  Vincent Selhorst-Jones

Graphs

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 0 answersPost by Hong Yang on September 25, 2020do you still check comments? 1 answerLast reply by: ChessdongdongFri Mar 20, 2020 12:25 PMPost by Chessdongdong on March 20, 2020For Example at 2:47 you wrote (-4,3) at the bottom instead of (-4,-3) at the bottom of the slide. Also, great lecture! 1 answer Last reply by: Professor Selhorst-JonesThu Jul 27, 2017 11:55 PMPost by Macy Li on July 27, 2017Do you still check these comments? 3 answersLast reply by: Kitt ParkerMon Feb 26, 2018 5:51 PMPost by Mohammed Jaweed on June 28, 2015you go way too fast! Please slow down. 1 answer Last reply by: Professor Selhorst-JonesTue Dec 23, 2014 11:33 AMPost by Jason Wilson on December 23, 2014In referencing example: Â  Is there always an assumption in problems like this that the -number is in parenthesis? Like this (-1)sq2 Â  Technically -1 squared is -(1 * 1) isn't it? THX in advance.[Edit: To clarify my previous question, is it assumed that the x squared is in parenthesis like this (x sqroot2) so if the value of X comes back negative, that negative number is inside the parenthesis like this(-x)squared (-1)squared ? That negative sign has to be in the parenthesis right?] 0 answersPost by Saadman Elman on September 20, 2014I saw your comment. Thanks for clarifying. Make sense now. 1 answer Last reply by: Professor Selhorst-JonesThu Sep 4, 2014 11:24 PMPost by Saadman Elman on September 4, 2014Example no.2, around 49 min ----50 min, The question was Estimate the X-Value where f(x)= -3. Your answer was x=-3.2, 0.6,1.7. You forgot to mention that in f (x) = -3 ; 0 is also an X-value for f(x)= -3. I spend a lot of time deeply thinking about it. I feel like it was just a subtle mistake. Please let me know via email if i am write or wrong and please explain your opinion. My email is XXXXXXXXXXXXX. I don't check the comment here. Â Over all, It was a great lecture. Appreciate it. 1 answer Last reply by: Professor Selhorst-JonesSun Aug 11, 2013 2:18 PMPost by Tami Cummins on August 10, 2013One thing I really wish Educator.com had is printable worksheets. 2 answers Last reply by: Professor Selhorst-JonesSun Sep 29, 2013 6:51 PMPost by Abdihakim Mohamed on July 4, 2013Never mind I hastened you corrected already. you are the greatest math teacher I have seen so far. 2 answers Last reply by: Professor Selhorst-JonesThu Sep 4, 2014 9:14 PMPost by Abdihakim Mohamed on July 4, 2013First example marking -1 I get as a function as 5 and not as 1Because -1squared is 1 and -3 times -1 is 3 plus 1 equals 5

### Graphs

• A graph visually represents a function or equation in math. It gives us an intuitive picture of how the function "works".
• There are two main ways to interpret what a graph means:
• Input ⇒ Output: The graph tells us what happens to each input value. "If I plug in some number for x, where will it go?" The input values are on the horizontal, the outputs are on the vertical.
• Location of Solutions: We can also interpret a graph as the location of all solutions to the equation. The graph of an equation is made up of all the points that make the equation true.
Between these two options, it's usually best to interpret it as the first one: how inputs are mapped to outputs. This gives us an intuitive way to see what happens as we change input values. However, the other way will occasionally come in handy, so don't forget about it entirely.
• Pay attention to the axes! The axes tell you where the graph is and what scale it has. Knowing this is important if you want to interpret what the graph means. [This is also called the graphing window.]
• In this course, we will not put arrows on the ends of our graphs. Instead, we assume we're all aware the graph keeps "going" past the edge. We won't use arrows because we know that most graphs are just a tiny window on a much larger function. [Caution: Some teachers might still want you draw arrows on the ends of your graphs. If that's the case, do what they say as long as you're in their class.]
• The easiest way to plot graphs is to plot points one-by-one. Make a table of values, calculate various inputs and outputs, then plot them on the graph. Once you have enough points to see the shape, draw it in.
• Almost always, the plotted points will connect with curves. As you see more and more functions, you'll start to learn the various shapes. Use this knowledge to help you draw graphs accurately.
• Anytime you're not sure how to draw in a graph, just plot more points. As you plot more points, you have more information. As you have more information, the picture becomes easier to see. This is always an option, even for the most confusing graphs.
• We can tell if a graph is the graph of a function with the Vertical Line Test. If a vertical line can be drawn that crosses the graph at more than one point, it is not a function. Why? Because this means a single input is mapped to two outputs, so it can't be a function.
• The domain of a function is all the inputs that a function can accept. Thus, every point on the x-axis that the graph is above or below is in the domain. However, if you can draw a vertical line on an x-value and it does not cross the graph, then that x is not in the domain. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
• The range of a function is all the possible outputs a function can create. Thus, every point on the y-axis that the graph is left or right of is in the range. However, if you can draw a horizontal line on a y-value and it does not cross the graph, then that y is not in the range. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
• If you haven't already noticed it, this is a great time to point out that this course has an appendix that's all about graphing calculators. Check out the appendix to learn more about graphing calculators, where you can find some free options, what they're good for, and how to use them.

### Graphs

What are the two main ways to interpret what a graph represents? Of those two ways, how do you normally want to interpret a graph?
• The most common way to make a graph is by creating a table of input values and corresponding output values. This is because a graph shows how input is transformed to output: given some x as input, what f(x) or y do we get as output?
• Alternatively, if we take any point on a graph and plug it in to the equation that created it, the equation will be true. Thus, every point on a graph is a solution to the equation: where the equation is true.
• In general, the first way is more useful for creating graphs and for understanding how they work. While the second way has its uses, it is usually more helpful to think in terms of input ⇒ output.
The two ways to interpret a graph are 1) How input (x-axis) is mapped to output (y-axis); 2) the location of all solutions to the equation (where it is true). Normally you want to think in terms of input ⇒ output.
Consider this graph of a function. Why does the graph stop at the edge of the graphing axes? [The space created by the axes for graphing is sometimes called the `graphing window'.] What happens beyond that edge? • For most functions, the function goes on forever. We can put in arbitrarily large numbers, but how could we possibly create a graph that shows every single number? We have to stop somewhere, so we usually stop at the edge of the axes.
• However, once again, most functions keep going beyond the edge of the axes. We had to stop because we can't draw forever, but the function keeps going past the edge of the graphing window.
• Just because we see only a portion of the graph does not mean the function stops at that edge (unless we are explicitly told it does stop). It is up to us to have a sense for how it behaves beyond that edge if we need to know what happens outside the graphing window.
The graph stops at the edge of the graphing axes (graphing window) because it has to stop somewhere. Most functions go on forever, so the graphing window is just a small window on a much larger thing. Beyond the edge, the graph continues on (unless we are explicitly told otherwise).
The following table of values is for the function f(x) = 2x2−8. Using the table of values, graph the function.
 x
 f(x)
 −3
 10
 −2
 0
 −1
 −6
 0
 −8
 1
 −6
 2
 0
 3
 10
• Begin by choosing an appropriate scale for the axes by considering the values in the table. The most extreme x-values are −3 and 3, so a scale of x:[−4, 4] would probably be good. The most extreme y-values are −8 and 10, so a scale of y:[−10,10] would probably be good (that way we have the same amount above and below the x-axis).
• Once the axes are set up, plot all the points. Remember, we interpret the table as a bunch of ordered points ( x,  f(x) ). The first coordinate is plotted on the horizontal axis, and the second coordinate is plotted on the vertical axis.
• Connect the points with curves. Notice how the vertical change increases more and more the farther we get away from x=0. Thus, the curves around x=0 are more gently sloped, while they become steeper as they get farther out. Graph the function f(x) = 0.5x + 2.
• Begin by creating a table of values. You could choose any x values to use, but it would probably be best to keep them small and near 0, since we probably want the graph to be centered around the origin.
• Creating the table of values, we get something like
 x
 f(x)
 −3
 0.5
 −2
 1
 −1
 1.5
 0
 2
 1
 2.5
 2
 3
 3
 3.5
Looking at the values, you might realize that you're graphing a line. You might also have realized it would be a line just from reading the function.
• Choose appropriate axes. In this case, square axes of [−5,5] on both sides would work nicely, but you could choose something else as well.
• Plot the points, connect them appropriately. This is particularly easy because the function is a line. Graph the equation x2 +y = 5.
• First, we need to put the equation into a format that we can easily graph: y=stuff. Moving things around in the equation, we obtain:
 y = 5 − x2.
• Create a table of values. You could choose any x values to use, but it would probably be best to keep them small and near 0, since we probably want the graph to be centered around the origin.
• Creating the table of values, we get something like
 x
 y
 −4
 −11
 −3
 −4
 −2
 1
 −1
 4
 0
 5
 1
 4
 2
 1
 3
 −4
 4
 −11
From the values and the function, we realize we are graphing a parabola. Knowing this will help us curve the graph appropriately.
• Choose appropriate axes. x:[−5,5] and y:[−10,10] would work well for this, although you could choose different axes.
• Plot the points, connect them appropriately. Notice that (−4,  −11) and (4,  −11) will be just outside the bounds of the graphing window. That's okay. You can plot these points just past the edge and have the graph pass out of the window, or when drawing the graph, you can just know what the curve is heading towards. Graph the function f(x) = √{|x+2|} − 2.
• Begin by noticing that the function has an absolute value wrapped in a square root. If you forgot that the vertical bars |  | represent absolute value, you'd want to begin by looking up what they mean. You'd find out that it turns whatever is inside of them positive.
• Create a table of values. You could choose any x values to use, but it would probably be best to keep them near 0, since we probably want the graph to be centered around the origin. Furthermore, we will get the "nicest" points when the value inside of the square root lines up with a perfect square. You might want to choose your x-values accordingly.
• Creating the table of values, we get something like
 x
 f(x)
 −18
 2
 −11
 1
 −6
 0
 −3
 −1
 −2
 −2
 −1
 −1
 2
 0
 7
 1
 14
 2
From the values and the function, we realize we are graphing something with the shape of a square root on both sides. Knowing this will help us curve the graph appropriately.
• Choose appropriate axes. Since the x-values get so large, we probably want to choose something like x:[−20,20]. For the y-axis, we could choose lots of things, but let's go with y:[−10,10] to show off how little height the function gains.
• Plot the points, connect them appropriately. Notice that our last points don't make it the edge of the graphing window. That's okay. Just keep drawing the graph out with the correct curve. You have enough information from the points you've seen so far to know how it will continue to curve. [If you want more information though, you can always add entries to your table of values.] Is it possible for this graph to be the graph of a function? • To check if a graph could be the graph of a function, we use the Vertical Line Test.
• The Vertical Line Test says that if a vertical line could intersect more than one point on a graph, it can not be the graph of a function.
• If we place a vertical line almost anywhere on this graph, it clearly cuts the graph in more than one place. Thus, it fails the Vertical Line Test and cannot be a function. (This is because the function would need one input to be mapped to two outputs, which a function is not allowed to do.)
No, because it fails the Vertical Line Test.
Is it possible for this graph to be the graph of a function? • To check if a graph could be the graph of a function, we use the Vertical Line Test.
• The Vertical Line Test says that if a vertical line could intersect more than one point on a graph, it can not be the graph of a function.
• While it may be hard to see at first, if we place a vertical line anywhere on this graph, it only cuts the graph once. There is no place we could place a vertical line that would cut it twice, so it passes the Vertical Line Test, so it could be a function.
Yes, because it passes the Vertical Line Test.
Assume that what we can see on the graph is the entirety of the function f. (In other words, there is nothing past the "edge" of the axes.) Estimate the domain and range of f from the graph. • The domain is all the inputs that the function can accept. Since every point on the graph that is above or below a given x-value accepted that as an input, those x-values make up the domain.
• Looking at the graph, the function accepts inputs of x:[−8,4]. Thus, the domain of the function is [−8, 4].
• The range is all the outputs that the function can create. Since every point on the graph that is left or right of a given y-value gave that as an output, those y-values make up the range.
• Looking at the graph, the function creates outputs of y:[−3,5]. Thus, the range of the function is [−3,5].
Domain: [−8,4];  Range: [−3,5]
Here is a graph of the function f(x) = x2+4x+1. Using the graph, determine the domain and range of the function. • The domain is all the inputs that the function can accept. Since every point on the graph that is above or below a given x-value accepted that as an input, those x-values make up the domain. However, the domain can be larger than just what we see on the graph, since the graph is only a window on a larger function.
• We don't actually get much help from the graph for finding the domain here. Instead, we have to realize that the function can accept any x-value as an input and work fine. Alternatively, we can realize that if we were to keep "zooming out" the graph, it would keep expanding to the left and right forever.
• The range is all the outputs that the function can create. Since every point on the graph that is left or right of a given y-value gave that as an output, those y-values make up the range. However, the range can be larger than just what we see on the graph, since the graph is only a window on a larger function.
• The graph helps us partially in figuring out the range. The graph lets us see the lowest value produced by the function: −3. However, to find the whole range, we have to realize that the function can keep going up forever. If we "zoom out" on the graph, it would continue up forever.
Domain: (−∞, ∞);  Range: [−3, ∞)

*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.

### Graphs

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 0:04
• How to Interpret Graphs 1:17
• Input / Independent Variable
• Output / Dependent Variable
• Graph as Input ⇒ Output 2:23
• One Way to Think of a Graph: See What Happened to Various Inputs
• Example
• Graph as Location of Solution 4:20
• A Way to See Solutions
• Example
• Which Way Should We Interpret? 7:13
• Easiest to Think In Terms of How Inputs Are Mapped to Outputs
• Sometimes It's Easier to Think In Terms of Solutions
• Pay Attention to Axes 9:50
• Axes Tell Where the Graph Is and What Scale It Has
• Often, The Axes Will Be Square
• Example
• Arrows or No Arrows? 16:07
• Will Not Use Arrows at the End of Our Graphs
• Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
• How to Graph 19:47
• Plot Points
• Connect with Curves
• If You Connect with Straight Lines
• Graphs of Functions are Smooth
• More Points ⇒ More Accurate
• 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
• Every Point on a Graph Tells Us Where the x-Value Below is Mapped
• Domain in Graphs 31:37
• The Domain is the Set of All Inputs That a Function Can Accept
• Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
• 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

### Transcription: Graphs

Hi--welcome back to Educator.com.0000

Today, we are going to talk about graphs.0002

A graph is a visual representation of a function or equation.0005

While perhaps not as precise as numbers and variables, a graph gives us an intuitive feel for how a function or equation works--how it looks.0009

This graph is able to convey a wealth of information in a single picture.0018

Now, just like functions, you have definitely been exposed to graphs by this point.0022

You have seen them in previous math courses; but you might not have fully grasped their meaning.0027

This lesson is going to crystallize our understanding of what is a graph is telling us about a function or an equation that it is representing.0032

It will tell us what it means, exactly--that is what this lesson is here for.0039

They get us all on the same base for graphs, so that we can move forward and understand everything that is going to come next.0043

Graphs can tell us a whole bunch of information very quickly; they come up all the time in math.0048

So, it is really, really important--we absolutely have to start by understanding what a graph represents,0054

because we are going to see them all the time in math, and in sciences, and in other things.0062

Having a really good understanding of what graphs mean is just going to matter for our understanding of a huge amount of other things.0066

So, we really want to start on the right foot.0073

All right, let's begin: when we have a graph, it shows how the input affects the output, or how one variable affects the other.0075

But what does that mean, and how should we interpret the pictures we see?0085

To answer that question, we are going to consider the graphs of f(x) = x + 1, and the equivalent graph of y = x + 1.0089

This graph over here is the same for both of those--either that function, f(x) = x + 1, or that equation, y = x + 1.0096

We are going to get that same graph on the right side.0104

Remember from past math classes: we always associate the horizontal axis with the input independent variable.0106

Our x is the input variable, the independent variable.0115

And the vertical axis gets connected to the output, or the dependent, variable, which is normally going to be f(x), or y.0119

So, over here, the vertical part connects to f(x), or y, while the independent input part connects to the x,0129

for this function and this equation that we are going to be talking about.0139

One way to think of a graph is as a way to see what happens to various inputs.0144

If I plug in some number for x, where does it go? What happens to this number?0148

The graph lets us see how different inputs are mapped to various outputs.0154

We get to see a whole bunch of inputs getting mapped to a whole bunch of outputs, all at the same time; that is what a graph is showing us.0160

So, let's interpret the graph of f(x) = x + 1 with this idea in mind.0167

The reason (2,3) (2 is the horizontal, and 3 the vertical, portion) is on the line is because,0171

if we use x = 2 as input, then if we plug in 2 into f, if we plug it in for that x, then we will get 2 + 1; and 2 + 1 is 3.0181

So, if we plug in 2, it gives out 3 right here.0192

That is where we are getting this graph from.0199

This line that we see is all of these possible inputs on this x-axis.0201

Each point on that line shows where the x-value directly below it is mapped.0205

If we look at 7, then it tells us that that came out as an 8; if we look at -4, it says that that came out as -3.0209

All right, we plug in a value from the x-axis, and it comes out on the y-axis; we get to see what this function does to that input value.0227

And that is how we are looking at a graph: the input goes in from the horizontal, and the output comes out on the vertical, axis.0235

It is a really great way of being able to see how the function affects many, many inputs,0242

all at the same time, as opposed to having to look at a table where each one takes up its own entry;0246

we can just see this nice curve, or this nice line, that explains many, many pieces of information very, very quickly and very, very succinctly.0251

We can take it in in a single look.0257

We can also think of it, though, as the location of solutions.0260

This is another way to interpret the graph that is kind of different than that other one.0263

They are connected, but they are also fairly different; and I think, in the way that we think about it, it has a really different meaning in our head.0267

The graph of an equation is made up of all of the points that make the equation true.0276

So, while that is the same thing as input to output in some ways,0280

we are going to see that we can also say that the reason why this point is here,0285

the reason why this point gets to be on our graph, is because it works with the equation; it is truth.0289

The points that aren't on our graph, the points that aren't highlighted in the graph, but they are just on our plane--0297

those don't make truth; those are false points, and so, since they would make the equation false, they don't get to be on the graph.0302

Only the points that would make the equation true get to be on the graph.0309

The graph is all of the truth points--all of the points that make our equation actually work.0312

Let's interpret the graph of y = x + 1 with this idea in mind.0320

The reason why (2,3) is on the graph...we go to (2,3)...the reason why this point here is on the graph0324

is because, if we set that into our equation, (2,3), then if we plug that in, here is the 3; 3 is y;0332

here is the x, which is 2; if we set that up as an equation, 3 = 2 + 1, yes, that is actually true; 3 does equal 2 + 1.0341

So, because 3 equals 2 + 1, it is true; (2,3) gets to be on the graph, because the equation that would connect to that,0349

3 = 2 + 1, is a true equation; every point on the line is a solution to the equation.0359

It is all of the true points, all of the points that would make the equation true.0365

8 = 7 + 1 gives us the point (7,8); -3 = -4 + 1 gives us the point (-4,3)...oops, not (-4,3), but (-4,-3); I'm sorry about that typo.0369

And that is what is going on right there.0385

If we were to put on some other point--let's just consider (0,10) for a second.0388

We consider the point (0,10): if we were to plug that in, we would get 10 = 0 + 1.0394

Wait, that is not true! 0 + 1 is not equal to 10; 10 does not equal 0 + 1, so this point here is a false point.0402

It doesn't get to be on our graph; and that is why the graph is just made up of that red line.0410

It is because those are all of the points that actually give us truth.0417

If we went with some point that was not on that line, it would actually end up making our equation false; so it doesn't get to be on the graph.0420

We can interpret the graph as the place of truth, the location of all of the solutions to the equation.0426

This gives us two very different ways to interpret, and they are both totally valid and useful.0433

That said, generally we are going to want to think in terms of the first one.0439

Mostly, the first one is going to be the easier way to think about what a graph is telling us.0443

For functions, it is almost always easiest to think in terms of how inputs are mapped to outputs.0448

For equations, it is not always best; but we can normally use it, as well.0453

We can normally use this method for equations, as long as they are in that form y = ....0456

If it is set up with a bunch of y's showing up in multiple places, we can't really use this,0462

because we do not have a good way to go from input to immediately showing us what the output has to be.0466

So, it has to really be in this form, y = ...; but that is really what we are used to.0472

When we see something like y = x2 + 3x + 1, it is set up in this form of y = ...(things involving x).0476

But in either case, as long as we are in this y = ..., or we are just looking at a straight function, f(x),0485

in either of these two cases, this interpretation is a great way to think about graphs.0491

We plug in an input, and then we get an output on the vertical.0495

We plug in a horizontal location as the input, and that gives out a vertical location as the output,0499

which gives us an ordered pair, which we can now plot on our plane, when all of those points put together make a graph.0505

This is a really useful way; it is really easy to grasp; it is very intuitive; and it works very, very well.0513

Still, at other times, it will actually be more useful to think in terms of solutions.0519

What point is a solution? Where is it true?0523

This idea is going to be especially important for certain types of equations that will get seen later on.0527

But it is also going to matter for when we want to talk about0534

where two equations or two functions intersect--where they have the same value at a certain point.0536

That idea of where it is true--two things being true at the same time--that is an interesting idea,0543

and useful for those locations, when we want to talk about intersection, or when we want to talk0547

about certain more complex equations that are not just in the form y = ..., but where y shows up on both sides,0552

or x and y are mixed up together...so sometimes we want to use that second form.0559

But mostly, we want to think in terms of that first way.0564

But the second way, we will occasionally use sometimes.0567

Think in terms of that first way; think in terms of "input goes to output."0571

But don't forget about the second way of "these are all of the places where it is true;0575

these are all of the locations of the solutions," because sometimes we will need to switch gears and think in terms of that,0579

because it will make things easier for us to understand at certain later points.0584

All right, now that we understand what it is about, let's talk about axes.0589

The axes are just the vertical axis and the horizontal axis--those lines that we are graphing on.0593

The location of a graph can be as important as its shape.0602

The location is set up by its axes; we want to pay attention to these axes.0605

The axes will tell us where the graph is and what scale it has.0609

Often, our axes are going to be square; that is to say, the x-axis is the same length as the y-axis.0614

For example, we might have -10 to positive 10 on our y-axis, and -10 to positive 10 on our x-axis.0619

This is a pretty common one; and this is square, because the x-axis is the same length as the y-axis.0628

So, when we look at the picture, it is square, which is sort of an odd idea.0634

But if we made it so that they had different lengths, but we had set them out as the same amount of line,0638

then we would have a sort of squished picture; it wouldn't be the natural picture,0644

where we think of width and length as meaning the exact same thing, in terms of length.0648

That is a little confusing, because we are using the words width and length...I mean width and height meaning the same thing, and how long it is.0653

So, as long as it is square, the graph isn't distorted from the square perspective we normally expect.0660

However, sometimes it is going to be useful to graph functions on axes that are different from each other,0665

where we are going to want to have a really, really big y-axis, but very small x-axis--0671

where the function grows very, very, very quickly, so we want to be able to show all of its ability to grow.0679

But since it does it so fast, we need a short x-axis.0684

So, this is another really important reason to pay attention to the axes.0688

You want to know how long they are, what amount of information is being represented in both of them,0691

and also how big it is and where we are located.0697

You want to have some sense of what the scale is: are they the same scale on both the x-axis and the y-axis?0700

And just where are we located? Are we located in a weird place--is it not centered on 0?--those sorts of things.0706

So, let's look at a single function: let's look at f(x) = x + 1 and see how many different graphs0712

we can get out of it, just by changing the axes.0720

Just by playing with the axes, we can get totally different-looking graphs.0723

Here is the standard graph, our -10 to 10, -10 to 10.0726

This top left graph here is basically what our standard graph would be.0730

We are nice and square; the y-axis and the x-axis are the same length--that is what means to be square.0735

It is from -10 to 10 and -10 to 10--numbers that we are used to and expecting.0742

And also, the origin is in the center; we have (0,0) in the center of the graph.0747

Now, let's consider the one below that--the bottom left.0752

In this one, we have still square axes, because we are going from -2 to...actually...they are still square technically...-2 to 15 and -1 to 16.0756

-2 to 15 means a length of 17; -1 to 16 means a length of 17; so even though they aren't putting down the exact same numbers,0769

it is still a square, because they have the same length, total.0776

-1 to 16 and -2 to 15 are both a length of 17; so it is still a square graph.0781

This one here is square; this one here is square, as well.0787

There is no distortion, no squishing in either the horizontal or in the vertical--no squishing of the graph.0795

And the origin, though, is in a totally different place than the center of the graph.0800

The origin is very, very bottom-left-corner; but it is still giving us the same x + 1.0805

It looks kind of different, in terms of where the axes are; but it is still pretty clearly the same function making the graph.0813

Let's look at another one: well, this one right here is actually not square.0819

Why? Well, we have totally different lengths here: -10 to 10 and -5 to 5--0824

that means the length of the horizontal is actually double the length of the vertical.0831

The origin is still in the middle, so that is nice; that is something we are used to.0836

But because we have a much shorter length, it ends up that we have more stuff in the horizontal than we do in the vertical.0839

That means we have to compress what we are doing in the horizontal; so it has gotten squished left/right, which has caused it to stretch up vertically.0847

This is not a square; these are not square axes right here.0855

Another one that is not square (and hopefully you can read the yellow)...0860

it is not too easy to read the yellow, but it is just me writing "not square" here.0864

Once again, it is from -10 to positive 2 and -20 to something, but -10 to 2 is a length of 12, and -20 to something greater than 0 means greater than 20.0869

So, once again, we have not-square axes; but this time, we have the vertical axis being longer than the horizontal axis.0881

The horizontal axis has a length of 12; the vertical axis has a length of more than 20.0889

So, that means that we have stretched it in the horizontal; as opposed to being squished horizontally,0894

it has been stretched horizontally, because now it has less stuff to have horizontally than vertically.0900

We have squished it vertically, because we are trying to cram in more vertical information while not having to cram in as much horizontal information.0905

It has been squished vertically; so we have very different things here--vertical squish has happened in the bottom right one;0912

and in the top one, we have horizontal squish, but it is not because of anything that has happened to the function.0924

The function is x + 1 for every single one of these graphs; but the squish can be caused based purely on how we set up the axes.0933

Setting up the axes, paying close attention to what the axes are telling us, is really important for us to actually understand what is going on in a function.0943

Unless we understand what the axes are telling us, we won't actually know what this picture means.0950

So, make sure you pay attention to axes; otherwise you can have no idea where you are.0955

You have to have a map before you can really make sense of what is going on, and the axes are the map that our graph lives on.0961

All right, one thing you might have noticed by this point is that the graphs in this course,0968

unlike this one to the right, do not have arrows on them.0972

I mean these arrows up here: at some point in the past, you have probably had a teacher who required you to draw arrows on the ends of your graph.0975

And that made sense; they were trying to get across a very specific point to you.0983

They were trying to remind you that the function keeps going on, even though we couldn't see it anymore.0987

In a way, we can think of the axes as sort of boxing in the function.0992

We don't get to see anything outside of the box of our axes.1000

But in reality, the function doesn't stop at 3; it doesn't stop at -3, necessarily; this is just a nice, normal parabola.1005

The function would keep going on; it would just continue off and off and off, and continue off and off and off and off.1012

It doesn't actually stop; so the reason those arrows were there is to remind us that it goes past the edge of our axes.1019

Just because the axes are here doesn't mean it stops; it is going to keep going.1026

So, that is what those arrows were for; at this point, though, I think you have probably gotten used to that idea.1031

We are not going to be using arrows at the ends of our graph in this course.1036

The ends of our graphs in this course are just going to stop on our graphs; but that doesn't mean that the function stops.1040

We are going to assume that we are all aware that the graph keeps going.1047

It doesn't stop once it hits the edge; it just keeps going, unless we have been very specifically told that the function stops at a certain location.1052

So, the graph is only stopping because the edge of the graphing axes stop.1061

It is the graphing axes that are stopping the function, not the function itself.1065

The function continues past the edge of our axes, unless in a very specific case, where we are told that it stops at some place.1071

So, when we see this lack of arrows, it doesn't mean that it stops; it just means that we have to remember that it keeps going past the edge.1078

The only reason it stops is because it has hit this boundary at the edge of it.1087

It is not stopping because it actually stops; it is not stopping because the function stops.1093

It is just stopping because we are looking through a window.1097

If you look out through a window, if you are in a house, and you look out through the window,1100

you can't necessarily see everything to the left and everything to the right.1105

You can only see what you are currently looking through in the window.1108

You have to move how you are looking through the window, or move the location of the window1111

(although that would require a sledgehammer, and is something no one that you live with is going to be very happy about)--1115

you can move the location of the window and be able to see different things outside; but the window fixes what you can see.1121

That is what the graphing axes are doing to us: they are fixing what we can see in space.1126

We are not going to use arrows in this course, because we know that graphs have to keep going.1132

We are just seeing a tiny window on a much larger function.1136

That said, even though, in this course, we are not going to use arrows, and we are all aware of it at this point,1138

I want to point out that there are some teachers out there, and some books,1145

that will still use arrows, and will still require you to use arrows.1149

So, just because I am here saying that you probably don't need to use them--you are probably used to them by this point--1153

doesn't mean that your teacher, if you are taking another course of the same type somewhere else--1162

that that teacher is going to be OK with it.1168

So, make sure that, if you have another teacher, if you have somebody else1170

who wants you to draw arrows--make sure you do what they are telling you to do.1173

So, do what they say as long as you are in their class.1176

For my class, you don't have to; we know what we are talking about.1179

But in somebody else's class, they might still want you to draw arrows, so be aware of that.1183

How do we actually graph? The easiest way to graph a function is by thinking in terms of that input-to-output.1189

Remember, you put in a number, and it gives out a number.1194

So, we choose a few x-values, and we figure out what y-values get mapped to those x-values, and then we plot those points.1198

For example, consider f(x) = x + 1, the one we keep working with.1203

If we plug in -2, that will give out -2 + 1, which is -1; so that gets us the point (-2,-1), right here.1207

If we plug in -1, that gets us 0; so that gets us the point (-1,0) right here.1215

If we plug in the point 0, then that gets us 1, 0 + 1, so that gets the point (0,1).1221

If we plug in 1, 1 + 1...we get 2, so we get (1,2); if we plug in 2, 2 + 1...we get 3, so that gets us the point (2,3).1226

And now we have a pretty clear idea: it is just a straight line; it is just going to keep going.1235

So at this point, we could come along, and we could draw in a straight line that just keeps going through all of these points.1239

And we know what is going on right here: we are able to figure out that these points tell us that that is what the shape of this graph is.1248

We don't have to graph all of the points perfectly in between, because it is pretty obvious,1255

at this point, that they would all just end up being on this graph, as well, if we were to keep going1258

with finer and finer steps, and how often we would check to see where inputs went to outputs.1263

However, straight lines are not necessarily the best way to connect all of our graphed points together.1270

In many ways, graphing is like playing a mathematical game of Connect the Dots.1275

But we don't necessarily want to connect with straight lines; we usually want to connect with curves.1281

For example, let's consider f(x) = x2.1286

Once again, here is a table that shows us input locations going to output locations, making points.1288

(-3,9), (-2,4),(-1,1), etc....we can see all of these points on this graph right now.1295

But let's look at what happens if we were to connect it all with straight lines.1301

If we connect with straight lines, we get this picture right here.1304

And while it is not a terrible representation of a parabola, it is not a very great representation of a parabola.1307

A real parabola has curves going on; it curves out; it curves out, as opposed to going out just in these straight, jagged lines.1314

So, we want to remember this fact: curves are normally what is going to connect our points, not straight lines.1323

The real f(x) = x2 is based on curves, so it looks like this picture right here.1329

It is based on these nice, smooth curves connecting all of these points together.1336

What about the fact that curves in one function are not necessarily going to look exactly like the curves in the next function?1341

That is true, but mostly, the graphs of functions are smooth; we want to connect points to each other through smooth curves.1346

So, whenever you are drawing a graph, make sure you are connecting things smoothly, without jagged, harsh connections.1352

Each function is going to curve in different ways.1358

Remember, the shape of a curve will be different: if we are using x2, x2 is going to give us1361

a totally different curve...well, not totally different, but it will be slightly different than x4,1365

which is going to be different than the cube root of x.1370

Each function that we graph will have a slightly different curve, or maybe a massively different curve.1373

But over time, you are going to become more familiar with the shapes of various functions.1379

As you graph more and more functions, as you see more and more functions,1383

you are going to think, "Oh, x2 should graph in this general way; √x should graph in this general way."1387

"The cube root of x, the x5...all of these things have curves that are slightly different."1394

It should curve a little faster, curve a little slower...those sorts of things.1400

Your previous experience with functions helps immensely, so just pay attention and think back:1404

when have I graphed something similar to what I am graphing right now?1408

And use that information to help you graph what you are working on at the moment.1412

Finally, the idea that more points make a more accurate graph: this is an important idea.1417

The more points you plot before drawing in your curves, the more accurate the graph becomes.1423

Each point on the graph is a piece of information.1428

So, it makes sense that, the more information we use to make our graph, the more accurate the graph is going to become.1431

Let's look at a specific example: Consider f(x) =...this complicated monster of a function, (x3 - 2x2 - 7x + 2)/x2 + 1.1439

And we plot it with various step sizes: what I mean is how big of a jump we have between the various test points that we are setting up.1451

We are going from -4 to 4; so we will start at -4, and then we will step forward by 2.1458

That is what I mean by a step size of 2; don't worry--this is Δx; it means change in x,1464

and it is just a way of saying how much we are changing x each time.1470

So, if we step forward 2, if we go from -4 here to -2 here, and then to 0 here,1474

and then to 2 here, and then to 4 here, we have stepped forward by 2 each time.1483

And we can evaluate...I am not putting the table down here, because it is just kind of a pain1488

for us to have to see all of the numbers that we are going to be going through soon.1492

But if we evaluated each one of these things, we get the following vertical locations.1494

-2 happens to be at 0; 0 happens to be at 2; 2 happens to be somewhere between -2 and -2.5; and so on, and so forth.1498

So, what happens if we increase the step size? We don't really have a very good idea of what this thing looks like.1508

It might go like this, but it could also go like this; it could maybe even do something crazy, like this.1512

We don't really have a good idea of what those points mean, because we haven't strung enough of them together to get a very good idea.1529

We are not used to this function, (x3 - 2x2 - 7x + 2)/(x2 + 1).1536

This is an unusual function; we are not used to graphing things like this, so we don't have a really good sense of what it is going to look like.1541

So, since we don't have a really good sense of what it is going to look like, we don't have the expectations;1547

we need more points down before we are going to be able to have a good sense of where it is going.1551

Let's consider a smaller step size--a step size of 1.1555

Now, we go from -4 to -3, then -2, then -1, then 0, etc.1558

Now, we are starting to get a better idea of what the curve of the function looks like.1562

We are starting to think, "Well, now we are starting to see what is happening."1565

There is still a little confusion; we are not really quite sure what happens between -2 and 1 horizontal locations.1568

But we are starting to get a better idea; let's make it an even smaller step size.1574

We are at .5; oh, now it is starting to come in much clearer--we can start to understand what is going on.1578

We go with .2; oh, now we are really starting to see what it is.1584

We now have a great idea; finally, we go to .01; now there are so many points down that it almost makes a continuous, smooth line.1588

The only place where it isn't quite smooth is this section in the middle right here,1596

where the function is changing so quickly that we can actually still see the space between these tiny points.1601

But when it is not changing that fast, like most of it here or here, we end up seeing that it strings together,1606

because we have put down so many points that it basically turns into a smooth line.1614

And that is exactly what happens when we make a graph.1618

We are putting down so many points that we are saying, "Oh, that is what the smooth line is that it is making."1621

That is what is happening when you use a graphing calculator, actually.1626

If you use a graphing calculator, the computer inside is basically saying, "Make a bunch of points."1629

It is now doing the same sort of thing; it is doing tiny, tiny steps, and then it is just stringing them all together with straight lines.1635

So, it makes a whole bunch of points, and then it just strings them together; and that is what we see in the end.1643

The way that you graph something is: you just keep using more and more points if you need more information.1649

If you have a pretty good sense of how it is going to curve, though, you just have to put down enough points1653

so that you can then put in the curves, because you have already had the experience of working with that function before.1656

All right, when we introduce the idea of a function, we discussed an important quality for functions.1662

For a given input, a function cannot produce more than one output.1667

So, for example, we said that if f(7) = -11, then it can't also be true that f(7) = 20.1671

Then that means that f(7) equals two things at once; and we said that, when you put something into a function, it always puts out the same output.1679

So, if we put in f(7) the first time, and it gets -11, then the second time, it has to give -11,1686

and the third time it has to give -11, and the fourth time it has to give -11.1690

It can't ever be the case that all of a sudden, things go crazy and it produces a different result.1694

No, we can trust our function; we can trust our transformation, our process, our map, our machine--whatever analogy we want to use.1698

We can trust the function to always give us the same output if we put in the same input.1704

So, if f(7) = -11, it can't be the case that f(7) equals something else, as well--something different than -11.1710

We can turn this idea into a thing that we can see in graphs.1718

We call this idea the vertical line test, and it says that if a vertical line could intersect1721

more than one point on a graph, it cannot be the graph of a function.1727

So, if we have a vertical line, and we bring it along like this,1732

if we put a vertical line on anything over here on the left, it ends up not being able to intersect at more than one point.1739

No matter where we bring a vertical line down on this graph on the left, it ends up passing the vertical line test.1748

This over here is a function; but if we deal with this one over here, pretty much any point we choose will end up hitting two points:1756

this one and this one--this one and this one; if we put it over here, it fails to hit any, but that doesn't necessarily mean it passes.1772

If we can do it at any place on the graph, even if there is only one place on the graph1780

where a vertical line hits the graph twice, then that means it is not a function.1784

If there is a vertical line that could intersect more than one point, it is not a function.1790

A vertical line--if it is able to intersect more than one location on the graph, it is not the graph of a function.1798

Why--why is this the case? Well, consider this.1808

Every point on a graph tells us where the x-value below is met.1811

The points on the graph are in the form (x,f(x)); the x that we put into the function, and the f(x),1814

the thing that the function puts out for that x--input and output put together.1821

So, for example, let's look at this graph: this is the graph of something like a square root function.1826

If on this graph we see, at x = 1, that we get f(1) = 2, we go to 1 on the horizontal; we bring it up, and we get to 2 on the vertical.1831

So, we get that f(1) = 2, which is coming from the fact that the point is (1,2).1844

So, we put in an input, and we get the output of 2.1849

But let's consider this other one: what if we had this graph instead?1852

On this graph, at x = 1, we get (1,2) and (1,-2); that means, since it is a graph,1857

that if it is the graph of a function, we have f(1) = 2 and f(1) = -2.1866

But that is not possible--a function cannot give out two different things.1872

We can't plug in 1 and get 2 and -2; if we plug in 1, it is not allowed to give out two different outputs.1877

That means we can't be looking at the graph of a function, because when we plug in one number, it gives out two things; so it fails the vertical line test.1885

This picture right here is not the graph of a function.1893

Remember, the domain is the set of all inputs the function can accept.1898

The domain is the set of all inputs that a function can accept; the domain is what the function can act on--the numbers that the function can do something to.1904

A graph shows where a function goes, so it means that we can see the domain in the graph.1914

Every point on the x-axis that the graph is above or below is in the domain.1921

So, every point on the x-axis that the graph is above or below has to be in the domain of that function.1926

However, if we can draw a line on an x-value, and it does not cross the graph, then that x is not in the domain.1933

A really quick example: if we had √x like this, then if we have tried drawing a vertical line here,1939

that means that this horizontal location has to be in the domain, because it ends up having an output.1951

If we plug in this horizontal, it comes out as this output; so that means that it must be in the domain.1958

But if we go over here, this horizontal location never shows up in our graph, so it must be the case that it is not included in the domain.1963

That horizontal location is not included in the domain.1975

So, if you can draw a vertical line on an x-value, and it does not cross the graph, then that x is not in the domain.1978

Remember, the domain is everything that the function can take in.1984

So, if a graph is above a point, then that means it had to be able to take it in, because it gives out something over that horizontal location.1989

This is a great way to visually notice the domain; but we have to be careful to remember1999

that our function probably continues past the edge of our viewing window.2003

Remember the axes that we had there; so if we are going to use this idea,2007

we have to remember that, just because it seems to stop,2010

or we don't see anything past the edge of the axis, that doesn't mean that the domain stops there.2014

We just need to remember that it might continue on; we have to have some sense for how it looks beyond the edge.2018

We need to have some familiarity; we need to think, "Where would this keep going to?2024

Would this keep picking up those points in its domain, or would it stop for some reason?"2028

Range is the set of all possible outputs a function can have.2034

It is all the numbers that our function could possibly produce; so domain is what could go in; range is what can come out.2040

Like the domain, we can see the range of a function in its graph.2046

Every point on the y-axis that the graph is left or right of is in the range.2050

However, if you can draw a horizontal line on a y-value, and it does not cross the graph, then that y is not in the range.2054

So, for example, let's consider x2; x2 looks something like this.2060

So, if we go to this horizontal location, we would be able to eventually go up and hit it; so it is in the domain.2066

Similarly, we can go to this vertical location, and if we cut horizontally, there must be some domain location that puts that out.2074

Now, it turns out that there are actually two different domain locations that put that out; but that is OK.2083

Multiple domain locations--multiple inputs--can give the same output.2089

f(22) is equal to (-2)2; that is perfectly fine...4 and 4.2093

It is OK that the same input gives the same output; but the fact that there is some input that gives that output2101

means that it must be in the range, because it can be an output.2107

So, we go to any location on our vertical axis, and if we draw a horizontal line and it cuts the graph,2112

then that must mean that there is something that can input and give that output.2118

Any location that is directly left or right of a vertical location means that that vertical location is in the range; that location, that number, is in the range.2124

If, on the other hand, we can draw a horizontal line on a vertical location, and it does not touch the graph--2137

that would not touch x2--then that means it is not in the range.2145

And that makes perfect sense: down here are the negative numbers.2149

So, can x2 give out negative numbers? No, it can't--there is no real number that we can plug in that will give out a negative number.2152

So, since there is no number that we can plug in to give out a negative number,2160

then that means that we can't output negative numbers, so they can't be in the range.2163

So, the range does not include any negative numbers, which is why, when we draw a horizontal line2166

in any of these negative numbers, it is not going to touch the graph,2171

because there is nothing that can make an output that would give a negative number.2174

Just like with the domain, we have to be careful to remember that our function probably continues past the edge of our viewing window.2179

That viewing window is just what we are looking through; so it is possible that your range is going to keep going, because the graph is going to keep going.2185

So, we have to have some feeling for how the function will look past the edges of what we are able to see.2193

Beyond the edge of our viewing window, we need to have some sense of what is going to keep going on.2198

If we have no idea, we need to expand our viewing window, so that we can have a better idea2204

and see, "Oh, yes, that would keep going," or "No, that actually stops."2208

Otherwise, we will not be able to figure out exactly where the range is.2212

Graphing calculators are really useful; if you haven't already noticed, this is a great time to point out2216

that there is an appendix to this course that is all about graphing calculators.2221

So, if you go the very bottom, and look at the appendix, there is an appendix about graphing calculators.2225

So, it might be at the end of the course, but that does not mean you should watch it last.2231

Graphing calculators are really, really useful for doing math.2235

And you can also use software for graphing on computers or tablets or phones.2239

And you can just start doing graphs on that really quickly and easily.2248

So, graphing calculators can be extremely helpful for getting a feel for how functions work.2251

If you are planning on taking calculus at some point, I definitely would recommend getting a graphing calculator in the near future.2256

You are almost certainly going to want a graphing calculator for calculus, and so it won't hurt to have it now in precalculus.2261

Even if you are not going to continue in math, you might find one useful for taking this course right now,2266

and maybe for other science courses that you are currently taking, or will take in the future.2272

So, if you are interested in getting a graphing calculator (and I would recommend it if you can afford it--2276

and even if you can't afford it, there are some alternatives that I am going to talk about that are free or extremely inexpensive)--2280

check out the appendix on graphing calculators; we are going to talk about all about2286

how you can use them, what they are good for, why you might want one,2290

what are some recommendations, things to look for, and that sort of thing.2295

So, check out the appendix; there is a whole lot of information on graphing calculators there.2298

It is really useful, and you are probably in a position where it is going to be useful for you to have a graphing calculator,2302

since you are taking this course, and there is a very good chance you will go on to take calculus.2306

I would definitely recommend to get a graphing calculator if you can afford it.2310

So, check out the appendix; there is lots of information there.2314

All right, we are ready for our examples: first, we are going to graph something.2318

Graph f(x) = x2 - 3x + 1: we have done this before, but let's just see a quick reminder.2322

We want to do this by plugging in points and getting outputs.2327

So, we are going to plug in x's, and we will get f(x)'s out.2332

We plug in...we are not quite sure what this looks like, so let's start with a simple number that we can be pretty sure is easy to do; let's plug in 0 first.2335

If we plug in 0, we get 02 - 3(0) + 1; that gets us 1.2344

If we plug in 1, then 12 - 3(1) + 1...well, 12 is 1, minus 3, plus 1...so we have 2 - 3; we have -1.2356

Keep going; we plug in 2; that will be 22 - 3(2) + 1; 22 is 4, minus 3(2); that is 6, so we have 4 - 6 + 1.2373

4 + 1 is 5; 5 - 6...we have -1, once again.2385

Let's try going in the other direction as well: let's plug in -1.2390

I am just going to start skipping directly to the numbers, because at this point, we should probably be able to do this in our heads,2394

or be able to do it on paper on your own, I'm sure; so we will just speed things up.2398

(-1)2 - 3(-1) + 1...that will get us positive 1.2403

We plug in -2: (-2)2 - 3(-2) + 1; (-2)2 gets us 4, minus 3(-2) gets us...2409

we should be able to do it in our head...that is ironic for me to have said that; maybe that would be a good reason to write it out.2425

So, (-1)2 - 3(-1) + 1...and this is also a good lesson in "never just trust yourself to immediately be able to do things in your head."2430

(-1)2 gets us positive 1; minus 3(-1) gets us positive 3; plus 1 gets us 5.2441

(-2)2 - 3(-2) + 1...we have 4 + 6 + 1; we have 11.2451

And if we go forward one more, at 3, we are going to see 32 - 3(3) + 1.2464

We would get 9 - 9 + 1, so we would get positive 1.2474

And one more: if we plug in 4, we would get 42 - 3(4) + 1, so 42 is 16, minus 3(4) is 12, plus 1.2479

So, 16 - 12 is 4; with 4 plus 1, we get 5.2490

All right, so we have a lot of information, but there is one thing that we might notice.2494

We might say, "Parabolas need a bottom"; we are graphing a quadratic, and while we haven't formally talked about them,2497

I am sure you have seen parabolas quite a few times by now.2505

We plug in 1; we get -1; we plug in 2; we get -1; we might realize that that doesn't actually give us a bottom.2509

That is going to give us sort of a flat bottom, so there is probably some point in between them that is even lower.2516

So, we want to have some sense of where it is going; so let's actually plug in a number in between them.2522

Let's plug in 1.5; if we plug in 1.5, f(1.5), we get -1.25; I will spare doing that here, but we would get -1.25.2526

I will actually do it here; so we plug in 1.5, so 1.52 - 3(1.5) + 1...2540

1.52...when we put that into a calculator or do it by hand, we get 2.25 - 3(1.5)...we get - 4.5 + 1.2548

So, we have 3.25 - 4.5; we get -1.25; great.2557

All right, so at this point, we have actually found something that seems like it could be the bottom; and it turns out that it actually is precisely the bottom.2563

But we don't know that technically; we haven't formally talked about it.2569

But at least it gives us a sense of where this is going to be bottoming out.2571

So now, let's actually set up our axes, and let's plot the thing.2575

Now, this never gets that low; it only gets down to -1.25; so let's make the bottom of our axis not that long.2578

So, we will go to -1, -2, because we never even reach -2; and we will go up 1, 2, 3, 4, 5, and it would keep going.2588

But we are going to top out, so we will never actually end up seeing the number 11,2603

because we can't make it up that high on these axes, if we are going to keep them at this reasonable size.2606

And let's keep it square; so the distance from the origin to a vertical one will be the same as the distance from the origin to a horizontal one.2610

So, this is approximately square; I am just roughly drawing it by hand, but it is pretty good.2617

1, 2, 3, 4, -1, -2, -3...and I would keep going to the left, but we know that we are never going to even see that point,2621

because -2 is already out of where we are going to be able to plot.2635

So, let's just plot our points now: let's see, 0 is at 1; we have 0 at 1, so (0,1)--we have that point.2639

Let's go to the left first; -1 manages to make it up to -5, and we are already going to be past the graph when we are going to -2; it is out here.2649

We plug in 1, and we are going to be at -1; we plug in 2, and we are going to be at -1.2658

Let's plug in the point in between them: 1.5 is going to be at -1.25, so it is just a little bit below.2665

3 is going to be at positive 1, and 4 is going to be at 5; so we curve this out, because we know it is a parabola.2673

So, we have some sense of how the curve looks.2680

All right, and it would keep going on and out; and it just keeps going, past the edge of our axis.2686

All right, and that is how we graph it; so this is pretty much how we can graph anything.2698

Plot some points on a T-table; plot some points on x and f(x), input and output.2702

Plot the points; figure out where they are going to go; then actually put them onto the graph.2708

Set up points, I mean; and then plot them onto the graph, and then connect it with curves, depending on how we know that kind of graph gets put together.2712

All right, this is the graph of f(x) = x3 + x2 - 6x.2720

Using it, we are going to estimate the values of f at -1.8, f at -1, f at 1, and f at 2.5.2725

Then, we are going to also estimate the values where f(x) = 0; and then finally, we will estimate the values where f(x) = -3.2732

So first, this part right here, f at -1.8...what we do is just go to -1, -1.8...well, that looks around here.2738

So, we go up; that looks like -1.8; we go up here, and we are about here, so it looks to be a little above the 8, somewhere between the 8 and the 9.2748

If that is the case, I would say that looks like around 8.3 to me, give or take.2763

We are just estimating, so we don't have to be absolutely, perfectly precise.2767

But I would say 8.5 is a pretty reasonable guess; 8.3 is probably a little closer, so let's go with 8.3.2769

f(-1.8) is equal to...it looks like 8.3; it is an estimate--it says "estimate"; it is a graph--2776

we are never going to be able to perfectly pull information from the graph.2784

Well, we might be able to in a few cases; but it is going to be normally something2787

where we are getting that we are pretty confident, but it might be slightly off by .01 or .01...2790

well, that is the same thing...by .1, lower by .1, these sorts of things.2796

It is hard to be absolutely, perfectly precise, since we are looking at a picture; but we can get a pretty good idea.2800

The same thing for everything else: for f(-1), we just go to -1; we go up; f at -1 seems to be about this high.2804

So, I would say probably about 5.7 or 5.6; so let's say it is 5.7.2814

f at 1...we are here; we drop down; and that one looks like it is really pretty much exactly -4; so f(1) = -4.2824

f(2.5)...plug in 2.5; we go up pretty high; that looks like it was a pretty good vertical...2836

look, I would say that looks like it is pretty close to being right on 8; so we will say that that is 8; great.2849

We have estimated the values for all of them; they might be a little bit inaccurate, but they are pretty close to right.2858

And that is what a graph gets us--it gets us a good way to get a really good sense of what is going on.2863

It might not be perfectly, absolutely, exactly right; but it will get us there pretty close, which is normally enough to be able to do stuff for lots of things.2867

Now, let's look at estimating the x-values for f(x) = 0; we will do this one in blue.2876

Estimate the x-value for f(x) = 0; so what is f(x) = 0?2882

Well, remember, the vertical axis is f(x); that is the output.2885

So, if that is the case, then we are looking for everything that is at the 0 height, which is the same thing as the x-axis.2889

If it is crossing the x-axis...it looks like here like it crosses the x-axis at 2, crosses the x-axis at 0, and crosses the x-axis at -3 precisely.2896

There is nothing else crossing there; so we can assume that we have found all of the x-values.2905

It seems that is going to be x = -3, 0, and 2; they all caused f(x) to come out as 0; great.2909

Finally, we will use red for the very last one; hopefully, that won't be too confusing.2924

Estimate the x-values for f(x) = -3; if that is the case, we go to where f(x) is -3.2928

f(x) is -3; we want to go and see...here is something; here is something; and that is pretty close to horizontal--not perfect; sorry.2934

And there is something; so f(x) = -3 at these three horizontal locations.2955

So, once again, it is not absolutely, precisely, absolutely perfect, but pretty good.2965

f(x) = -3 is going to be at the x's that are...the first one, I would say, is a little past -3, but not by much, so probably -3.2.2972

And then, the next one looks like it is around...just a little past where the...here is positive 1; that is right here, so this is 0.5.2982

That is right here, so I would say that is just a hair past 0.5, so let's say that is 0.6.2998

And then finally, here it is just a little past 1.5; I would say it is a little bit more past it, though; so probably 1.6, or maybe 1.65.3004

Let's go with...let's say 1.7; maybe 1.6, maybe 1.7; but it is a little past 1.5, and we are sure of that.3016

That is how we use a graph to figure out things from it.3025

We can estimate values given an input, or we can estimate values given an output.3027

We figure out what makes that output or where that input would get mapped to.3032

What would that input get output as?3038

Vertical line test: Which of the below is not the graph of a function?3042

This one is not too hard: if the v's are the entirety that we are seeing, we just have to use the vertical line test.3046

If we come along this one, and we put a vertical line on this one, it is pretty easy to see that it is not going to fail at any point.3053

The vertical line is never going to cross it at anything.3060

The only place where you might be a little curious is right here where it curves up.3062

But it never really continues on in such a way that we can be sure.3065

Any vertical line that we are making seems to cut it just once.3069

Now, it does have this part where it sort of curves like this, but that is inaccurate.3073

It looks like that, but the graph is actually curving a little more like this.3078

And the reason why it looks like it is stacked on top of itself is because we have to add thickness to our line.3082

In reality, the line is actually thinner than that; and is even thinner than that, because a point is infinitely thin.3088

So, there is no stacking, because of that infinite thinness; it is only because of that thickness of our line3094

that it ends up looking like there is something stacked.3099

So in reality, if we come along with the vertical line test, since the vertical line is also infinitely thin,3101

it is not going to cut it twice, because it doesn't really curve back on itself; it is only going to hit one thing at one point.3106

So, this is a function.3112

If we cut in the middle, it is going to hit a bunch of times.3121

It cuts here, here, and here; so that is more than 1 intersection.3127

If we go on the far sides, it will pass; but all we need is one place of failure, once place where cuts across multiple times.3131

So, in the middle, it manages to fail being a function, because one input manages to simultaneously have three outputs.3139

So, it is not a function.3148

Finally, this one over here is the same idea as the left side.3154

Even though it looks like it is starting to get vertical, it is never actually vertical at any point.3159

It just needs to be an infinitely thin line to really understand what is going on, and a vertical line has to be infinitely thin, as well.3164

So, we have to think about this, beyond just saying, "Well, it looks kind of stacked, so it must be."3171

No, we have to think, "Oh, that is really only approximating where the graph is, because the line,3175

while we can't see infinitely thin things...that is what the line is representing."3180

So, it is the case that this one is also a function, because there is nothing where it clearly cuts two places at once.3185

Great; that is how we use the vertical line test.3191

Just drop vertical lines, and if there is any place where it clearly cuts the graph more than once, then it is not a function.3193

If we can drop vertical lines everywhere, and it would never cut the function more than once, then it is a function.3201

Final example: Prove that there is no function that could produce a circle as its graph.3209

This might seem a little complicated at first; so what we want to do is think, "Well, how could we prove this?"3213

Well, if we want to prove it, we need to show something involving circles as graphs.3218

We get stuck too much on trying to think, "What is the right way to do this?"3224

We might never get any progress.3227

But if we think, "Well, what does a circle look like?" a circle has things stacked--it would fail the vertical line test.3228

So, we know we can prove this by contradiction.3237

Proof by contradiction: we are going to start by assuming that there is such a thing.3241

So, proof by contradiction: assume such a function exists.3249

If there is a function that could produce a circle, then look at its graph.3257

Since it is a circle, we know what the graph of a circle looks like; who knows where it is going to show up on the graph, but we know it has to show up somewhere.3268

So, here is a circle; and while it is not a perfect circle--I am but a mortal--it is a good idea.3278

We can say, "Look, just take this and cut it at any place; any place inside of the circle, we are going to fail the vertical line test."3284

The graph must fail the vertical line test; therefore, it is not a function--it cannot be a function.3294

The graph cannot be a function; but it was the graph of a function.3314

So, since the graph cannot be a function, it must be that no such function exists.3322

So, our assumption was that the function did exist; since the graph cannot be a function,3335

but it was just the graph of a function, then there is a contradiction.3340

The function cannot exist, so it must be that no such function exists; and we are done--that is our proof.3343

All right, assume that what we can see on the graph below is the entirety of the function f.3350

In other words, there is nothing past the edge of the axes.3355

We are looking through that window, but we have been told that there is nothing interesting past the edges of the window.3358

So, this graph here is the entirety of the function f.3363

Now, we want to estimate the domain and range of f from the graph.3367

Now remember, the domain was everything that can be input.3370

So, if we go to, say, 0, look: 0 shows up in the graph.3374

Well, what about -3? -3 never shows up in the graph; there is nothing that it gets graphed to--nothing that it gets output as.3381

It looks like the edge is -2; it looks like -2 is the very edge; and over here, 3 gets put in; 4 doesn't get put in;3390

but it looks, probably, like 3.5 gets put in, so we would say that the domain is going to go from -2 to 3.5.3399

What about the range? Range is everything that can be output.3410

Is there anything that can output at 1? Yes, 1 manages to touch here, and manages to touch here.3416

There is some input that gives out 1; if we put in an input here, we can see that it connects here.3426

But if we go to 3 and we cut across, 3 horizontally never touches the graph, so it must be the case that there is no input that produces 3.3433

So, 3 is not in the range; the highest that we manage to get to is right here.3442

So, it looks like 1.5 is the highest that we managed to get to with the graph.3448

It never shows up over here, but that is OK, because it shows up somewhere.3453

And then finally, it looks like the lowest we manage to get to is -2.3457

So, our range: the lowest location on our range is -2, and the highest location that we manage to make it to is 1.5.3462

And we hit everything in between: if you go to any higher location in between, it shows up.3471

So, our range is everything in between -2 and 1.5, because all of them have something that they are able to contact; great.3477

All right, I hope you have understood what is going on here; I hope it has really crystallized the idea of a graph.3484

Graphs are so important; they are going to show up in so many things in math.3489

And they are also going to show up in science, and even if you just look in a newspaper.3493

Graphs make up a really, really big part of mathematics,3496

so it is really important that we understand what is going on with them now,3499

because we are going to see a lot of them as we go on.3502

All right, we will see you at Educator.com later--goodbye!3503