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Vincent Selhorst-Jones

Vincent Selhorst-Jones

Transformation of Functions

Slide Duration:

Table of Contents

I. Introduction
Introduction to Precalculus

10m 3s

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

45m 11s

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

35m 31s

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

35m 2s

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

48m 43s

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

56m 31s

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

39m 54s

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

58m 26s

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

48m 49s

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

29m 20s

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

48m 35s

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

33m 24s

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

51m 42s

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

49m 37s

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

28m 49s

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

38m 41s

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

41m 7s

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

39m 43s

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

45m 34s

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

46m 8s

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

45m 36s

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

19m 9s

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

33m 22s

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

34m 16s

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

49m 7s

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

44m 56s

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

35m 17s

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

47m 4s

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

40m 31s

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

42m 33s

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

34m 10s

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

48m 46s

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

39m 5s

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

43m 16s

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

33m 5s

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

52m 3s

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

36m 4s

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

27m 18s

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

32m 58s

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

31m 8s

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

19m 11s

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

23m 16s

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

52m 52s

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

29m 5s

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

43m 55s

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

25m 43s

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

56m 40s

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

49m 5s

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

27m 37s

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

34m 25s

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

55m 40s

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

1h 13s

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

41m 1s

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

1h 9m 31s

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

35m 20s

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

54m 7s

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

47m 12s

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

58m 34s

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

53m 33s

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

48m 7s

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

38m 16s

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

40m 43s

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

57m 37s

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

31m 36s

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

44m 3s

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

36m 58s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

57m 45s

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

40m 27s

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

31m 36s

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

39m 27s

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

49m 53s

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

1h 13m 13s

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

40m 22s

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

57m 11s

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

32m 40s

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

32m 43s

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

32m 49s

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

51m 13s

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

45m 26s

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

10m 41s

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

10m 51s

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

10m 38s

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

9m 45s

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

7m 8s

Intro
0:00
Change Graph Type
0:08
Located in General 'Settings'
0:16
Graphing in Parametric
1:06
Set Up Both Horizontal Function and Vertical Function
1:08
For Example
2:04
Graphing in Polar
4:00
For Example
4:28
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Lecture Comments (15)

1 answer

Last reply by: Professor Selhorst-Jones
Tue Jan 6, 2015 12:36 PM

Post by Andrew Demidenko on January 4, 2015

Professor, I am still not clear about Stacking Transformation.

2 answers

Last reply by: John K
Fri Aug 22, 2014 5:13 AM

Post by John K on August 21, 2014

Professor,
Is vertical stretch and horizontal shrink the same thing?

1 answer

Last reply by: Professor Selhorst-Jones
Sat Nov 9, 2013 4:12 PM

Post by Damien O Byrne on November 9, 2013

If you horizontally stretch a function does that mean you vertically stretch the functions graph and visa versa. for example the inverse function 1/x transformed to 1/2x could this be interpreted as multiplying f(x) by 1/2 ( vertical shrink) and also interpretted as f(2x) horizontal shrink. is this just coincidence ?

its just visually if i stretch a graph horizontally shouldn't the y values decrease as in a vertical shrink and if I shrink horizontally shouldnt the y value increase.

1 answer

Last reply by: Professor Selhorst-Jones
Mon Oct 28, 2013 9:59 AM

Post by Charles Reinmuth on October 27, 2013

The vertical stretch and horizontal stretch look very similar to me. I see a difference in that there are parentheticals around the horizontal (eg. f(x) = (3x)^2 ...vs... f(x) = 3x^2)

Still, I don't think I understand fully what is going on. What exactly is the difference? Perhaps I missed something. Thankyou so much!!

1 answer

Last reply by: Professor Selhorst-Jones
Sun Jul 28, 2013 9:11 PM

Post by Jason Todd on July 26, 2013

Professor, in example 2 how did you differentiate vertical vs. horizontal flip possibilities? Thanks in advance.

1 answer

Last reply by: Professor Selhorst-Jones
Thu Jul 11, 2013 1:11 PM

Post by Sarawut Chaiyadech on June 28, 2013

Thanks

1 answer

Last reply by: Professor Selhorst-Jones
Thu May 23, 2013 10:47 AM

Post by Matthew Chantry on May 22, 2013

These questions are for Example 2:

1. Shouldn't everything in the h(x) function after the - be in brackets?
2. Could this be seen as a horizontal flip as well? Would that look different?

Thank-you

Transformation of Functions

  • We often have to work with functions that are similar to ones we already know, but not precisely the same. Many times, this difference is the result of a transformation. A transformation is a shift, stretch, or flip of a function.
  • A vertical shift moves a function up or down by some amount. If we want to shift a function f by k units, we use
    f(x) + k.
    [If k is positive, it moves up. If negative, down.]
  • A vertical stretch/shrink "pulls/pushes" the function away from/toward the x-axis by some multiplicative factor. If we want to vertically stretch/shrink a function by a multiplicative factor a, we use
    a ·f(x).
    [If a > 1, the function stretches. If 0 < a < 1, it shrinks. If a=1, nothing happens.]
  • A horizontal shift moves a function left or right by some amount. If we want to shift a function f by k units, we use
    f(x+k).
    [If k is positive, the graph moves left. If k is negative, the graph moves right. (This may seem counter-intuitive, but remember that the shift is being caused by how f "sees" (x+k). Check out the video for an in-depth explanation of what's going on.)]
  • A horizontal stretch/shrink changes how fast the function "sees" the x−axis. If we want to horizontally stretch/shrink a function by a multiplicative factor a, we use
    f(a ·x).
    [If a > 1, it shrinks horizontally ("speeds up"). If 0 < a < 1, it stretches horizontally ("slows down"). (This may seem counter-intuitive, but remember that the stretch/shrink is being caused by how f "sees" (a·x). Check out the video for an in-depth explanation of what's going on.)]
  • To vertically flip a graph (mirror over the x-axis), we need to swap every output for the negative version. If we want to vertically flip, we use
    −f(x).
  • To horizontally flip a graph (mirror over the y-axis), we need to "flip" how f "sees" the x-axis. We do this by plugging in −x (which is effectively a "flipped" x). If we want to horizontally flip a function, we use
    f(−x).
  • If you want to do multiple transformations, just apply one transformation after another. However, order matters, so start by deciding on the order you want the transformations to occur in. Then apply them to the base function in that order.

Transformation of Functions

Let f(x) = x2. Create a new function g(x) that is a transformation of f(x) after being shifted two units to the right and one unit up.
  • The first transformation given is a horizontal shift. Horizontal shifts are obtained by f(x+k), where a positive k causes it to move left and a negative k causes it to move right. To obtain a shift of two units to the right, we will use
    f(x−2).
  • The second transformation given is a vertical shift. Vertical shifts are obtained by f(x) + k, where a positive k causes it to move up and a negative k causes it to move down. But remember, the order of transformations is important. We want to stack this transformation on top of what we got from the previous step: f(x−2). To obtain a subsequent shift of one unit up, we will use
    f(x−2) + 1.
  • We have found that g(x) = f(x−2) + 1. Now we substitute appropriately to find a formula for g(x):
    g(x) = (x−2)2 + 1.
g(x) = (x−2)2 + 1
[It would also be reasonable to expand it: g(x) = x2 − 4x + 5.]
Let f(x) = 2x+3. Create a new function g(x) that is a transformation of f(x) after being shifted four units down then vertically stretched by a factor of three.
  • The first transformation given is a vertical shift. Vertical shifts are obtained by f(x) + k, where a positive k causes it to move up and a negative k causes it to move down. To obtain a shift of four units down, we will use
    f(x) − 4.
  • The second transformation given is a vertical stretch. Vertical stretches are obtained by a·f(x) , where a positive a > 1 causes it to stretch and 1 > a > 0 causes it to shrink. But remember, the order of transformations is important. We want to stack this transformation on top of what we got from the previous step: f(x) − 4. To obtain a subsequent vertical stretch with a factor of three, we will use
    3·[f(x) − 4].
  • We have now found that g(x) = 3·[f(x) − 4]. Now we substitute appropriately to find a formula for g(x):
    g(x) = 3 ·[(2x+3) − 4].
    From here, simplify to obtain g(x). [If you don't see why the order of transformation is important, try it in the other order and notice how you get a different result.]
g(x) = 6x−3
Let f(x) = x−√{3−2x}. Create a new function g(x) that is a transformation of f(x) after being flipped horizontally, then flipped vertically, then shifted to the left by two units.
  • The first transformation given is a horizontal flip. Horizontal flips are obtained by f(−x) . Thus we will use
    f(−x).
  • The second transformation given is a vertical flip. Vertical flips are obtained by − f(x). But remember, the order of transformations is important. We want to stack this transformation on top of what we got from the previous step: f(−x). Thus, to obtain a subsequent vertical flip, we will use
    − f(−x).
  • The third transformation is a horizontal shift. Horizontal shifts are obtained by f(x+k), where a positive k causes it to move left and a negative k causes it to move right. But remember, the order of transformations is important. We want to stack this transformation on top of what we got from the previous step: −f(−x). To obtain a shift of two units to the left, we will use
    − f
    − [x+2]
    .
  • We have now found that g(x) = − f(− [x+2]). Now we substitute appropriately to find a formula for g(x):
    g(x) = −

    −[x+2]
      ⎛


    3−2
    −[x+2]
     

    .
    From here, simplify to obtain g(x). [If you don't see why the order of transformation is important, try it with the horizontal shift first and notice how you get a different result.]
g(x) = x+2 +√{2x+7}
The parent function in red is f(x) = |x|. Give a function for the graph in blue.

  • Notice the difference between the two graphs. The shape of the function is the same, the only difference is that its location has been changed. It has been shifted to the right by two units and shifted down by four units.
  • Horizontal shifts are obtained by f(x+k), where a positive k causes it to move left and a negative k causes it to move right. Vertical shifts are obtained by f(x) + k, where a positive k causes it to move up and a negative k causes it to move down.
  • In this case, the order of the transformations doesn't matter because we get the same result whether it's right-down or down-right. Thus, we have
    g(x) = f(x−2) − 4.
g(x) = |x−2| −4
The parent function in red is f(x) = x3. Give a function for the graph in blue.

  • Notice the difference between the two graphs. The shape of the function is the same, but the graph has been flipped vertically then moved to the left by two units and up by one. [The vertical flip could also be seen as a horizontal flip.]
  • Vertical flips are obtained by −f(x). Horizontal shifts are obtained by f(x+k), where a positive k causes it to move left and a negative k causes it to move right. Vertical shifts are obtained by f(x) + k, where a positive k causes it to move up and a negative k causes it to move down.
  • In this case, the order of the transformations matters, so we have to be careful to do the transformations in order to f(x). Doing the steps in order gives us:
    f(x)  ⇒ − f(x)  ⇒ −f(x+2)  ⇒ −f(x+2) +1
  • Thus, g(x) is the result in the final step:
    g(x) = −f(x+2) +1.
g(x) = − (x+2)3 + 1
The graph of f(x) = x2 is below. Using your knowledge of transformations, graph g(x) = −x2 + 4. [Do this without making a table of values and plotting points.]
  • The two changes to f(x) are the negative that has been applied to x2 and the addition of 4. Notice that the negative has to be applied first, because if it was applied to the whole function after the 4 had been added, it would have become −4.
  • Since the negative must have been applied first, that is the first transformation. To get from x2 to −x2, we apply a negative to the function:
    −f(x).
    A negative applied to the entire function means a vertical flip.
  • Adding 4 is applied second, so that is the second transformation. To get from −x2 to −x2+4, we add a 4:
    −f(x) + 4.
    Adding 4 to the entire thing causes it to be vertically shifted up by four units.
  • Taking these two transformations, we can adjust the initial graph of f(x) to create g(x): flip the graph vertically, then move it up four units.
The graph of f(x) = √x is below. Using your knowledge of transformations, graph g(x) = √{−x+4} + 1. [Do this without making a table of values and plotting points.]
  • The three changes to f(x) are the negative that has been applied to x, the addition of 4 inside the root, and the addition of 1 outside the root.
  • Notice that there are two ways of getting a root of √{−x+4} using transformations:


     

    (−x) + 4
     
        or    

     

    −(x−4)
     
    The first way translates to a horizontal shift of four units to the left then a horizontal flip, while the second way translates to a horizontal flip then a shift of four units to the right. The second way is slightly easier to visualize, so we will work with that one. [With both of these, it can be difficult to see how the transformations are ordered just from the math. Figuring it out comes from understanding which transformation goes into another.]
  • No matter how we look at the horizontal transformations, we can see that the +1 at the end of g(x) causes a vertical shift upward of one unit.
  • Taking these three transformations, we can adjust the initial graph of f(x) to create g(x): flip the graph horizontally, then move it right four units, and finally up one unit
Let g(x) = (x−6)2 + 47. What is the parent function f(x) and what were the transformations applied to f(x) to create g(x)?
  • We see the ( )2 in g(x), which alerts us it to being based off the parent function f(x) = x2.
  • From f(x) = x2, the x has been replaced by (x−6). That's the same as f(x−6), which means a horizontal shift to the right of six units.
  • The other change is the +47, which we would obtain from adding 47 to f(x). Thus g(x) also has a vertical shift upward of 47 units.
    f(x−6) + 47 = (x−6)2+47 = g(x)
The parent function is the square function: f(x) = x2.
The transformations are a shift to the right of six units and an upward shift of 47 units.
Let g(x) = −3·|x+1|. What is the parent function f(x) and what were the transformations applied to f(x) to create g(x)?
  • We see the | | in g(x), which alerts us it to being based off the parent function f(x) = |x|.
  • From f(x) = |x|, the x has been replaced by (x+1). That's the same as f(x+1), which means a horizontal shift to the left of one unit.
  • The other change is the absolute value being multiplied by −3, which we obtain by multiplying −3 and f(x+1). This is actually two transformations rolled into one operation: a vertical flip and a vertical stretch by a factor of three.
    −3 ·f(x+1) = −3 ·|x+1| = g(x)
  • Notice that in this case, the order of the transformations will not affect the result. This is not always true, but in the case of this problem, it does not matter what order the transformations were applied in.
The parent function is the absolute value function: f(x) = |x|.
The transformations are a shift to the right of one unit, a vertical flip, and a vertical stretch by a factor of three.
Below is the graph of g(x). What is the parent function g(x) comes from? Using your knowledge of transformations, write out g(x).
  • Looking at the graph, we see that both ends go off in the same direction and that as it goes farther left or right from its center, its slope becomes larger. Thus it is based on the square function f(x) = x2, whose graph is below:
  • Comparing it to the graph of f(x) = x2, we see that it has been flipped vertically, then moved four units to the left and 8 units up.
  • As steps for transforming a function, the three transformations above give
    f(x)  ⇒ −f(x)  ⇒ −f( x+4)  ⇒ −f(x+4) + 8
    Therefore, g(x) = −f(x+4) + 8.
The parent function is the square function f(x) = x2.
g(x) = − (x+4)2 + 8

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Transformation of 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 0:04
  • Vertical Shift 1:12
    • Graphical Example
    • A Further Explanation
  • Vertical Stretch/Shrink 3:34
    • Graph Shrinks
    • Graph Stretches
    • A Further Explanation
  • Horizontal Shift 6:49
    • Moving the Graph to the Right
    • Moving the Graph to the Left
    • A Further Explanation
    • Understanding Movement on the x-axis
  • Horizontal Stretch/Shrink 12:59
    • Shrinking the Graph
    • Stretching the Graph
    • A Further Explanation
    • Understanding Stretches from the x-axis
  • Vertical Flip (aka Mirror) 16:55
    • Example Graph
    • Multiplying the Vertical Component by -1
  • Horizontal Flip (aka Mirror) 18:43
    • Example Graph
    • Multiplying the Horizontal Component by -1
  • Summary of Transformations 22:11
  • Stacking Transformations 24:46
    • Order Matters
    • Transformation Example
  • Example 1 29:21
  • Example 2 34:44
  • Example 3 38:10
  • Example 4 43:46

Transcription: Transformation of Functions

Hi--welcome back to Educator.com.0000

Today, we are going to talk about transformations of functions.0002

By now, we are familiar with a variety of different functions--things like x, x2, √x, |x|, etc., etc.0005

We have seen a bunch of different fundamental parent functions--that function petting zoo we visited in the last lesson.0013

However, we often have to work with or graph functions that are similar to these fundamental functions, but not precisely the same.0019

They are not the ones we are already familiar with, exactly.0025

Many times, the difference is the result of a transformation.0027

A transformation is a shift, a stretch, or a flip of a function.0031

And when I say that, I mean that graphically; it has been moved either left/right or up/down;0036

it has been stretched either horizontally or vertically; or it has been flipped vertically or horizontally.0041

Understanding transformations is useful for working with functions or building our own functions, if we want to build one from scratch.0049

For this lesson, we will begin by looking at vertical shifts and stretches, because they are the easiest ones to graph.0056

It is easiest to understand moving things around vertically.0061

Then, we will learn how horizontal shifts and stretches work, and what we have understood in vertical0064

will give us a slightly better understanding of what is going on horizontally; and finally, we will look at flips; all right, let's go!0067

The very first one is vertical shift; this is the easiest one of all--to shift vertically, you simply add to a function.0073

Positive numbers shift up; negatives shift down; so let's see an example.0079

Consider f(x) = x2 graphed with f(x) + 2 = x2 + 2 and f(x) - 7, which equals x2 - 7.0083

f(x) = x2, our base function, is the red part of our graph.0092

We want to see what f(x) + 2 is; that is the blue graph; and finally, the green graph is f(x) - 7.0096

Notice: if we just take a function, and we add 2 to it, it gets raised by 2 units; the height goes up to 2; it goes up to here.0104

But if we take the function and subtract 7, f(x) - 7, which would be x2 - 7, it goes down by 7.0113

It goes lower off its base of normally starting at (0,0); x2 has its home base, in a way, at (0,0).0121

It drops down by 7; or we can raise it up; so we can move it up and down with this vertical shift.0129

What is going on? Think about it like this: if we wanted to move one point vertically, we would add k to its vertical component, its y-value.0135

So, let's say, hypothetically, we want to change the point (1,2); we want to move it up by 5.0141

So, if we want to move it up by 5, we would just add 5; we would have (1,2 + 5), which would be (1,7).0148

That is what we would get if we wanted to move it up by 5.0158

To move up one point, we would just add k or subtract k; but let's think of it as adding a negative k.0160

We add k, and if k is positive, it goes up; if k is negative, it goes down; that moves one point up.0167

If we want to move all of the points in a graph, then we have to add this to all of the vertical components.0172

Now remember: the vertical components of a function's graph are the function's output.0179

So, the function's output...we just need to make the output be k more, or k less, if it is a negative number.0184

So, we add k to it; we just change the output everywhere by adding k to the function.0190

So, if we start with f(x), and we want to vertically shift by f, we just use f(x) + k.0194

If k is positive, it moves up; if k is negative, it moves down--as simple as that.0201

So, f(x) + k: we take our original function, and we add k to it, and we have a vertical shift of k units; great.0207

Next, the vertical stretch/shrink: we want to vertically stretch or shrink a graph--we want to pull it/stretch it,0214

or we want to shrink it--we want to squish it.0221

We multiply the function by a multiplicative factor, a: if 0 is less than a, which is less than 1 (a is between 0 and 1), the graph will shrink.0223

If a > 1, the graph stretches.0232

Let's see an example: consider f(x) = x2; that is the one in red--that is our basic function,0235

that we are starting with to have a sense of how things are going to go.0241

And then, we compare that to 3 times f(x): it has a multiplicative factor of 3 hitting it.0244

So, in this case, we have the graph stretching, because it is 3 times f(x); so a is greater than 1.0250

And indeed, this one right here has been stretched up; we have the parabola normally,0258

but we have grabbed it and pulled it up higher.0263

If we look at this, every point here is 3 higher; here is 1, 2, 3 higher to get up to there.0266

If we compare that to the shrinking in the green graph, we have a = 1/3; so 1/3 is between 0 and 1, so it has been shrunk; it has been squished down.0276

Where we are at the red...we go to 1/3 of where we are at the red, and we find ourselves on the green one.0289

It has been squished by a factor of 1/3.0296

We can either stretch it with a larger-than-one factor, or squish it with a less-than-one factor, but not in the negatives.0299

We will talk about negatives later.0305

What is going on? First, when we say stretch/shrink, it means we are grabbing a point and pulling or pushing it away from or toward the x-axis.0308

So, for example, let's say we have the point (1,2), and we want to apply some multiplicative factor to the point's height.0317

Its height is just 2; so if a = 1/2, then we get (1,2(1/2)), so we get (1,1).0326

We have halved the point's height; if we were at (1,10), it would become (1,5).0338

We are squishing by a factor of 1/2; we could also expand by a factor of a = 7, and it would get multiplied by 7.0344

So, we have this multiplicative factor either pulling it apart if it is greater than 1, or squishing it together if it is less than 1.0351

If we want to do this to all of the points in the graph of a function, we need to apply that multiplicative factor to all of the function's outputs.0357

To do it to all of the function's output, we need to multiply the function itself by a.0363

So, that will apply it to all of the outputs, because the function tells us where the outputs go.0371

So, if we multiply something against the function, it will have done it to all of the possible outputs that would come out of that function.0374

So, to stretch/shrink a function by a multiplicative factor, a, we use a times our function, a times f(x).0380

If a is greater than 1, the function stretches; if a is between 0 and 1, it shrinks.0387

And finally, if a is equal to 1, then it doesn't do anything; it has no effect, because we are between stretching and shrinking.0395

We are exactly on the middle, and we are just left with the function as it was before; it has no effect.0400

We have grabbed it, and then just immediately let go, instead of pulling it and pushing it together.0405

Horizontal shift: this is a little more complex than vertical shift, but we are ready to talk about this now.0409

To shift horizontally, we need to change where the function sees x = 0.0415

We do this by plugging something different than x into the function.0419

What I mean by that: take, for example, a normal line, a normal f(x) = x kind of line.0422

Well, if we wanted, we could say that in a way, its home base is this 0 point where it crosses that axis.0431

It is where it sees 0 on the x-axis; but we could also talk about another graph where it is the same line, but shifted over to the right.0439

What has happened there is: we have taken this home base, and we have shifted it over by some amount.0448

So, we have the same picture, but it has been moved to the right.0459

We are seeing the home base, the effective x = 0, in a different place; that is the idea we want to bring to this.0461

Consider f(x) = x2 (once again, we see that in red) graphed with f(x) - 4 = (x - 4)2.0469

That one is in blue; it has been shifted four units to the right.0478

And f(x) + 2...that one is in green: (x + 2)2 has been shifted two units to the left.0483

Now, that might seem counterintuitive at first: -4 causes us to shift to the right, but + 2 causes us to shift to the left.0492

To understand what is going on, we need to think about this a little more deeply.0498

What is going on? Think about it like this: the graph of a function is a way to look at how the function sees the x-axis.0501

A graph is where an input gets placed as an output; that is at least one way to interpret a graph, and it is how we are doing it with functions.0508

If we are looking at a graph, it is how it sees the entire x-axis--at least, all of the x-axis in our viewing window--at once.0516

That x-axis is mapped to some sort of curve; it takes in each x-value in the x-axis, and it outputs a y-value.0522

A graph shows how the function is seeing all of the x-axis at the same time.0529

Now, normally we plug in just x; so the x-axis looks like your normal number line: 0 in the middle, 1, 2, 3 out, -1, 2, 3 out in the other direction.0532

1, 2, 3; -1, 2, 3; it is exactly what we are used to; it is your normal number line.0542

But we could move this number line around; we can move this home base.0547

Currently, our home base of 0 (think about it as a home base for now)--0 is in the middle when we plug in just x.0550

How do we move around that home base? We move it around like this.0557

Normally, once again, our normal plane, x--the normal number line--we can move it around by plugging in x + k.0561

For example, if k = -2, then our normal home base now is x - 2, so it is 0 - 2; and we get -2.0569

1 - 2...we get -1; 2 - 2...we get 0; look, our home base has shifted 2 units to the right.0579

We have gone over 1, 2 clicks over to the right.0590

And everything has been moved by this amount: we have the -2 on everything, and so that is how we are getting all of these new points.0593

But what we have gotten by doing this is: we have effectively shifted over the location of home base.0601

We have effectively shifted over the location of x = 0 by subtracting 2 from everything.0607

Since everything is now 2 lower, we had to go where we originally had 2 to now just have 0--to be back in our usual home base.0613

So, I want to say this once again: The graph shifts to the right with a negative.0621

So, if we plug in a negative for k, this x + k here...if we plug in a negative, we get shifting to the right,0627

because we are taking that much away from all of the numbers.0632

And so, if we are taking them away, it has to be the high numbers, the traditionally right-side numbers,0635

that are now going to have our new home base in them.0640

Shifting to the right: horizontal shifts to the right happen with a negative value for k.0643

What if we wanted to shift to the left? Well, we could plug in a different x + k.0649

If we plug in k = 3, then if we have + 3, we now have 3 for our old 0.0655

We have to go to -3 for us to get our home base back.0662

So, it is now 1, 2, 3 over to the left for us to get from our old home base to our new home base.0666

And everything is going to get hit by this + 3, which is why we have 6 over here and all these sorts of things.0675

So, by adding a positive number, it is the lower numbers, the negative numbers,0680

that are now going to end up taking the place of having the home base be on that side.0684

The home base will move to the left if we have a positive k.0688

If we want to shift to the left, we use positive for k in that x + k; great.0691

Therefore, we can shift around where the function sees x = 0...this idea of seeing x = 0,0699

seeing our home base, and the rest of the x-axis in turn, is by plugging in x + k instead of just plugging in x.0705

By shifting around the perceived x-axis, this perceived home base, the graph will move horizontally (move to the left/move to the right).0712

Now, notice: this doesn't actually change the x-axis; it just changes the way the function sees it.0720

The x-axis is still going to look totally normal to us; it will be that normal x-axis that we are used to.0725

But the way that the function will interact with it is now based off of this new home base, because of plugging in that x + k.0730

So, to horizontally shift the function by k units, we use f(x + k).0737

And if k is a positive, we shift left; if k is a negative, we shift right.0741

Now, remember: that seems a little bit counterintuitive at first; but if you think about why it is the case--0749

if we put in a positive number, it has to be the negative side that establishes the new home base, the new 0--0753

if we put in a negative number, then it has to be the positive side, the right side,0759

that establishes the new home base, the new 0--that seems a little counterintuitive,0763

but if you think about those slides, those ideas we just saw, you will think, "oh, yes, it makes sense0768

that I am plugging in the positive to go left and plugging in the negative to go right."0773

All right, horizontal stretch: this idea is similarly complex.0778

But now that we understand how horizontal shifting works, this will probably make more sense.0782

To horizontally stretch or shrink a function--that is, to pull it apart or to squish it together--0785

we need to change how fast the function sees the x-axis.0790

Once again, it is not really, literally seeing it; the function isn't a living, breathing thing.0794

But the way that it is going to interact with it, we can effectively personify it and pretend that it is alive for this.0799

Effectively, we need to stretch/shrink how the function perceives x.0805

We need to change the way that the function will interact with that x-axis.0808

Let's look at some examples first: f(x) = x2, our normal red graph--let's try if we put in 3 times x instead.0813

We have put that multiplicative factor on the x, and we get (3x)2 on our blue graph.0822

We plug in 1/2 times x...(1/2x)2 on our green graph.0828

All right, so let's understand what is going on here--what is going on?0833

This idea is very much like what we did with horizontal shift.0836

We are playing with how the function sees the x-axis.0838

Once again, remember: if we plug in just x, we get this one, our normal number line with 0 in the middle, -1, -2, -3 to the left, and 1, 2, 3 to the right.0840

Great, that is just like normal.0850

But we can speed this up; we can speed up or slow down the experience of this number line.0851

If we plug in an x-axis that has been stretched or shrunk by a multiplicative factor, a;0859

if we have a times x--for example, if we have 3 times x--we speed it up.0863

Instead of 0 to 1, it is now 0 to 3; so it is 0 to 3, and then 3 to 6; and of course, 1 and 2 are still in there.0869

But they have gotten shrunk down; we are speeding up how fast we are moving through the numbers.0879

So, each number effectively has been multiplied by 3; we are moving through the numbers faster, which will condense the graph.0884

The graph will be happening faster horizontally, so it is going to go through what it would do normally, faster.0890

We can also expand it by slowing it down with a small a.0897

If we put in a small a, like 1/2 times x, we go from 0 to 1; and now, we are going 0 to 1/2.0901

We have to take two steps forward before we even manage to make it to 1; so now, we are going at a speed that is 1/2 the speed of originally.0908

We are slowed down by a factor of 2.0914

OK, by applying a multiplicative factor, a, to x, we can change how fast the x-axis looks to the function.0917

This change in horizontal speed either stretches or shrinks the graph horizontally.0925

If it looks faster, the graph will compress, because it has the same amount of things happening in a shorter amount of x-time.0929

If it is stretched, then if we slow it down, it will be stretched, because it takes more x-time to be able to get through the same information.0936

And like before, this doesn't actually change the x-axis that we see; it is just how the function will interact with the x-axis.0945

So, to horizontally stretch/shrink a function by a factor of a, we use f(ax); we play with how that x-axis works.0951

We change around the speed that that x is moving at.0959

So, if a is greater than 1, it will shrink horizontally, because we have sped up.0962

So, a > 1: it shrinks horizontally, because it is speeding up how fast the x-axis goes.0968

We want to think about that in terms of speeding up; it makes it easier to understand what is going on.0973

So, if we speed it up by putting in a large a, a > 1, we are going to shrink horizontally,0977

because more stuff will happen in the same period of "time."0984

0 < a < 1: we are going to stretch horizontally; it will slow down the x-axis,0987

because we now have to go through a longer interval, a longer amount of x-time, for us to be able to get the same information through.0994

a > 1 shrinks horizontally; it speeds up; if a is between 0 and 1, it stretches horizontally; it slows down.1003

All right, vertical flip, which we might also call mirroring vertically, or a vertical mirror:1015

to vertically flip a graph around the x-axis, we simply multiply the function by -1; it is as simple as that.1019

We just multiply the function; so if we have f(x) = x2 in red, then we can flip it vertically by just multiplying by a -1.1025

-f(x), which is -x2...it flips to pointing in the opposite direction.1034

What is going on here? Well, if we wanted to flip a single point around the x-axis--say we have (1,2) again--1039

if we wanted to flip it around the x-axis to the opposite height, then we would just multiply the vertical component,1045

the y-value, by -1; so vertical times -1 would become (1,2(-1)), or (1,-2).1051

We have flipped that point to the opposite vertical location, the opposite height.1063

This sends it to the opposite side; but it still has the same height in terms of distance from the x-axis.1068

It is now a negative height; or if it started negative, it will now be positive.1074

For example, we could have, say, (3,-7); and that would flip to (3,7); so we are flipping from one side to the other side.1077

If we want to do this to all the points on the graph of a function, we need to apply it to the entire function.1089

The vertical components are the outputs of the function, so we need to make the function output the negative version everywhere.1094

We do this by just multiplying the whole function by -1.1100

So, to flip a function vertically around the x-axis, that is if we have smiley-face here, then it will become reverse-smiley-face here;1103

it is flipped vertically; so to flip a function vertically around the x-axis, we use -f(x); great.1115

To horizontally flip a graph around the y-axis, we change how the function sees the x-axis to its opposite.1124

We need to flip its perception of the x-axis, just like we changed perception of the x-axis with horizontal shift and horizontal stretch.1130

We are going to do that for horizontal flip.1137

We do this by plugging in -x; let's see an example--consider f(x) = √x.1139

If we have √x, and that is the red graph (we couldn't use x2, because its horizontal flip1145

will just look like the exact same thing); we will graph that with f(-x).1150

So, we plug in -x, and we will get -√x, which ends up pointing in the exact opposite direction.1157

Why is it pointing in the exact opposite direction?1162

Well, if we tried to plug in a positive number, like, say, positive 6...√-6...if we plug in x = 6, it is going to get us √-6, which does not exist.1164

So, it doesn't exist on the right side, just like √x, normal square root of positive x, doesn't exist on the left side.1177

So, our blue graph has to go in the opposite direction, because it sees -6 as being the same height as the red one sees +6.1185

All right, what is going on here--how is this working?1194

We are reversing how the function sees the x-axis.1197

Normally, once again, we see x going off to the right and to the left, just like usual.1200

But if we plug in -x, it reverses; let's put some color here, so we can see what I am talking about.1205

So, if we have, on our normal, positive x version, that it goes to the right in red, and it goes to the left in blue,1210

when we plug in -x, we see that it goes to the right in blue, and the left in red.1217

If we hit 3 by -1, it becomes -3; -3 by -1 becomes positive 3; so we have flipped the order that the x-axis occurs in.1225

As opposed to going from negative to positive, it now goes from positive to negative; we have flipped the order that it occurs in.1236

To flip a point horizontally around the y-axis, we need to just multiply the horizontal component by -1.1244

For example, our point (1,2): if we want to flip horizontally, then we are just going to look at the negative version of the x-axis.1249

So, we would go to (-1,2); so this will move the point to the opposite horizontal location.1260

If we plug -x into a function, it reverses how it sees the x-axis throughout.1268

So, we will be plugging in opposite horizontal locations everywhere; so everything will flip to the opposite horizontal location.1272

All of the points are going to show up in the opposite horizontal location, because we have plugged in this -x.1279

To flip a function horizontally around the y-axis, we use f(-x).1284

What does that mean? Once again, say we have some smiley-face over here.1289

Smiley-face, sadly, doesn't have anything left-right; he is a perfect left/right thing.1295

So, let's make smiley-face--it is now Ms. Smiley-face, and Ms. Smiley-face has a little bow.1304

So, if we flip her around the y-axis--we flip her horizontally--she will show up on the other side.1311

And her face will look the same, because her face is mirror-symmetric horizontally; but now her bow is going to be on the opposite side.1319

All right, so she shows up on the opposite side now; she has been mirrored horizontally around the y-axis.1325

Here is a summary of transformations; I know it is a lot of transformations that we have seen at this point.1332

So, don't worry if you have to refer back to this list later on.1336

Also, if you currently have some sort of book that you are working on along with this course in,1340

or if you have another teacher who is working in a book, you are almost certainly1344

going to be easily able to find a table of these in any section where they would be teaching the same things in that book.1347

This table is really useful, because it can be a little hard to remember all of them immediately.1352

Vertical shift is f(x) + k; k is positive; that causes us to go up; k is negative--that causes us to go down.1356

Vertical stretch is a(f(x)); if a is between 0 and 1, it shrinks it; if a is greater than 1, it stretches it.1362

Horizontal shift is plugging in x + k; if k is positive, we go to the left; if k is negative, we go to the right.1369

Horizontal stretch is...0 to 1 means we slow down, and slowing down means we stretch out.1378

a greater than 1...did I say horizontal stretch?...f(a) times x...I am not quite sure I said that...1384

a greater than 1 causes us to go faster, which means we squish together.1390

Vertical flip: we flip over vertical; that is -f(x); horizontal flip, f(-x), causes us to flip horizontally.1394

One thing to notice: all the vertical stuff happens outside the function.1402

If it is vertical shift, it is f(x) + k; if it is vertical stretch, it is a(f(x)); if it is vertical flip, it is negative times f(x).1415

Everything is doing it on the outside of the function.1427

However, horizontal things happen inside: horizontal shift is where you plug in x + k.1430

x + k goes into the function; horizontal stretch is a times x, which goes into the function.1440

Horizontal flip is where -x goes into the function.1447

So, horizontal things will happen inside the function; it happens to what we are plugging into the function,1452

whereas vertical things happen on the outside of the function--we don't have to worry about it being plugged in.1457

OK, that is a summary of transformations; don't worry if you have to refer to this.1461

But you can also probably think about this sort of thing, now that we have an understanding of where this stuff is coming from.1465

You can probably actually figure out that it makes sense, and just re-figure it out, re-derive it, in your own head,1469

without even having to refer to these lists.1474

Horizontal shift, horizontal stretch, horizontal flip--they might be a little bit more difficult.1476

But remember that idea where we are shifting around our home base--we are shifting around the experience of the x-axis.1480

That is what we are shifting with those.1485

Stacking transformations: if you want to do multiple transformations on one function, you just apply one transformation after another.1488

But order matters; unlike when we multiply and divide and multiply...if I multiply 5 by 3 by 7 by 8, I will multiply all of those numbers together.1495

It doesn't matter what order I multiply them in; but in transformations, it matters what order you put the transformations on in.1504

Be careful: the order you apply your transformations in can affect the results.1510

The order you apply will affect how it comes out--not always, but a lot of the time.1516

Decide on the order you want before you do it; decide on the order, then apply them to your base function in that order.1521

The order that they hit that base function--the order that the do their transformations in--will change what happens.1528

There are some cases where it won't matter what order you put it on in.1535

But other times, you are going to get totally different results; and we will look at an example in just a second.1538

This means you have to think about order when doing multiple transformations.1542

So, make sure you think about order if you are doing multiple transformations,1545

because if you don't think about it, you can really get confused and get completely the wrong answer.1549

All right, let's look at an example of why it matters how we stack our transformations--the order that we put our transformations on in.1553

For example, let's consider f(x) = √x.1558

And just in case you have forgotten what that looks like, it starts with the origin and just goes up like that, and slowly increases the farther it goes up.1561

All right, let's say we want to move it two units right and flip it horizontally.1567

We move it two units right by plugging in x - 2 into where we have x.1573

And we flip horizontally by plugging that -x in here as well.1579

That is how we do the two different things: we plug in -x into the function, or we plug in x - 2.1583

Look at how order matters; we get very different things, depending on the order we put this in.1588

So, if we move, and then we flip, first we would move; we would plug in x - 2 first, so we would get √(x - 2).1592

And then, the next action is flipping; so we then plug in -x next; so -x will go into where we have x, so we will get √(-x - 2).1599

That gives us the function g(x) = √(-x - 2), which would look like this graph right here.1611

And that makes sense, because I think this is the direction of right for you (for me, it is my left, but oh well).1617

If we have a square root going out like this, and then we pick it up and we move it over,1625

well, the middle is still here; it used to be coming directly out, but we picked it up, and we moved it over.1630

So, when we flip it, it is going to be a farther distance over now, and going out in the opposite direction.1634

And that is what we see here on this red graph.1639

We see that it is away from the y-axis, because it moved away from the y-axis, and then it flipped.1642

It turned all the way over; it basically grabbed it like a pole and spun to the other side.1647

So, now it is 2 away, but -2.1651

What if we did flip and then move? If we did flip and then move, the first thing we would do is plug in the -x.1655

So, -x would go in first; and then, into that x, we would plug in x - 2.1662

x - 2 goes in there; so we have x - 2, which is now replacing the x, because we are plugging into the function.1668

We are just plugging inside of that x; remember, it is just a placeholder.1676

It doesn't really mean that we have just x allowed to be there; it is just a placeholder for f(_) = √_.1681

So, if we want to flip horizontally, we plug in -x; if we want to move units right, we move x - 2.1688

Flip, then move: we get √-x; and then, the next thing--we put in the move, so we plug into that x an x - 2 instead of x.1694

So, we have -(x - 2); that gives us the function h(x) = √(-x + 2), because the negative cancels out the minus sign.1702

That is going to start at positive 2, and then move off to the right.1713

It starts at positive 2 and moves to the right; and once again, this makes sense.1717

We have this sort of centerpost of the y-axis, and it moves off to the right--the normal square root moves off to the right.1719

So, if we start by flipping it so that it is going this way, and then we move it two units to the right, we are going to still being going right of that y-axis.1726

We will move right of that y-axis.1735

So, when we move and then flip, we move this way, but then we flip into this location.1739

But when we flip and then move, we start going this way, but then we flip into back this way; but then we move after that.1745

So, that is why we are seeing two totally different things; so the order we put the transformation on...1754

we get totally different answers; so order really matters with our transformations.1757

All right, let's start working some examples.1761

We want to give three transformations of f(x) = x3 - 2x + 2: first, we shift it down by 5.1763

So, we will do this in red: shift it down by 5; remember, f(x) - 5...f(x) + k, so if we want to go down 5, it is -5 that we plug in for k.1771

So, f(x) - 5 will give us some new function; let's name this g(x) is equal to...1786

I will rewrite it, so we can see more easily what is going on.1797

g(x) = f(x) - 5, which would be x3 - 2x + 2, our normal function, and then just - 5.1800

So, we have g(x) = x3 - 2x - 3 (two minus five); and that is what we get for shifting down by 5.1813

If we want to shift right by 2, then let's make a new function; we will call this one h(x), and h(x) is going to be equal1826

to f, our original function, shifted right by 2; we do that by x + k; going to the right causes a negative k,1835

because we have to get that 0 to show up on the right now.1843

So, x + k is x - 2, since we are shifting to the right; that equals...we plug in for our old x.1847

Our old placeholder is now replaced by x - 2; so (x - 2)3 - 2(x - 2) + 2.1856

And if we wanted to, we could expand and simplify; but that is really not what the point of this lesson is about.1865

Expanding and simplification--I am pretty sure you can handle that.1874

And if you can't, we will have other lessons where we are doing that more carefully in polynomials.1876

We could expand if we wanted to--if we had to for the problem.1881

Then finally, in green, we shift it left by 1, then up by 4, then flip vertically.1885

This is the most complicated one of all: we are going to start by using g and h, and then finally we will make it to k.1891

We will treat each of these as one function after another.1897

We start at f(x) = x3 - 2x + 2.1900

All right, now we are going to have a transformation that is going to be left by 1.1906

We do that by plugging in x + 1, positive k, to move left (if we are plugging in horizontally).1914

So, we have a new function, g(x), that is equal to f plugged in...x + 1; and this is what happens if we just shift to the left.1926

It equals (x + 1)3 - 2(x + 1) + 2; great.1934

We could expand if we wanted to; we are not going to worry about that right now.1942

Next, we move up by 4; so up by 4 is function + 4.1946

Let's call it a new function; so it is h(x), and now we are doing this to g(x), so h(x) = g(x) + 4.1956

And since g(x) was equal to f(x) + 1, it is f(x) + 1 + 4, so we have what we had before, (f(x) + 1)3 - 2(x + 1) + 2.1964

Great; the final one--we are going to name this function k; and now it is a flip vertically.1979

It is a negative version of the function--just multiplying the function by -1.1987

So, k(x) is the vertical flip of h(x); remember, this has to happen in order, so we are doing it to h(x).1993

So, it is -h(x); now, what was h(x)? Well, h(x)...it is negative quantity..what did we have for h(x)?2000

We had g(x) + 4; so it is -(g(x) + 4); and then, what was g(x) + 4?2008

That was -(f(x) + 1 + 4); so we get, finally, -((x + 1)3 - 2(x + 1)...2019

oops, on the very one above that--sorry about that--I forgot to add on the 4; so we get + 2 + 4, or + 6; sorry about that one.2039

So, plus 6; so it is - 2(x + 1 + 6); great.2048

And there we are; we could continue to expand that if we wanted to; but that is ultimately what it is going to be.2055

We would have to expand a bunch of things--expand the cube in there, and then distribute out that negative sign.2059

But that is pretty much what is going on; we just now have to do that all in the order that we are supposed to.2065

But it is important that we do it in the order of shift to the left by 1, then the shift up by 4, then flip vertically.2069

If we break that order, it is not going to end up working out; we are going to get a different answer.2077

Great; all right, the next example: we have a parent function of cube root of x, and that is this one on the left.2081

Now, we want to give the function for the graph on the right.2090

We need to figure out what things happen to the graph on the right.2093

So, the first thing it looks like to me is...notice how we have that the right side is down and the left side is up.2096

The way we would do that is: at first, we have a vertical flip; and what comes after that?2104

The "home base" moves; where was it originally?2111

It was originally at (0,0); we will make a home and say, "That seems like a reasonable place to say where its home used to be."2119

Its home used to be at (0,0); and over the course of becoming the second graph, it goes to...where is its home?2125

Here is (2,1); so it is at (2,1); so the home moves to (2,1), which means that we have two things coming out of this.2132

We have a shift right by 2, and a shift up by 1.2142

So, we can come up with a function for this by just applying these transformations.2154

First, we have f(x)...let's do our first transformation, g(x) =...the vertical flip is -f(x) = cube root of x.2160

This is our first part; but next, we have the second part that comes in.2170

The second part...we will call this one h(x), so that will be what our final function actually is.2176

h(x) equals...we have shifting right; shift right happens by x - 2; remember, it is x + k, but we put in negatives to shift to the right.2180

Shift up is just our function + 1; those two we can actually put in in any order; they will never end up interacting with each other,2189

since the + 1 happens completely outside of the function--outside of everything.2198

So, -f(x)...we have here that h(x) is equal to g; plugging in x - 2, and also shifting it up by 1--2203

now, if g(x) is equal to -f(x), we have -f(x); g(x) becomes f(x), so this here is just the same thing as -f(x).2214

So, it is going to be -f(x) is 3√x (oops, it should be a negative sign there).2236

So, g(x - 2) is -f(x - 2) + 1, which is equal to -3√(x - 2) + 1.2244

So, the function that we are seeing over here is h(x) = -3√(x - 2) + 1; great.2276

For this example, we want to give the parent function, f, and what transformations were applied, and the order they were applied in to create g.2292

So, g(x) = 7 - x3 for the first one; so the first one...we will do it in red; what is the parent function that makes this up?2299

Well, the parent function for g(x) in red is going to be f(x) = x3.2307

We see that x3 there, so it seems reasonable that that is going to be it.2314

What had to happen to be able to get 7 - x3?2317

Well, the first thing that had to happen is a vertical flip, and then a vertical shift up.2320

We could also have a vertical shift down, and then a vertical flip of everything.2325

But it is easier to see it as a vertical flip, and then the second thing as shift up by 7.2329

And that is why we get -x3 from the vertical flip, and then shift up 7 will be -x3 + 7; so we get 7 - x3.2341

And that is how we get our red g(x).2349

Now, our blue g(x), 10√(x + 5); this f(x) will start from the basic function of √x.2353

That is our fundamental function; so what has happened in here?2361

Let's say that the first thing...it seems like it is easier to move horizontally than to have to do a vertical stretch first.2364

And actually, it won't matter what order we do it in.2372

But let's say we will do it in first order of shift; so we have x + 5 in there.2374

So, x + 5...x + k means that it is a positive, so shift by 5 left.2382

It is a horizontal shift, and it goes to the left, because it is a positive k.2391

And then, the second thing we do is multiply the entire function by 10; so it is a vertical stretch by a = 10.2396

We actually could do that in the completely opposite order; we could do vertical stretch by a = 10,2409

and we get 10√x, and then plug in x + 5; and we get 10(√x + 5).2413

So, 1 or 2...it doesn't actually matter which one goes first, unlike the red function, which actually...2418

it did matter that we had a vertical flip, and then shifted up by 7.2424

We would have to do a slightly different thing if we wanted to do it in a different order for the red function.2426

But the blue function--anything works.2430

Finally, the green function: its basic parent function, what is creating it, its base function, is |x|.2431

Now, for this one, it is a little bit harder to see which one this has to be.2440

It is much easier to start by shifting to the right by 3, because we see right here this 2x.2444

So, 2x means that how the x-axis is being affected is that it has been "sped up."2450

But if we speed up, and then move by a different thing...we are used to moving;2455

all of our theory about moving is based off of "move first"; our theory of moving how we experience the x-axis2459

was all done on the principle of it starting as x, and not starting as 2x or 5x or 1/2x.2466

It was all based on x + k, not 2x + k.2474

So, we want to start by shifting--doing our shifting--dealing with that first.2477

We will shift right by 6; and we know it is to the right, because we have a -6.2482

Now, here is actually the thing: it is not by 6; this is a confusion.2491

It seems to be 6 at first, because of that + k; but notice what is really there.2495

2x - 6...once again, we have things in the form x + k; 2x - 6 is not in the form x + k, because it has 2x; it is not just 1x.2500

So, we have to get it to 1x first; so we pull out the 2, and we get 2 times (x - 3).2512

The shift to the right is actually by this 3 here; so we shift right by 3, because we have k at -3.2520

And then, our second one is a horizontal speed-up by a = 2, which is to say it will squish to half of its original horizontal length.2528

Any horizontal interval will squish to half.2545

And then finally, we have this + 1 here; so it shifts up by 1.2548

Now, it actually turns out that the 3 could be at the top, or it could be at the bottom; it doesn't matter.2555

Shifting up by 1--that could happen at the very beginning; it could happen at the very end.2561

But because of the shift right and the horizontal speed-up, we have to have it in this x + k form.2564

We can't get it out of 2x - 6, because it is 2x + k; that is not the same form.2570

We have to have it as x + k; so we have to pull that 2 out first.2575

There is a way where we could have the horizontal speed-up go first, and then shift.2578

But it is much easier to think in terms of the shift right, and then the horizontal speed-up.2581

And if that one seems a little confusing, I wouldn't worry about it too much.2585

That is probably the absolute hardest kind of question of this type that you would ever see, at least for the next couple of years,2587

until you are in college--or if you are not just in college, but taking an advanced-level math class in college.2593

So, don't really worry about this right here; this is a fairly difficult kind of problem.2599

But this is the sort of thing you want to be thinking about it with.2605

You want to be thinking in terms of "What do I have to do here if I am following that formula table--if I am following that table?"2607

And you have to follow it carefully; what does it have to fit in?2612

It has to fit in things of the form x + k; and you notice, 2x is different--it is not in that same form.2615

So, you have to get it into that form before you can use these things that we talked about before.2621

All right, the final example: How is vertically stretching the graph of f(x) = x2 the same as horizontally stretching it?2625

Remember, a vertical stretch is done by a times f(x); a horizontal stretch is done by f(a times x).2632

Now, I think it is a little bit confusing to use a in two places.2656

So instead, we are going to call this b; so we say we are just using b to prevent confusion.2660

Don't worry about the fact that it is not what we were seeing before; it is not the same a multiplicative factor that we saw before.2670

It means the exact same thing; a and b are both just constants.2675

So, a and b are multiplicative constants; they are just how much we are stretching by--2680

whether it is a horizontal stretch or it is a vertical stretch--it is just how much we are stretching by.2693

All right, so let's see how this works on f(x) = x2.2698

a times f(x)...that is going to be a times x2; f(bx)...that means b times x will plug in, instead of the x;2702

so we will get bx; it is the quantity, squared.2710

So, over here we have ax2 and b2x2, because the "squared" will get put onto both of them.2714

So, we have these two things; how is it that they are the same?2724

Well, how is it that they are similar--what is the connection between them?2728

Well, think about this: a is just a constant, and b is just a constant.2733

a and b are both constants; but if b is a constant, then that means that b2 is also just a constant.2738

If b is 4, then b2 is just 16; so it will be a larger constant than b, but it is still just a constant.2751

It is not allowed to vary around; so b2 is also a constant.2758

So, what that means here is that in either case, whether it is a vertical stretch or a horizontal stretch,2767

it just has the effect of multiplying x2 by a constant.2777

So, what we are seeing here: the reason why, if we do a vertical stretch, and we do a horizontal stretch,2796

and if you go back and you look at what you saw when we saw a vertical stretch example and we saw the horizontal stretch example...2802

you will notice that they actually looked basically the exact same.2808

There were slight differences, but it is the same sort of stretching going on,2811

because when we compress it horizontally, it causes it to just sort of squirt up vertically.2814

And when we stretch it out vertically, it is the same thing as if we had compressed it horizontally.2819

So, in either case--whether it is a vertical stretch or a horizontal stretch--it is just the same thing as multiplying by a constant.2822

So, that is why they are so similar.2827

We, in actually, I think, all of the fundamental functions that we are used to using by this point...2830

all of them are already things where this is just the horizontal and the vertical stretch...it will end up having the same effect.2836

It will do it by different amounts; but ultimately, it is just putting a constant into the mix--multiplying things by a constant.2843

The first time that you will end up seeing things--and right now, if you have even taken trigonometry,2850

the only thing you would see where you would be able to see the difference between a horizontal and a vertical stretch--2854

is trigonometric functions; if you look at sine and cosine, it actually is possible to horizontally stretch those2859

and vertically stretch those, and you will get totally different-looking things out of a vertical stretch versus a horizontal stretch.2864

Now, it is OK that we haven't really talked about trigonometric functions yet.2870

And you haven't seen them yet, probably; don't worry about that--that is OK.2873

And if you have seen trigonometric functions, or you have taken some trigonometry, that is all the better.2876

You probably are already exposed to this.2880

But just know that later on, you will see cases where there is a difference between horizontal stretch and vertical stretch.2882

But for some other functions, like f(x) = x2, it ends up being that there is not really a difference at all.2888

All right, I hope you now have a good understanding of all of the different transformations that are available to us.2892

I know that there are a lot of them; but if you think through what you are doing with each one,2896

you can probably figure it out without even having to resort to the table.2899

These are really useful, because they let us build a bunch of different functions2902

and understand how to graph functions that seem complex at first,2905

but are really just some basic function we are used to graphing, that has been stretched and squished and moved around.2908

All right, we will see you at Educator.com later--goodbye!2913