x.1305

We subtract that whole thing and distribute our negative: minus, plus...7x2 - 7x2 is 0; 7x + 21x is 28x.1311

Bring down the 8; we get + 8 here; at this point, we see 28x--how many times can x2 go into 28x?1321

It can't go in anymore because of our degrees; so we have a remainder of 28x + 8.1330

And so, notice how these are the same thing: 3x + 7 is what we got as the result, 3x + 7 here.1337

And then, plus our remainder, 28x + 8, divided by what we started doing our division with...1344

that is how we get polynomial division; and indeed, 3x + 7 + (28x + 8)1353

divided by (x2 - 3x) is what our initial rational function is equal to; so we can break it down.1358

Break the function into two parts: a portion with no denominator (the portion with no denominator is 3x + 7--it doesn't have a denominator).1368

So, that is what our slant asymptote is, because 3x + 7 describes a line that is in the form mx + b.1377

So, 3x + 7 is a line, and the remainder to our division is our 28x + 8; it goes over the denominator x2 - 3x,1388

back over the denominator that we started with; and notice, (28x + 8)/(x2 - 3x), because it is the remainder...1404

the remainder is always going to have a degree less than what we started dividing with.1410

So, we have 28x1 divided by (x2 - 3x); so we have a bigger degree on the bottom than we do on the top.1414

In the long run, the denominator is going to crush the numerator; and this whole thing will go to 0, and we will be left with just 3x + 7.1421

So, in the long run, we will end up having a portion that has no denominator, and the portion that has a denominator,1429

because the degree on the bottom is now less after polynomial division--it will go to 0 in the long term.1435

Now, this method of polynomial division will also work to find horizontal asymptotes.1441

So, you can also use this method if you want to find the horizontal asymptote.1446

It is just that, instead of getting a line here, you will just get a constant--it will just be a constant value,1449

if it is a horizontal asymptote, as opposed to a slant asymptote.1458

So, you can also use polynomial division to find horizontal asymptotes.1463

But we have that other method for finding horizontal asymptotes that was pretty fast.1466

So, it is normally easiest to just use polynomial division when you want slant asymptotes.1469

All right, let's go over some examples.1474

f(x) = (10x5 + 3x4 + 8x2 - 2)/(2x5 + 27x3 + 12x).1475

Is there a horizontal or a slant asymptote?1483

What we do is compare the degree on the top to the degree on the bottom.1485

They are the same degree; so if they are the same degree, we have a horizontal asymptote.1489

Now, if we want to figure out what the horizontal asymptote is, well, we will figure that out.1497

And we just look at its ratio of leading coefficients; what is the leading coefficient for the top and the bottom?1502

The top has a leading coefficient of 10; the bottom has a leading coefficient of 2; we simplify that, and we get 5.1511

So, 5 is what our horizontal asymptote will be; so y = 5 is the horizontal asymptote.1521

Great; once we see that the degree on the top and the degree on the bottom are the same,1531

we know we have a horizontal asymptote that is not just going to be a 0.1539

And now, we figure it out by looking at the ratio of the leading coefficients,1542

because ultimately, the ratio of the leading coefficients on our biggest exponent, x's,1545

is going to be what determines what happens to these functions in the long run.1550

The next one: Using the graph of the function, determine a.1554

We have (12x3 + 5x2 - 10x + 8)/(ax3 + 2x - 2).1558

Now, we notice that there are 3 and 3; and here is a horizontal asymptote.1564

We have a horizontal asymptote, because the degrees are the same.1571

Also, we can see that we have a horizontal asymptote in our graph,1575

so it had better be the fact that the degree on the numerator and the denominator is the same.1579

At this point, we know...what is our horizontal asymptote? It is y = 3.1582

We see that by looking at where it goes; it cuts evenly between the 2 and the 4, so it must be at y = 3.1587

So, if that is the case, we know that the ratio of the leading coefficients, 12/a1592

(the leading coefficient on top is 12; on the bottom, it is a), must be equal to 3, our horizontal asymptote.1598

We work this out: 12 = 3a; divide by 3, and we get 4 = a, so a is 4; great.1607

We have (28x3 + 110x2 - 47x + 55)/(0.2x4 - 5x2 + 4).1616

Is there a horizontal or a slant asymptote? What is it?1624

In this case, the first thing, as always: we look to see what are our degrees.1627

The numerator degree is 3; the denominator degree is a 4.1631

So, if the denominator degree is larger, it crushes everything.1635

It is going to be able to eventually, in the long run, overtake the numerator.1639

It will run faster than the numerator and grow larger, and it will crush everything to 0.1643

So, there is a horizontal asymptote, but it is going to be the most boring horizontal asymptote of all,1647

but at the same time, kind of interesting: y = 0.1652

The denominator crushes that puny numerator; that numerator is just too small,1657

so the denominator crushes the puny numerator, because it has a larger degree.1663

And I want to point out that it seems at first...well, we have .2x4 and 28 times x3;1668

and there are also...110 and these other big numbers up here on the top.1674

But on the bottom, it all seems like pretty small, insignificant numbers.1678

So, why is it that the bottom is going to be bigger?...because it has a larger degree.1681

Ultimately, how a polynomial behaves in the long run is really determined by that degree.1686

The coefficients have effects; they affect things; but being really dominant is just determined by having the biggest degree.1691

x4, no matter how small that coefficient at the front is,1699

will eventually be able to outrun x3, no matter how big its coefficient is at the front.1702

So, that x4 will crush the numerator, because the numerator only has a degree of 3.1707

All right, the final example: at this point, we have (-4x4 + 7x3 + 23x2 - 43x + 5)/(x3 - 5x).1712

We are asked, "Is there a horizontal or slant asymptote?" and then, "What is it?"1720

The first thing we do is look at our degrees: we have 4 on the numerator and 3 on the denominator.1723

So, that means that the degree of the numerator is exactly 1 greater than the denominator.1728

If it is 1 greater, then that means we have a slant asymptote; and that makes sense,1741

because one degree larger than something else...if we divide them out...x50/x49...1747

that is going to become something along the lines of x.1753

So, it is going to have a nice linear form; it is going to become a line.1755

Degree 1 is a linear function, so that is why we see a slant asymptote, a line, coming out of it.1759

How do we figure it out? We figure it out by using polynomial division.1765

So, (x3 - 5x)...and notice: that is 5x1, not 5x2;1769

that is going to have a slight effect when we are doing our division.1775

-4x4 + 7x3 + 23x2 - 43x + 5.1779

x3: how many times does it go into -4x4?1791

It is going to go in -4x times, because -4x times x3 gets us -4x4.1793

-4x times -5x gets us 20x2; but the next thing we have is 7x3, so that is not going to go on the 7x3.1800

It is going to go on the 23x2 column.1808

-4x times -5x gets us 20x2; so 20x2 lines up there.1812

And it is positive; so at this point, we subtract by this amount.1818

So, we distribute our negative; that becomes positive; that becomes negative.1821

-4x4 + 4x4 becomes 0 (what it should be when we are doing polynomial division--the first part should always cancel to 0).1825

23x2 - 20x2 gets us positive 3x2.1831

The next step: we bring down the other things that we will end up using.1836

We bring down the 7x3; we bring down the -43x.1841

How many times does x3 go into 7x3? It will go in 7 times.1848

7 times x3 gets us 7x3; 7 on -5x gets us -35x.1851

We subtract by this amount and distribute our negative; it will give us a positive, so 7x3 - 7x3 gets us 0.1859

-43x + 35x gets us -8x; bring down 3x2; bring down 5.1867

We have 3x2 - 8x + 5; at this point, can x3 go into 3x2? No.1876

x3 has a larger degree than 3x2, so we are left with a remainder of 3x2 - 8x + 5.1883

We are left with that remainder; so at this point, we know that our original function, f(x)...1893

through division, we have just shown that it is the same thing as -4x + 7,1897

plus the remainder, 3x2 - 8x + 5, divided by our original denominator, x3 - 5x.1903

So, our answer for what our slant asymptote will end up being is going to be the part in the front, -4x + 7; that is our slant asymptote.1915

Now, we can also check our work, at this point, by just making sure that if we combine these, we get back to our original function.1926

So, let's put them over a common denominator: we have (-4x + 7) times (x3 - 5x), over (x3 - 5x).1933

So, what will that end up being? This is just the same thing here.1948

So, -4x times x3 gets us -4x4; -4x and -5x get us +20x;1951

+7 times x3 gets us 7x3; 7 times -5x gets us -35x; all over x3 - 5x.1962

And we can add on our thing from our other one, because they are now on a common denominator.1974

+ 3x2 - 8x + 5: what does that end up becoming?1978

We are starting to run out of space, so I will do this vertically.1983

We have -4x4 right here; great, that checks out.1986

20...next, x3; 7x3...we have no other x3, so + 7x3; that checks out.1994

3x2...any other x2's?...oops, I accidentally didn't write the squared; -4x times -5x became positive 20x2.2002

So, we have + 3x2 and + 20x2; it becomes + 23x2; that checks out.2010

-35x - 8x...that becomes - 43x; that checks out.2017

And then, + 5...+ 5, and that checks out, because this whole thing is still over that denominator of x3 - 5x.2024

We have the exact same thing that we started with, so what we just did checks out.2035

So, we know for sure that -4x + 7 is good; it is definitely our answer.2041

All right, we will see you next time when we talk about graphing rational functions in general,2045

being able to use these vertical and horizontal asymptotes together to be able to quickly make the graphs to these kinds of functions.2049

All right, see you at Educator.com later--goodbye!2055