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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Properties of Quadratic Functions

Slide Duration:

Table of Contents

I. Introduction
Introduction to Math Analysis

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
XV. Sequences, Series, & Induction
Introduction to Sequences

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

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

1 answer

Last reply by: Professor Selhorst-Jones
Wed Mar 25, 2015 11:26 PM

Post by thelma clarke on March 25, 2015

why is this video skipping and going backwards and repeating I have been watching for over an hour and it has not complete because of all the problems with the vidoe

Properties of Quadratic Functions

  • The graph of a quadratic function gives a parabola: a symmetric, cup-shaped figure.
  • Since the parabola is symmetric, we can draw a line that the parabola is symmetric around. This is called the axis of symmetry.
  • The point where the axis of symmetry crosses the graph is the vertex. Informally, we can think of this point as where the parabola "turns"-the place where the graph changes directions.
  • While parabolas can be very different from one another, they are still fundamentally similar. Any parabola can be turned into another parabola through transformations (see the lesson on Transformations of Functions).
  • With this realization in mind, we can convert any quadratic into the form
    a ·(x−h)2 + k.
    This form shows all the transformations that have been applied to the fundamental square function of x2.
  • We can convert into this format by completing the square (see the previous lesson, Completing the Square and the Quadratic Formula, for more information).
  • In this new form, it's easy to find the vertex. From transformations, we see the the vertex has been shifted horizontally by h and vertically by k. So, in this form, our vertex is at (h, k).
  • If our quadratic is in the standard form of f(x) = ax2+bx+c, the vertex is at



     
    b

    2a


     
    ,   f

     
    b

    2a


     


     



     
    .
  • By knowing the vertex of a parabola, we can find the minimum or maximum that the quadratic attains.
  • Once we know the location of the vertex, it's extremely easy to find the axis of symmetry. The axis of symmetry runs through the vertex, so if the vertex is at (h,k), then the axis of symmetry is the vertical line x=h.

Properties of Quadratic Functions

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

  • Intro 0:00
  • Introduction 0:05
  • Parabolas 0:35
    • Examples of Different Parabolas
  • Axis of Symmetry and Vertex 1:28
    • Drawing an Axis of Symmetry
    • Placing the Vertex
    • Looking at the Axis of Symmetry and Vertex for other Parabolas
  • Transformations 4:18
    • Reviewing Transformation Rules
    • Note the Different Horizontal Shift Form
  • An Alternate Form to Quadratics 8:54
    • The Constants: k, h, a
    • Transformations Formed
    • Analyzing Different Parabolas
  • Switching Forms by Completing the Square 11:43
  • Vertex of a Parabola 16:30
    • Vertex at (h, k)
    • Vertex in Terms of a, b, and c Coefficients
  • Minimum/Maximum at Vertex 18:19
    • When a is Positive
    • When a is Negative
  • 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

Transcription: Properties of Quadratic Functions

Hi--welcome back to Educator.com.0000

Today we are going to talk about the properties of quadratic functions.0002

In the previous lesson, we worked on finding the roots of quadratics.0005

To that end, we learned how to complete the square, and we derived the quadratic formula, which tells us the roots of any quadratic polynomial.0008

In this lesson, we will continue working with quadratic polynomials.0014

We will explore another important feature of quadratics, the vertex, along with its connection to the shape of the graph.0017

In addition, we will also see a new alternative form to write quadratics in that will give us a lot more information0023

than that general form, ax2 + bx + c, that we have been used to using so far; let's go!0028

Let's take a look at the shape of a quadratic function's graph; we call this shape a parabola.0036

The parabola has this nice curved shape on either side; and we are used to making these--we have been making them0040

since probably pretty much right after we learned how to graph in algebra.0046

We have been learning how to make parabolas--parabolic shapes--so we are pretty used to this.0050

And you see it a lot in nature: if you throw a ball up in the air, it makes a parabolic arc.0054

In fact, that is where the roots of the word parabola come from--they have to do with throwing, back either in Latin or Greek--I am not quite sure.0058

Anyway, we see this kind of shape in every single quadratic function that we graph: we get this parabolic shape, a parabola.0065

No matter the specific quadratic, this shape is always there in some form.0073

It may have been moved; it may have been stretched; it might have been flipped.0077

But it is still a parabola--it still has this cup shape that is symmetric.0080

The graph of any quadratic will give us the symmetric, cup-shaped figure.0084

As we just mentioned, a parabola is symmetric: the two sides of a parabola are always mirror images.0089

If we look at the left side, and we look at the right side, they are doing the same thing all the way to this bottom part.0094

And if we go up, we keep seeing the same thing.0101

We don't have arrows continuing it up; but it is going to be true forever; as long as we are going up, it is also going to be true on this thing.0104

Formally, we can draw a line called the axis of symmetry that the parabola is symmetric around.0111

So, if we go to the right at some height, and we go to the left at the same height, we will notice that the distance here is equal to the distance here.0117

If we do that at a different height (like, say, here), these two are going to be equal, as well.0127

So, that is what we have for symmetry: there is this axis that we can draw down the middle, this imaginary line0134

that we can put down the middle, that is going to make it so that each side is exactly the same as the other--0139

that each side is a mirror reflection--that they are symmetric around that axis of symmetry.0144

We call the point where the axis of symmetry crosses the graph the vertex.0150

So, where that imaginary line intersects our parabola, we call that the vertex.0153

Informally, we can think of this as the point where the parabola turns.0158

It is going down; it is going down; it is going down; and then, at some point, it changes back, and it starts to go up.0162

That point in changing, where it switches the direction it is going, whether up or down,0166

or if it is coming from the bottom, going up, to going down--that is the place of turning.0170

It is changing its direction; and we call that the vertex.0175

Formally, it is the midpoint for the symmetry breakdown; but we can also think of it as where it changes directions, the turning--0178

sort of a corner, as much as a nice, smooth curve can have a corner.0185

Just as a graph of every quadratic has a parabola, we can now see that every quadratic has an axis of symmetry and a vertex.0191

If we go back through all of our previous ones, our original function, f(x)...that has a vertex down here;0198

and then, it has an axis of symmetry through the middle.0205

Our g(x) right here...that one has its vertex down at the bottom, and an axis of symmetry, as well.0208

Our h(x) in green has its vertex right there; and while it is not quite even on the halves,0218

because of just the nature of the graphing window we are looking at, it still has that same axis of symmetry.0225

If we could see what happened farther out to the left, we would see that it is, indeed, symmetric.0231

And finally, I don't actually have the color purple, so I will use yellow, awkwardly, for this purple color right here.0235

It, too, has the same vertex, and it has this axis of symmetry.0244

So, whenever we draw a parabola, we are going to have this vertex, and we are going to have this axis of symmetry.0247

Now, how can we find where they are located--where does this vertex occur; where does this axis of symmetry occur?0252

We know where the vertex and axis of symmetry are for the fundamental parabola, our normal square function, x2.0260

That one is going to be easy; the vertex is at (0,0); we just saw that; and the axis of symmetry will be a vertical line, x = 0.0265

A really quick tangent: why is it called x = 0--why does that give us a vertical line?0272

Well, remember: one way to think of a graph is all of the solutions that are possible--all of the things that would be true.0279

Well, if we have x = 0 here, all that that means is that every point on this graph where the x component is 0.0285

So, that is going to be everything on this vertical thing.0293

So, we are going to have everything on the vertical axis end up being x = 0.0297

We might have (0,-5) or (0,5) or (0,1) or (0,0); but they all end up being the case, that x = 0 for each one of these.0303

So, that is why we end up seeing that a vertical line is defined as x = something,0318

because we are fixing the x, but we are letting the y component, the vertical component, have free rein.0324

All right, that is the end of my tangent; what about those other different parabolas,0329

the ones that aren't our normal square function that we are used to, where it is pretty easy to figure out where it is going to be?0333

Well, we noticed earlier, when we were just looking at all of those parabolas side-by-side on the same graph--0339

we noticed earlier that all parabolas are similar to each other.0344

It is just a matter of being shifted, stretched, or flipped, compared to f(x) = x2, that fundamental square function.0348

Now, shifted/stretched/flipped--that sounds exactly like transformations.0357

Remember when we learned about transformations, when we were first talking about functions?0361

So, we go back to our lesson on transformations, and we refresh ourselves.0365

Let's pretend that we just went back, and we grabbed the transformations for how vertical shifts, horizontal shifts, stretches, and vertical flips work.0369

And if any of this stuff is really confusing in this lesson, go back and watch that there.0375

And it will give you a refresher, and you will think, "Oh, that makes sense, how these things are all coming into being."0378

It all gets fully explained in that lesson; so if you didn't watch it before, it might be a good thing to check out now.0381

So, we go back; we grab that information; it is always useful to think in terms of "I learned this before; that would be useful now."0386

Then, you just go back and look it up in a book; you find it on the Internet; you watch a lesson like this at Educator.com;0392

you re-learn what you need for what you are doing right now.0397

Reviewing our work in transformations, we get that vertical shift is f(x) + k,0400

so we have that f(x) is just x2, because that is our basic, fundamental square function.0406

And we add k, and that just shifts it up and down by k.0412

Horizontal shift is f(x - h); so that is going to be x - h plugging into that square...so it will be (x - h), and then that whole thing gets squared.0416

And that is going to shift it by h; so positive h will end up going to the right; -h will end up going to the left.0429

Stretch: if we want to stretch it, we multiply by a, a times f(x).0434

So, as a gets larger and larger, it will be more stretched vertically, because it is taking it up.0439

And as a gets smaller and smaller (closer and closer to 0), it is going to squish it down, because it is taking it and making any given value output smaller.0444

And then finally, vertical flip: vertical flip is just based on the sign,0453

so it is going to be a negative in front that will cause us to go from being up to being down,0457

because everything will get flipped to its negative output.0462

Now, one quick note: we are now using a slightly different form to shift horizontally.0465

Previously, we were using f(x + k); but now we are using f(x - h); f(x - h) is what we are using right now, (x - h)2.0470

And that is going to help us with parabolas; and we will see why in just a moment.0483

It is going to just make it a little bit easier for us to write out a formula for the vertex.0486

Now, for x + k, positive k shifted to the left; but now, when we are using x - h, positive h will shift to the right.0490

Now, why is that? Remember: if for x + k, when we plugged in the -1, it caused us to shift to the right;0501

then if we plug in a positive h, that is just going to be the same thing as a negative k.0507

A positive h is equivalent to a negative k; so a positive h goes right; a negative k goes right.0512

A negative h goes left; a positive k goes left.0518

For now, we can just switch our thinking entirely to this (x - h)2 form, because that is what we will be working with right now.0521

But it is useful to see what the connection is between when we first introduced the idea of transformations,0527

in our previous lesson, to what we are working with now.0531

All right, we can combine all of this together into a new form for quadratic polynomials: a(x - h)2 + k.0534

This shows all of the transformations a parabola can have.0543

Our vertical shift is the + k over here; the horizontal shift is the - h portion inside of the squared.0546

The stretch or squeeze on it is the a right there; and then, the vertical flip is also shown by the a, just based on the sign of the a.0556

If the a is positive, then we are pointed up; if the a is negative, then we are pointed down.0565

So, all of the information about a parabola can be put into just these three values: k, h, and a.0569

And this also makes sense, because, in our normal form, ax2 + bx + c,0574

all of the information about a parabola can be put in 1, 2, 3 (a, b, c).0579

All of the information you need to make up a quadratic can be put into a, b, and c.0585

So, clearly, it just takes three pieces of information to say exactly what your parabola is.0589

So, it shouldn't be too surprising that there are 1, 2, 3 pieces of information if we swap things around into a different order of looking at it.0594

We can convert any quadratic we have into this form, this a(x - h)2 + k form.0601

For example, all of the various parabolas we saw earlier...f(x) is the one in red; g(x) is the one in blue; h(x) is the one in green;0610

and j(x) is the one in purple that I confusingly I am using yellow to highlight (sorry about that).0620

So, if we look at this, 1 times (x - 0)2 + 0...this means a shift horizontal of 0, a shift vertical of 0, and it is a non-stretched, normal-looking thing.0626

And that is exactly what we have right here with our red parabola; it seems pretty reasonable.0645

The blue one: x - 2 means that we have shifted to the right by 2, and we have shifted down by 3.0650

And that is exactly what has happened here; we are at 2 in terms of horizontal location, and 3 in terms of height.0659

And then, the 4 here means that we have stretched up, which seems to make sense,0668

because it seems to be pulled up more than when we compare it to the green one.0672

Similarly, we have similar things with green; it has been moved; the -1/5 causes it to flip and sort of stretch out.0676

It has been squished out, so it is not quite as long.0683

And then, once again, our confusing yellow is purple: -11 has been flipped down, and it is even more stretched out than the blue one is.0688

It is even more stretched out, because it has a larger value (11 versus 4).0696

The negative just causes it to flip.0700

We see how this stuff connects; how do we convert from our general form, ax2 + bx + c, into this new alternate form?0702

We are used to getting things in the form ax2 + bx + c; we are using to working with things in the form ax2 + bx + c.0710

So, we might want to be able to convert from ax2 + bx + c into this new form that seems to give us all of this information.0717

It is very easy to graph with this new form, because we just set a vertex, and then we know how much to squish it or stretch it.0722

Now, how do we do this?--we do it through completing the square.0729

So, if you don't quite remember how we did completing the square--if you didn't watch the previous lesson,0733

and you are confused by what is about to happen--just go and check out the previous lesson.0736

It will get explained pretty clearly as you work through that one.0739

So, assuming you understand completing the square, let's see it get used here.0742

We have ax2 + bx + c right here; and so, what we do is take the c,0745

and we just sort of move it off to the side, so we don't have to work with it right now.0750

And then, we put parentheses around this, and we pull the a out of those parentheses.0753

So, since we divide this part by a--since we divide the ax2 by a to pull it out right here--0758

then we also have to divide it from the b, as well; so we get b/a.0764

We have a(x2 + b/a(x)) + c.0768

So, if you multiply that out, you will see that, yes, you have the exact same thing that you just started with.0773

The next step: remember, when you are completing the square, you want to take this thing right here;0777

and then you want to divide that by 2, square it, and then plug it right back in.0783

So, if we do work through this, we get b/a, over 2, would become b/2a; and we would square that, and we would get b2/4a2.0789

So, that is what we are plugging in right here: b2/4a2.0799

But since we are putting something in somewhere, we have to keep the scales balanced.0804

If we put one thing into some place, we can't just have nothing else counteract that.0808

For example, if I had 5, and I wanted to put in a 3 (just, for some reason, I felt like putting in a 3):0813

5 is totally different than 5 + 3; so you have to put in something else to counteract that.0820

What will counteract + 3? Minus 3 will counteract that.0825

Now, notice, though: we have this a standing in front; so if we put b2/4a2 right here,0829

this a is going to multiply it, so it is effectively like we put in more than that.0836

Going back to 5 + 3, if we had a 2 standing out front, and we wanted to have this be just the same thing as 2 times 5--0841

we wanted this to be the same as this--then 2(5 + 3)...well, this may be the same thing as a 6 inside of it.0849

So, we would have to put a -6 on the outside; 2 times 3; we are having this thing on the outside that has to be dealt with on what we just added in.0857

Otherwise, we are not going to have equal scales.0868

We have this thing on the outside, this extra weight modifying things, this a on the outside of our quantity.0871

And so, if we don't deal with that, when we figure out how much to put on the outside part (-6 or -3),0877

it is going to end up not being equal on the two sides anymore.0886

So, if that is the case, then we have -b2/4a, because if we think about it,0889

we have the yellow a (just because yellow is a little bit hard to read...) times this thing right here, b2/4a2.0897

So, that ends up being the a here and the a2...the a2 cancels to just a, and that cancels out a right there.0911

So, we have b2/4a; and so, that right there is what we are going to need to subtract by.0916

We will subtract by that, and we will end up having it still be equal.0922

And if you work that out, if you multiply that a out into that entire quantity and check it out,0925

you will see that we haven't changed it at all from ax2 + bx + c; it is still equivalent.0930

So, at this point, we can now collapse this thing right here into a squared form: (x + b/2a)2.0934

Let's check and see that that actually makes sense still: (x + b/2a)2.0942

So, we get x2, x(b/2a) + (b/2a)x...so that becomes b/a(x), because they get added together twice; + b2/4a2.0946

So sure enough, that checks out; that is the same thing.0958

-b2/4a + c...we just want to put that over a common denominator.0960

So, if we have c, that is going to be the same as 4ac, all over 4a; so now they are over a common denominator.0965

So, this + c here becomes the 4ac right there; so we have (-b2 + 4ac)/a.0972

And now, we are back in that original alternate form...sorry, not in that original alternate form...0977

We have gone from our original, normal form of ax2 + bx + c into our new alternate form.0983

We will see how it parallels in just a moment.0988

In our new form, it is really easy to find the vertex; we have a(x - h)2 + k;0991

so, from our information about transformations, we see that the vertex has been shifted horizontally by h:0996

a horizontal shift of h, and then vertically by k.1002

So, in this form, our vertex is at (h,k); it is as simple as that.1007

What about ax2 + bx + c, if we start them that way?1013

Well, we just figured out that ax2 + bx + c is just the same thing as this right here.1016

Now, that is kind of a big thing; but we can see how this parallels right here.1021

So, we have -h here; we have +b/2a here; so with h, it must be the case that we have -b/2a,1025

because otherwise we wouldn't be able to deal with that + sign right there.1035

Then, we also have the +(-b2 + 4ac), all over 4a.1038

And that is the exact same thing as our k.1042

Since they are both plus signs here, they end up saying the same value.1044

We have -b2 + 4ac, over 4a; so that gives us our vertex.1048

Now, I think it is pretty difficult to remember -b2 + 4ac, all over 4a, since it is also different than our quadratic formula.1052

But it is a lot like it, so we might get the two confused.1060

So, what I would recommend is: just remember that the horizontal location of the vertex occurs at -b/2a.1062

And then, if you want to figure out what the vertical location is, just plug it into your function.1070

You have to have your function to know what your polynomial is; so you have -b/2a; that will tell you what the horizontal location of the vertex is.1075

And then, you just plug that right into your function, and that will, after you work it through, give out what the vertical location is.1082

It seems pretty easy to me--and much easier, I think, than trying to memorize this complicated formula.1088

So, I would just remember that vertices occur horizontally at -b/2a.1093

Great; by knowing the vertex of a parabola, we also know the minimum or maximum that the quadratic attains.1099

So, if a is positive, it cups up; it goes up; so if it cups up, then we are going to have a minimum at the bottom.1105

So, if a is positive, it cups up; then we will have a minimum at x = -b/2a, because that is where the vertex is.1112

And clearly, that is going to be the most extreme point; the vertex is going to be the extreme point of our parabola, whether it is high or low, depending.1125

Then, if we have that a is negative, then that means we are cupping down.1132

It points down; so if we are cupping down, then it must be the high point that is going to be the vertex.1139

So, it is a maximum if we have an a that is negative, because we are cupping down; so it will be x = -b/2a.1144

Now, you might have a little difficulty remembering that a is positive means minimum; that seems a little bit counterintuitive, probably.1151

a is negative means maximum...what I would recommend is just to think, "a is positive; that means it is going to have to cup up."1157

Oh, the vertex has to come at the bottom.1163

If a is negative, it means it has to cup down; oh, the vertex is going to come at the maximum.1166

Remember it in terms of that, just honestly making a picture in front of your face,1171

being able to just use your hands and gesticulate in the air in front of yourself; and see what you are making in the air.1174

That makes it very easy to be able to remember this stuff.1180

Just trying to memorize it as cold, dry facts is not as easy as just being able to think, "Oh, I see it--that makes sense!"1182

You are remembering primary concepts; and working from there, it is much easier to work things out.1188

All right, also, if we know the vertex, it is pretty easy to find the axis of symmetry.1193

The axis of symmetry runs through the vertex; since it has to go through the vertex,1199

and the vertex is at a horizontal location of h, then it is just going to end up being a vertical line, x = h.1204

So, for example, if we had f(x) = (x - 1)2 - 1.5, then we would know that that would give us a vertex of x - h + k.1210

So, it would give us a vertex of...-1 becomes h; it is just 1; and then k...since it is + k, it means that k must be -1.5.1222

We don't really care about the -1.5, since we are looking for a vertical line that is just x = h.1232

So, we take that x = h, and we make x = 1; so we get an axis of symmetry that runs right through the parabola at a horizontal location of 1.1237

And sure enough, that splits it right down the middle--a nice axis of symmetry.1248

And here is an incredibly minor note on grammar--this is really minor, but a lot of people are confused by this grammar point.1253

So, I just want you to have this clear, so you don't accidentally say the wrong thing, and it ends up being embarrassing.1259

You might as well know what it is: the singular form, if you want to just talk about one, is "vertex";1264

the singular form, if you want to just talk about one, is "axis."1270

On the other hand, if you want to talk about multiple of a vertex, that is going to be "vertices"; multiple vertex...multiple vertices is what it becomes.1273

Vertexes is kind of hard to say, so that is why it transforms into vertices.1284

Axis becomes axes; so axises, once again, sounds a little bit awkward; so we make axis become axes.1289

So, it is not just one vertex, but many vertices; the earth has one axis, but the plane has two axes.1297

So, it is a really, really minor thing; but you might as well know it, because you occasionally have to say this stuff out loud.1304

All right, we are ready for our first example.1309

If we have this polynomial, f(x) = -3x2 - 24x - 55,1311

and we want to put this in the form a(x - h)2 + k, then we will identify what a, h, and k are.1316

So, how do we end up doing this? We have to complete the square.1323

So, we have to complete the square; and we are going to complete the square on our polynomial.1326

-3x2 - 24x - 55; all right, so once again, if you don't quite remember how to complete the square,1333

you can also get a chance to see more completing the square in the previous lesson.1341

But you also might be able to just pick it up right here.1344

So, the first thing we have to do is separate out that -55, so we can see things a little more easily.1346

We aren't actually going to do anything to it; we are just going to shift it, literally shift it to the side,1352

just so we can see things and keep our head clear of the -55.1356

It is still part of the expression; we are just moving it over right now.1359

Now, we want to pull out the -3 to clear things out.1362

So, if we are pulling out -3 on the left, we have to pull out -3...1365

If we are pulling -3 out of -3x2, we also have to pull it out of -24x.1369

If we are going to do that--if we are pulling out -3 out front, dividing into -3x2: -3x2,1373

divided by -3, becomes just positive x2; great; and then -24x/-3 becomes + 8x; minus 55 still.1380

Now, we might be a little bit unsure of this, doing the distribution property in reverse.1390

It might be a little bit worrisome; we are not used to doing that yet.1395

So, we might want to check this: -3 times (x2 + 8x); -3x2 - 24x; great; that checks out, just like what we originally had.1398

So, -3(x2 + 8x) - 55 is the exact same thing as what we started with right here.1408

So, checking like this--you will probably find that you can just do it in your head.1415

You could do -3 times that quantity in your head, and think, "Yes, this makes sense."1418

But if it is an exam, if you have something like that, you definitely want to check under those situations.1421

Make sure that if it is something really important, you are really checking and thinking about your work,1425

because it is really easy to make mistakes, especially if it is new to you.1429

All right, -3(x2 + 8x)--how do we get this to collapse into a square?1432

How do we get this to actually pull into a square?1438

Well, if you remember, we were talking about completing the squares; we want to take this number,1440

and we want to add in 8 divided by 2, and then squared: (8/2)2 is 42, which is equal to 16.1445

So, we want to add 16 inside; so we have -3(x2 + 8x + 16); so we are adding 16 in.1455

But now, remember, if we are putting something into the expression, we have to make sure that we keep those scales balanced.1466

If you put 16 in, we have to take away however much we just put in.1472

Now, we didn't just put 16 in; there is also this -3 up front, so we put 16 into the quantity, but that gets multiplied by -3, as well.1477

So, what did we put, in total, into the expression?1484

We put -3 times 16, in total, into this expression.1487

-3 times 16...3 times 10 is 30; 3 times 6 is 18; so it is -48 in total.1491

So, if we put -48 into the expression, we need to take -48 out of the expression.1498

What is the opposite to -48? Positive 48; so if we put -48 in, and we put in positive 48 at the same time, it is going to be as if we had done nothing.1505

We have that positive 16 inside of the quantity; but because of that -3 out front,1515

it is as if we had put in a -48; so we put in a positive 48 outside of the parentheses, and it is as if we had had no effect at all.1521

So, it is still equivalent to what we started with: + 48 - 55...at this point, we just finish things up and collapse the stuff.1529

x2 + 8x + 16 becomes (x + 4)2; we know that because it is going to end up being 8/2, which is 4.1535

Let's check it really quickly: (x + 4)2: x2; x(4) + 4(x) is 8x; 4 times 4 is 16; great, it checks out.1543

And then, plus 48 minus 55; 48 - 55 becomes + -7; great.1552

At this point, we are ready to identify: we have a = -3, because that one is in front of our multiplication.1559

Then, h is equal to -4, because it is the one right here; but notice it is -h, and what we have is + 4.1567

So, since we have + 4, it has to be h = -4; otherwise we are breaking from that form.1579

And then finally, k is equal to -7.1585

Great; and we have everything we need to be able to put it in that form.1591

All right, the next example: Give a function for the parabola graph below.1594

We have this great new form; and look at that--it looks to me very like we have the vertex right here.1599

It is pretty clearly the lowest point on that parabola; so it must be the vertex,1605

if it is the absolute lowest point on a parabola that is pointing up.1609

So, f(x) = a(x - h)2 + k.1612

Great; so what is our (h,k)? Well, (h,k) is going to come out of this, so (h,k) is what our vertex is, which is (2,1).1621

So, h = 2; k = 1; we plug that information in; and we are going to get that f(x) is now equal to a(x - 2)2 + 1.1631

All right, so we are close, but we still don't know what a is.1645

But look over here: we have this other point over here with additional information.1648

We know that, when we plug in -1, we get out 4; f(-1) = 4.1654

So now, if f(-1) = 4, then we can say that 4 = a(-1 - 2)2 + 1.1661

4 = a(-3)2 + 1; 4 = a(9) + 1; we subtract the 1 from both sides; we get 3 = a(9).1676

We divide out the 9, and we get 1/3 (3/9 becomes 1/3) equals a.1689

So finally, our function is f(x) = 1/3(x - 2)2 + 1; great.1695

And if the problem had asked us to put it in that general form, ax2 + bx + c, that we are used to,1708

at this point you could also just expand 1/3(x - 2)2 + 1; you would be able to expand it1713

and work through it, and you would also be able to get into that form, ax2 + bx + c.1719

Remember: you can just switch from one to the other.1722

To switch from this new form into our old, general form, you just expand.1724

If you want to go from the old general form, you complete the square, like we just did in the previous example.1730

All right, the third example: f(x) = 6x2 - 18x + 5; does f have a global minimum or a global maximum?1735

And then, which one and at what point?1742

So, if f(x) = 6x2, then notice a: ax2 + bx + c...they end up being the same in f(x) = a(x - h)2 + k.1746

So, a = 6; if a equals 6, then a is positive; if a is positive, we have that it cups up; if it cups up, then it is going to have a minimum.1756

What it has is a global minimum on it.1774

Now, where is it going to happen? The vertex horizontally is x = -b/2a.1779

What is our b? Our b is -18 (remember, because it is ax2 + bx...so if it is -18, then b is -18);1793

so we have negative...what is b? -18, over 2 times...what was our a? 6.1802

We simplify this out; so we get positive 18 (when those negatives cancel), over 12, equals 3/2.1808

So, our x location is at 3/2; and now, what is our y going to be at?1814

Well, we can figure that out by f of...plug in 3/2...equals 6(3/2)2 - 18(3/2) + 5.1821

We work through that: 6(9/4) (we square both the top and the bottom) - 18(3/2)...well, we can knock that out, and this becomes a 9.1835

So, we have 9(3), or -27 + 5 equals...6(9/4)...well, this is 2(2); 6 is 3(2); so we knock out the 2's;1845

and we have 3(9) up top; 27/2 minus...what is 27 + 5? That becomes 22.1858

So, we can put -22 over a common denominator with 27/2; we get 27/2 - 44/2; (27 - 44)/2; 27 - 44...we get -17/2.1866

So, that is our y location; so that means that the point where we have our minimum is point (3/2,-17/2).1886

And there is the point of our global minimum.1896

Great; all right, the final one: this one is a big word problem, but it is a really great problem.1900

A Norman window has the shape of a semicircle on top of a rectangle.1905

If the perimeter of the window is 6 meters in total, what height (h) and width (w) will give a window with maximum area?1910

At this point, how do we do this? The first thing we want to do is that we want to understand what is going on.1917

This seems to make sense: we have a semicircle--that is a semicircle right here.1922

What is a semicircle? A semicircle is just half of a circle; and yes, that looks like half of a circle, on top of a rectangle.1928

It is on top of a rectangle, and we have our rectangle right here; so that seems to make sense, right?1935

We have a semicircle on top of a rectangle box; OK, that idea makes sense.1940

The perimeter of the window is 6 meters; now, you might have forgotten what perimeter is; but perimeter is just all of the outside edge put together.1945

A nice, hard-to-see yellow...we go outside: perimeter, perimeter, perimeter, perimeter, perimeter, perimeter, perimeter, perimeter, perimeter...1952

It is going to be all of the outside of that box, but it is not going to go across now;1959

it is going to also have to be the outside perimeter of the top part of the window.1964

We have that dashed line there to indicate that that is where we split into the semicircle.1968

But there is no actual material there; the perimeter is just the very outside part.1971

The perimeter will be the top part of that semicircle, and then three sides of our box.1975

The fourth side doesn't actually exist; it is just connected into the semicircle.1980

All right, that makes sense; we know that it is 6 meters, and then we want to ask what height and width will give a window with maximum area.1984

So, if we are looking for area, we are going to need to use area; so A = area--we will define that idea.1991

If we also know information about the perimeter, then we will say P = perimeter.1998

OK, let's work these things out: area is equal to the area of this part, plus the area of our box;2008

so it is the area of our semicircle, plus the area of our box.2016

So, area of our semicircle...well, what is the area of our circle?2019

Area for a circle is equal to πr2; so a semicircle: if it is a half-circle,2026

that is going to be area = 1/2πr2, because it is half of a circle.2034

So, we have 1/2πr2; well, we don't have an r yet, so we had better introduce an r.2041

So, we will say that here is the middle; r is from the middle of our circle, out like that.2047

Area equals πr2 divided by 2 (because, remember, it is a semicircle, so it is half of it), plus...what is the area of our box?2053

That is height times width.2062

Now, we don't really want to have to have r; the fewer variables, the better, if we are going to try to solve this.2066

So, how does r connect to the rest of it?2071

Well, r...look, that is connected to our w, because it is just half of that side; so r = w/2.2073

We can change this formula into area = π...r2 is (w/2)2...over 2, plus height times width.2083

Now, we have managed to get rid of the radius there; so let's simplify that out a bit more.2093

Area = π...the w/2 gets us w2/4, over 2, plus hw.2097

We have this whole thing divided by 2; so area equals πw2, and we are dividing...2107

so with the 4 and the 2, since they are both dividing on π and that stuff, they are going to combine into dividing by 8...plus hw.2113

Great; so we have that area equals πw2 + hw.2123

All right, what about perimeter--what can we say about the perimeter?2129

Well, the perimeter is going to be equal to...well, there is an h here; there is a w here; what is here? Just another h.2132

And then, what is this portion here? Right away, we see that we have two h's, plus a w, plus something else.2139

Circumference for a circle: if we want the perimeter of a circle, that is circumference.2148

Circle circumference is equal to 2πr; but if it is a semicircle, then it is half a circle; so it is going to be 2πr/2.2153

So, perimeter for a semicircle will be 1/2(2πr), which is just going to be πr.2168

We have 2h + w + (the amount of perimeter that our semicircle puts in is) πr.2177

Now, once again, we don't really want to have extra variables floating around.2183

So, we want to get rid of that; so perimeter equals 2h + w + π(w/2).2186

Great; at this point, we see that we have h and w; we have area.2195

So, if we could turn this into...we have that area is unknown; we don't really know what the maximum area is,2201

or what area we are dealing with; really, it is a function that is going to give area when we plug in h and w.2205

So, it is not even something that has a fixed value, necessarily.2210

What about perimeter, though? Perimeter is something we do know.2213

Remember: perimeter is 6 meters; so we can plug in 6 = 2h + w + π(w/2).2216

Now, it seems like we have more w's than we have h's, in both our area and perimeter stuff.2225

So, let's try to get rid of area; we will figure out what h is in terms of w, so we can substitute out the h and switch it in for stuff about w.2232

We will move everything over, and we have 6 - w - π(w/2) = 2h.2242

We divide by 2 on both sides; we get 3 - w/2 - π(w/4) = h.2251

And now, let's just pull those things together, so it will be easier to plug in later, into our one over here,2259

because we want to swap out the h here for it.2264

3 - 2w/4 + πw...now, that part might be a little confusing; but notice that this has minus, and this has minus;2267

so when we combine them together, they are just one minus, because they are actually working together before they do their subtraction.2279

It equals h; and we can even pull out the w onto the outside; so we get 3 - (2 + π)/4 times w = h.2284

Great; now we have an expression about area, and we have an expression about h and w.2296

At this point, we can plug in what we know about h, and we can plug it in for the h in the hw in our area.2304

And we will have area equals...just stuff involving w; and since it is just stuff involving w, we will have just one variable.2312

Maybe we can figure things out; maybe it looks like something--maybe it looks like a quadratic,2319

since, after all, this was all about how quadratics work.2323

So (student logic), it seems pretty likely that it is going to end up looking like a quadratic.2326

And we can apply the knowledge that we just learned.2329

So, the maximum area is at what h and w?2332

We have area = πw2/8 + wh; and h = 3 - (2 + π)/4 times w.2335

OK, great; so at this point, we take h here, and we plug it in over here.2344

We have area equals πw2/8 (that is still the same thing), plus w times h,2349

so w times...what is h?...3 - (2 + π)/4 times w.2358

We expand that out; our area is equal to πw2/8, plus w times [3 - ((2 + π)/4)w].2366

So, we get 3w minus...let's keep it as (2 + π)/4, and it just combines with that other w; we have w2.2375

So now, we have a w2 here and a w2 here; so let's make them talk to each other.2382

We get πw2/8; let's actually pull that down--we will make it as2386

(π/8)w2 - [(2 + π)/4]w2 + 3w.2396

So now, we have π/8, and we can bring this stuff to bear, so it will be minus 2 + π, but it used to be a 4.2407

So, it is going to be...to become an 8, we are going to multiply by 2 here, to keep it the same.2416

2 times (2 + π) becomes 4 + 2π; so we have [π - (4 + 2π)]w2 + 3w.2422

We simplify this out: π - 4...so the -4 will come through, and -2π will become -π, over 8, w2 + 3w.2432

Look, if area equals this, then this right here--this whole thing--is a quadratic.2444

If it is a quadratic, then we can use the stuff that we know about where vertices show up.2454

Where do vertices show up on the parabola?2460

Now, we are looking for the maximum area; so we had better hope that our parabola points down, so it does have a maximum at its vertex.2462

Sure enough, we have a negative here and a negative here.2469

And since that is on the w2, that means we can pull out the negative, and our first a is going to be -(4 + π)/8.2473

That whole thing ends up being a negative number; so sure enough, it is going to have a vertex at the top,2481

so it will have a maximum area out of this--it is cupping down.2487

At this point, we have figured out--remember--the vertex is at our horizontal location (in this case,2492

horizontal would be just our w); vertex is going to be at w = -b/2a.2501

So, what is that? It is going to be w =...what is our b? That is going to be 3, so we have -3, over...2508

what is a? a is this whole thing, (-4 - π)/8.2517

This is a little bit confusing; but we have a fraction over a fraction, so if it is like this,2526

we can multiply the top and the bottom by 8; 8 on top; 8 on the bottom; we will get -24, and the 8's down here will just cancel out.2529

So, we will get -24 - 4 - π we see that there are negatives everywhere, so we can cancel out all of our negatives.2538

Multiply the top and the bottom by -1; we get positive 24, over (4 + π).2546

The maximum is going to happen at 24/(4 + π); there is our...2552

Oh, sorry: one thing that I just realized--I made one tiny mistake: it is 2a; so we have a 2 here.2562

So, it is still a 2 up front; so it is not 4 + π.2568

It is always important to think about what you are doing.2576

24/2(4 + π) cancels to be 12/(4 + π).2578

Now, that was clearly a very long, very difficult problem; but we see that it is actually really similar to the previous example that we just did.2587

All we are looking for is where the vertex is; the only thing is that it is couched inside of a word problem.2594

So, we just have to be carefully thinking: how do we build equations?2599

Once we have our equations built, how do we put them together?2601

How do we get this to look like something where we can apply what we just learned in this lesson?2604

How can I apply this stuff about quadratic properties?2608

So, we find out that the maximum occurs when w is equal to 12/(4 + π).2611

Now, they asked what h and w; so since we have to figure that out, we know h is equal to 3 - (2 + π)/4 times w.2616

So, h is going to be at its maximum, as well, since it is h and w.2627

3 - (2 + π)/4 times (12/(4 + π)) (as our w): we notice that 4 can take out the 12, and we will get 3 up top.2632

We get that h is equal to 3 - [3(2 + π)] over (4 + π).2647

We want to put them over common denominators, so we can get the two pieces talking to each other.2656

We have 3(4 + π)/(4 + π); and then it is going to be minus 3(2 + π), so - 6 - 3π.2659

We have 12 + 3π, minus 6 - 3π, all over 4 + π.2672

So finally, our h is going to be...the -3π's cancel each other out; for 12 - 6, we get 6, over 4 + π.2681

And that is our value for what our maximum h will be with our maximum width.2691

So, the maximum area will occur when our width is 12/(4 + π), and our h is 6/(4 + π).2695

They will both be in units of meters, because meters is what we started with for our perimeter.2702

All right, I hope you have a sense of how quadratics work, what their shape is, and this idea of the vertex,2706

and that being where maximum and minimum are located.2711

Now, remember: you just have to remember that the horizontal location for maximum or minimum--2713

the horizontal location for the vertex--is going to occur at -b/2a, when we have it in that standard form of ax2 + bx + c.2717

As long as you remember -b/2a, you can just plug it in any time that you need to find what the vertical location going along for that vertex is.2725

All right, we will see you at Educator.com later--goodbye!2732