INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Analyzing Graphs of Polynomial Functions

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
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
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (5)

1 answer

Last reply by: Hong Yang
Sun Nov 3, 2019 8:22 AM

Post by Hong Yang on November 3, 2019

fokmecok

2 answers

Last reply by: Hong Yang
Sun Nov 3, 2019 8:20 AM

Post by Daniel Cuellar on October 30, 2012

by looking at a polynomial of the degree of n, can we have "at most" n-1 relative max & mins? or is it that we WILL have n-1 relative max & mins. Thanks

Analyzing Graphs of Polynomial Functions

  • If f(a) < 0 and f(b) > 0, then f has a zero between a and b.
  • The graph of a polynomial function of degree n has at most n – 1 local maximums and local minimums.

Analyzing Graphs of Polynomial Functions

Graph f(x) = x3 − 4x
  • Find the zeros
  • 0 = x(x2 − 4) = x(x − 2)(x + 2)
  • Using the Zero Product Property you'll have three equations
  • x = 0, x − 2 = 0, and x + 2 = 0
  • Solve and plot the roots/zeros
  • x = 0, x = 2, and x = − 2
  • Plot Some Points
  • x-3-113
    y    
  • x-3-113
    y-153-315
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coefficient is positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞ , y approaches − ∞
Graph f(x) = x3 − 3x2 − 4x
  • Find the zeros
  • 0 = x(x2 − 3x − 4) = x(x − 4)(x + 1)
  • Using the Zero Product Property you'll have three equations
  • x = 0, x − 4 = 0, and x + 1 = 0
  • Solve and plot the roots/zeros
  • x = 0, x = 4, and x = − 1
  • Plot Some Points
  • x-2235
    y    
  • x-2235
    y-12-12-1230
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coesfficient is positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞ , y approaches − ∞
Graph f(x) = − x3 − x2 + 6x
  • Find the zeros
  • 0 = − x(x2 + x − 6) = x(x − 2)(x + 3)
  • Using the Zero Product Property you'll have three equations
  • x = 0, x − 2 = 0, and x + 3 = 0
  • Solve and plot the roots/zeros
  • x = 0, x = 2, and x = − 3
  • Plot Some Points
  • x-4-213
    y    
  • x-4-213
    y24-84-18
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coesfficient is Negative, the graph will have the following properties:
  • As x approaches + ∞ , y approaches − ∞
  • As x approaches − ∞ , y approaches + ∞
Graph f(x) = x4 − 5x2 + 4
  • Find the zeros
  • 0 = (x2)2 − 5(x)2 + 4
  • We know how to factor x2 − 5x + 4 = 0
  • x2 − 5x + 4 = 0
  • (x − 4)(x − 1) = 0 Since what we really want is to factor out is the polynomial to the 4th power,
  • all we have to do is replace the x with x2
  • (x2 − 4)(x2 − 1) = 0
  • Here we have difference of squares, so we'll use that pattern to factor out completely.
  • (x − 2)(x + 2)(x − 1)(x + 1) = 0
  • Using the Zero Product Property you'll have 4 equations
  • x − 2 = 0, x + 2 = 0, x − 1 = 0, and x + 1 = 0
  • Solve and plot the roots/zeros
  • x = 2, x = − 2, x = 1, and x = − 1
  • Plot Some Points
  • x-3−[3/2]0[3/2]3
    y     
  • x-3−[3/2]0[3/2]3
    y40-2.194-2.1940
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Even Degree and Leading Coesfficient is Positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞ , y approaches + ∞
Graph f(x) = − x4 + 3x2 − 2
  • Find the zeros
  • 0 = − ((x2)2 − 3(x)2 + 2)
  • We know how to factor x2 − 3x + 2 = 0
  • x2 − 3x + 2 = 0
  • (x − 2)(x − 1) = 0 Since what we really want is to factor out is the polynomial to the 4th power,
  • all we have to do is replace the x with x2
  • − (x2 − 2)(x2 − 1) = 0
  • Here we have difference of squares, so we'll use that pattern to factor out completely.
  • − (x2 − 2)(x − 1)(x + 1) = 0
  • Using the Zero Product Property you'll have 4 equations. For the first equation, solve by taking the square root of both sides.
  • x2 − 2 = 0, x − 1 = 0, and x + 1 = 0
  • Solve and plot the roots/zeros
  • x = + √2 = 1.4, x = − √2 = − 1.4, x = 1, and x = − 1
  • Plot Some Points
  • x-2−[5/4]0[5/4]2
    y     
  • x-2−[5/4]0[5/4]2
    y-6[1/4]-2[1/4]-6
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Even Degree and Leading Coesfficient is Negative, the graph will have the following properties:
  • As x approaches + ∞ , y approaches − ∞
  • As x approaches − ∞ , y approaches − ∞
Graph f(x) = x6 − 5x4 + 4x2
  • Find the zeros
  • 0 = x2(x4 − 5x2 + 4) = x2((x2)2 − 5(x)2 + 4)
  • We know how to factor x2 − 5x + 4 = 0
  • x2 − 5x + 4 = 0
  • (x − 4)(x − 1) = 0 Since what we really want is to factor out is the polynomial to the 4th power,
  • all we have to do is replace the x with x2
  • x2(x2 − 4)(x2 − 1) = 0
  • Here we have difference of squares, so we'll use that pattern to factor out completely.
  • x2(x − 2)(x + 2)(x − 1)(x + 1) = 0
  • Using the Zero Product Property you'll have 5 equations.
  • x2 = 0, x − 2 = 0, x + 2 = 0, x − 1 = 0, and x + 1 = 0
  • Solve and plot the roots/zeros
  • x = 0, x = 2, x = − 2, x = 1, and x = − 1
  • Plot Some Points
  • x-2.5-1.5-0.50.51.52.5
    y      
  • x-2.5-1.5-0.50.51.52.5
    y73.83-4.920.70.7-4.9273.83
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Even Degree and Leading Coesfficient is Positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞, y approaches + ∞
Graph f(x) = x3 − 2x2 − 2x + 4
  • Find the zeros, factor by grouping
  • 0 = (x3 − 2x2) + ( − 2x + 4)
  • 0 = x2(x − 2) − 2(x − 2)
  • 0 = (x2 − 2)(x − 2)
  • Using the Zero Product Property you'll have 3 equations.
  • x2 − 2 = 0 and x − 2 = 0
  • Solve and plot the roots/zeros. You must use the square root to solve the first equation.
  • x = + √2 = 1.41, x = − √2 = − 1.41, and x = 2
  • Plot Some Points
  • x-2-1013
    y     
  • x-2-1013
    y-83417
  • Sketch graph using end behavior by using the table below
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coesfficient is Positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞ , y approaches − ∞
Sketch the graph of f(x) = − x3 + x2 + 7x − 7 by finding the zeros and using the end - behavior only
  • Find the zeros, factor by grouping
  • 0 = ( − x3 + x2) + (7x − 7)
  • 0 = − x2(x − 1) + 7(x − 1)
  • 0 = (7 − x2)(x − 1) = − (x2 − 7)(x − 1)
  • Using the Zero Product Property you'll have 3 equations.
  • x2 − 7 = 0 and x − 1 = 0
  • Solve and plot the roots/zeros. You must use the square root to solve the first equation.
  • x = + √7 = 2.65, x = − √7 = − 2.65, and x = 1
  • Analyze the end behavior and use it to sketch the graph.
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coefficient is Negative, the graph will have the following properties:
  • As x approaches + ∞, y approaches − ∞
  • As x approaches − ∞, y approaches + ∞
  • With your pencil, start somewhere high on Quadrant II and draw a smooth curve going down passing through
  • the zero ( − 2.65,0) going down and then start going up going through the zero (1,0) and then start going down and pass through the
  • last zero at (2.65,0) until you are in Quadrant IV. Your lines should be smooth and the graph must pass through those three points.
  • The rest is just a sketch.
  • Compare your graph with the actual graph. How close/far were you?
Sketch the graph of f(x) = x3 − 6x2 − 4x + 24 by finding the zeros, f(0), and end - behavior only.
  • Find the zeros, factor by grouping
  • 0 = (x3 − 6x2) + ( − 4x + 24)
  • 0 = x2(x − 6) − 4(x − 6)
  • 0 = (x2 − 4)(x − 6)
  • Using the difference of Squares, factor completely
  • 0 = (x − 2)(x + 2)(x − 6)
  • Using the Zero Product Property you'll have 3 equations.
  • x − 2 = 0, x + 2 = 0, adn x − 6 = 0
  • Solve and plot the roots/zeros.
  • x = 2, x = − 2, and x = 6
  • Find f(0) and plot it
  • f(0) = (0)3 − 6(0)2 − 4(0) + 24 = 24
  • Analyze the end behavior and use it to sketch the graph.
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Odd Degree and Leading Coefficient is Positive, the graph will have the following properties:
  • As x approaches + ∞ , y approaches + ∞
  • As x approaches − ∞, y approaches − ∞
  • With your pencil, start somewhere low on Quadrant III and draw a smooth curve going up passing through
  • the zero ( − 2,0) going up and then start going down as you pass (0,24), then continue going down as you pass through (2,0) and continue going down about
  • 24 down then start going up as you cross (6,0) and continue going up until you're in Quadrant I. . Your lines should be smooth and the graph must
  • pass through those 4 points. The rest is just a sketch.
  • Compare your graph with the actual graph. How close/far were you?
Sketch the graph of f(x) = − x4 + 5x2 − 4 by finding the zeros, f(0), and end - behavior only.
  • Find the zeros, imagine you're factoring x2 and then replace x with x2 as in previous examples
  • 0 = − (x4 − 5x2 + 4)
  • 0 = − (x − 4)(x − 1)
  • 0 = − (x2 − 4)(x2 − 1)
  • Using the difference of Squares, factor completely
  • 0 = − (x − 2)(x + 2)(x − 1)(x + 1)
  • Using the Zero Product Property you'll have 4 equations.
  • x − 2 = 0, x + 2 = 0, x − 1 = 0, and x + 1 = 0
  • Solve and plot the roots/zeros.
  • x = 2, x = − 2, x = 1, and x = − 1
  • Find f(0) and plot it
  • f(0) = − (0)4 + 5(0)2 − 4 = − 4
  • Analyze the end behavior and use it to sketch the graph.
  • End Behavior ChartEven DegreeOdd Degree
    + leading coefficient1. As x approaches +∞, y approaches +∞1. As x approaches +∞, y approaches +∞
     2. As x approaches -∞, y approaches +∞2. As x approaches -∞, y approaches -∞
    - leading coefficient1. As x approaches +∞, y approaches -∞1. As x approaches +∞, y approaches -∞
     2. As x approaches -∞, y approaches -∞2. As x approaches -∞, y approaches +∞
  • Since the graph is of Even Degree and Leading Coefficient is Negative, the graph will have the following properties:
  • As x approaches + ∞ , y approaches − ∞
  • As x approaches − ∞ , y approaches − ∞
  • With your pencil, start on Quadrant III and go up and pass through ( − 2,0) and continue going up a little bit then start
  • coming down and go through point ( − 1,0). Continue going down and then start going up as you cross (0, − 4) heading towards (1,0) going up.
  • Continue going up and then start going down as you cross the point (2,0) and continue going down. You should end on Quadrant IV.
  • Compare your graph with the actual graph. How close/far were you?

*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

Analyzing Graphs of Polynomial 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
  • Graphing Polynomial Functions 0:11
    • Example: Table and End Behavior
  • Location Principle 4:43
    • Zero Between Two Points
    • Example: Location Principle
  • Maximum and Minimum Points 8:40
    • Relative Maximum and Relative Minimum
    • Example: Number of Relative Max/Min
  • Example 1: Graph Polynomial Function 11:57
  • Example 2: Graph Polynomial Function 16:19
  • Example 3: Graph Polynomial Function 23:27
  • Example 4: Graph Polynomial Function 28:35

Transcription: Analyzing Graphs of Polynomial Functions

Welcome to Educator.com.0000

In a previous lecture, we introduced the concept of the graphs of polynomial functions.0002

Now, we are going to go further and actually talk about how to develop those graphs.0006

OK, in order to obtain the graph of a polynomial function, you make a table of values,0012

connect these points with a curve, and then use information about the end behavior of the function.0017

Earlier on, when you had to graph linear and quadratic functions, you used the technique of making a table.0024

So, it is the same idea here, but with the additional concept of thinking about end behavior.0032

For example, if you were to graph a function such as f(x) = 3x3 - 2x2 + x,0039

you would approach this by finding some x and f(x) values and graphing those out.0054

Then, look at the areas of the function that tell you about end behavior.0063

I am not going to go ahead and make the table on this one.0070

But let's say that you were to graph it out, and it came out something like this.0072

OK, using points, and also finding the zeroes, can help you graph.0080

In addition to just finding various points, sometimes you can approach making a graph of a polynomial function by also finding the zeroes0088

(and recall that the zeroes are where the graph crosses the x-axis, so they are the x-intercepts), and then finding some other points.0097

And then, you think about what happens to the graph at the ends.0109

Well, recall that, if you have an odd-degree polynomial, such as this one, the ends are going to go in different directions.0112

So, one will increase; the other will decrease; and then you have to look at the leading coefficient.0124

The leading coefficient: if it is positive (if it is greater than 0), as x increases, y will also increase.0131

So, this is odd degree with a positive leading coefficient: x and y are both increasing, and as x gets very, very small,0140

y is going to get very, very small, as well; so this is odd degree, where a of n, the leading coefficient, is greater than 0.0152

If you have an odd degree where the leading coefficient is negative, then the graph is going to pretty much be the opposite,0168

where you are going to have y decreasing (f(x) decreasing) as x gets very large.0177

And when x gets very small, f(x) is going to increase; this is odd degree with a negative value for the leading coefficient.0184

Let's look at even degrees right here.0199

If you are talking about a function with an even degree, such as the largest exponent of the fourth power or the sixth power,0207

so it's an even degree, both ends go in the same direction--they both face up, or they both face down.0222

And the leading coefficient tells you which direction that is.0228

With a leading coefficient that is positive, you are going to get a graph where both ends are facing up.0231

So, as x gets very large, or as x gets very small, y is going to get very large; this is even with a leading coefficient that is positive.0238

Now, if you have an even degree where the leading coefficient is negative, then both ends will face down.0249

So, this could be an example of even with a negative leading coefficient.0260

This is review from a previous lecture; but recall that, in addition to finding what is going on0265

here in the middle of the graph, and finding your zeroes--finding various points--0270

you also need to know about end behavior in order to develop a graph of a polynomial function.0275

OK, I mentioned talking about finding the zeroes.0283

Well, sometimes it can be difficult or time-consuming to find the exact location of a zero.0286

But if you are just trying to sketch out a graph, you can use the location principle0292

to estimate where a zero will be--where a graph is going to cross the x-axis.0296

So, let's look at what this is saying: this is saying that, if you have a polynomial function f(x),0300

suppose that there are numbers a and b, such that f(a) < 0 and f(b) > 0.0306

Then, there is a zero, f(x), between these two points, a and b.0314

So, let's make this concrete: let's say that this is my point a--that I am going to let a equal 1.0318

And this is saying that if f(a) is less than 0 (I find f(a)--let's say it is somewhere down here: f(a) = -3), a is at 1, and f(a) is -3;0328

at (1,-3), that is (a,f(a)) on the graph; there is a negative value to the function.0349

OK, now let's say I find another value b, and let b equal 3.0362

So, here I had a equal 1, and let b equal 3; f(a) = -3; let's let f(b) equal 4.0367

Now, what this is telling me is that, somewhere between this point and this point,0382

we went from a negative value of the function to a positive value of the function.0394

That tells me that there has to be a zero somewhere...actually, it could even be over here; but somewhere between 1 and 3, this is going to cross.0398

So, I don't know that it is exactly here; but I know that somewhere between 1 and 3, there is a zero,0410

at an x-value somewhere between x = 1 and x = 3.0420

And I know that because the function changed from a negative value to a positive value.0432

Now, there actually could be more than one zero between those points.0442

For example, I could have a graph right here that goes like this.0446

And if I just graphed this point out here, I would see that the function is negative at this point, at -4: at f(-4), I have a negative value.0452

And then I say, "OK, let's say over here, at f(-2), I have a positive value."0464

I know that there is a zero in here; but as it turns out, there is actually more than one zero.0474

I know that there is at least one zero here, but there also could be more than one.0476

So, when you go ahead and make your table of values, x and f(x), if you see that you have,0480

say, positive, positive, and then it goes to negative, I know that at an x-value somewhere between here and here, there is a zero.0487

And then, I might have negative and negative, and then switch to positive again.0498

And I know that, in here somewhere, there is a zero.0501

So again, the idea is that, if you see the value of the function switch from positive to negative,0508

or negative to positive, you know that the graph has to pass through the x-axis, and that there is a zero in that location.0512

OK, maximum and minimum points: recall that, when we worked with quadratic functions, we talked about maximums and minimums.0521

And here, today, with polynomials, we are talking about relative maximums.0528

When we talked about quadratics that were downward-facing parabolas, we said, "OK, the vertex is a maximum."0533

We didn't say relative; we just said maximum, because this is the highest value that the function could achieve.0540

If we were dealing with a minimum value, we didn't say relative; we just said, "OK, this is the minimum,"0546

because this is the smallest value that the function could achieve.0552

However, when we talk about more complicated, or higher-degree, polynomials, we talk about relative maximums and minimums.0556

And what we mean is relative to the x-values around those.0564

For example, if I have a graph that looks like this, I could say that this right here--this point (we will call that a,0569

and then, right here, the y-value is going to be f(a)--f(a) would be right there,0581

so this point is going to be (a,f(a)))--this is a relative maximum.0589

And the reason it is a relative maximum is: if I look at any value of x around in this region, the function value is lower than it is right here.0597

So, it is the maximum, relative to the values of x near this point.0607

A relative minimum is the same idea: if I look at this point here, this gives me a relative minimum.0613

And that means that this value of the function (I will call this f(b), so this would be b, and then the y-value would be f(b),0621

so this is at (b,f(b)))--this is the smallest value for the function in that region, for these x's.0636

But you see here: the graph actually goes lower way over here.0645

So, it is not an absolute minimum; it is just relative to that region of the graph.0648

And there might be a maximum that is higher up at some area in the graph; but this is a relative maximum for that region.0652

The other important point is that the graph of a polynomial function of a certain degree,0660

degree n, can have at most n - 1 points that are relative maximums or relative minimums.0665

So, if I have a function--let's say f(x) = 4x5 + 3x3 - 2x + 6,0672

then here, the degree, n is 5 in this case, so the maximum number of relative maximums and relative minimums is 5 - 1, which equals 4.0682

So, in a graph like this, degree 5, I can have at most 4 of these maximums and minimums.0707

OK, putting together what we have discussed about finding points, the zeroes, using end behavior,0718

and relative maximums and minimums, we can develop a graph of some polynomials.0724

OK, looking at this first one, this is x3 - 2x; and I am just going to start this one by plotting out some points.0729

OK, so let's let x equal -2; this would equal -8; and then, -2 times -2 is 4; -8 + 4 is going to give me -4.0739

Here I have -1 for my x-value, and that is going to give me a -1 here, and then -1 times -2 is going to give me 2, so this will be 1.0756

Now, notice something: my value for the function switched from negative to positive.0767

That means that somewhere between -2 and -1, there is a zero.0774

Using the location principle, I know that, somewhere between x = -2 and x = -1, this graph of this function crosses the x-axis.0781

When x is 0, f(x) is 0; well, if f(x) is 0, then this point lies on the x-axis, and I actually found a zero--this is a zero.0793

Now, when x is 1, that gives me 1, minus 2; that is -1.0808

When x is 2, 23 is 8, minus 2 times 2 (that is minus 4)--that is going to give me 4.0815

Oh, and notice also, here, another zero.0823

There is a zero between these values (between the corresponding x-values); somewhere between 1 and 2, there is a zero.0835

I have a zero between -2 and -1; I know I have a zero right here; and then, I have another zero between 1 and 2.0841

So, I am going to go ahead and plot these out; when x is -2, the function is -4.0848

When x is -1, the function is 1; when x is 0, the value of the function is 0;0854

when x is 1, we have a -1 right here; and when x is 2, the function value is 2.0862

I can see, between here and here: this had to cross the x-axis in between here and here.0870

Now, I connect these with a smooth line; I found my points; I know that there is a zero0875

somewhere in here; that there is one here; and that there is one here.0889

Now, I am going to think about end behavior; and the degree equals 3, and that is odd.0894

The leading coefficient, a of n, is 1, so that is positive.0902

And what that tells me is: it confirms that my graph is correct, because if it is an odd degree, the two ends are going to face in different directions.0910

And since it is a positive leading coefficient, that means that, as x gets very large, y gets very large.0920

As x gets very small, y gets very small.0927

The other thing I know is that the number of relative maximums and relative minimums is going to be one less than the degree of the polynomial.0930

And since the degree is 3, I can have, at most, 2 relative maximums and relative minimums.0939

And I see right here: I do have a relative maximum--this is the largest function value for the x-values in this region.0952

And right here, I have a relative minimum; so I found both relative maximums and minimums.0962

So, this is a pretty good graph of this polynomial, just based on using knowledge that I had and finding a few points on the graph.0969

In the second example, again, we have a polynomial of degree 3; but notice that it has a negative leading coefficient.0980

That is going to give the graph a different shape.0988

In the previous example, I just went ahead and graphed some points; I am going to take a slightly different approach here.0991

And I am going to find the zeroes; last time, I found one of the zeroes by luck at point (0,0),0997

and I also used the location principle to find the approximate location of two more zeroes.1003

Here, I am just going to go ahead and find them directly.1008

So, what I am going to do is look at the corresponding equation, -x3 + x2 + 6x.1011

And I am going to set that equal to 0 to find values of x for which the function's value is 0; that will give me the zeroes.1019

So, I am going to approach this by factoring; and I see that I have a common factor of x in all three terms.1028

And I am actually going to factor out -x, so what I have left behind is simpler to factor.1035

If I pull a -x here, I am going to have an x2 left.1040

If I pull a -x from x2, that will leave me -x, because -x times -x is positive x2.1043

Here, I am also going to pull out a negative x, leaving behind -6, and checking that -x and -6 is + 6x.1052

OK, now I can look here and see that I can factor this further.1062

So, I know that this is going to have the general form (x + something) (x - something).1070

And I know that, because I have a negative sign here; so a positive and a negative is going to give me a negative.1077

Factors of 6 are 1 and 6, and 2 and 3; and I need those factors when one is positive and one is negative.1084

I am going to make one positive and one negative; I want them to add up to the middle term of -1.1096

Well, these two are too far apart; now, if I take -3 + 2, that is going to equal -1.1102

So, I know that this is the right set of factors, and that I want 3 to be negative and 2 to be positive.1110

If I use FOIL to check this, I get x2 - 3x + 2x (is going to give me -x) + 2 times -3 (is going to give me -6).1116

So, this is factored out as far as it can go.1133

And then, recall the zero product property that says that, if, say, a times b equals 0, then either a equals 0 or b equals 0, or they both equal 0.1135

So, any of these factors could equal 0, and that would give me a total value of 0.1147

And don't forget about this -x; that is part of it; x + 2 could equal 0, or x - 3 could equal 0, to make this equation true.1154

Dividing both sides by -1 just gives me x = 0; here, subtracting 2 from both sides gives me x = -2.1166

And adding 3 to both sides, I get x = 3.1176

These are the zeroes; these are the points at which the graph crosses the x-intercept.1179

So, x equals 0; x equals -2; and x equals 3.1185

OK, now, to flesh out this graph some more, I am going to find some other points.1192

I found three points--I found my zeroes; now I am just going to find some other points on the graph in this region.1195

When x is -1, if you work this out, it comes out to f(x) is -4.1203

When x is 1, this will give you -1 + 1 + 6, so that would give you 6.1212

When x is 2, working this out, it will give you f(x) is 8.1221

All right, so when x is -1, f(x) is 4; when x is 1, f(x) is 6 (about up here).1229

And notice here that this tells me that there is a zero in here--that between x is -1 and x is 1, there is a zero right in here.1244

And I defined that zero between x is -1 and x is 1; there is a zero that lies in here.1254

Let's see, when x is 2, f(x) is way up here; it is going to be way up here somewhere--it is at 8, so we will just put it right there.1264

All right--oh, and when x is...actually, I wrote that incorrectly; when x is -1, this is going to be -4.1282

And that is why, when you get the switch from negative to positive, there is a zero right in here (in the switch from here to here).1294

So, I have some points graphed, and I have my zeroes; so I am going to go ahead and connect these points.1305

I am also going to use end behavior; and since this is an odd degree, one end is going to go up, and the other is going to go down.1324

And since it is a leading coefficient that is negative, then I am going to have--as x becomes very large, y would be very small;1336

and as x becomes very small, over here, y becomes very large.1349

So, end behavior helped me figure out these portions of the graph.1356

Also, I know that, because my degree is 3, the greatest number of relative maximums and minimums I can have is 2.1360

2 relative maximums and minimums at most--that is the greatest number I can have.1369

And I do have a relative minimum here and a relative maximum there.1376

So, graphing is based on finding the zeroes, graphing a few points, and then thinking about end behavior.1384

And notice, also, that here, when my graph switched from a value of -4 (a negative value for the function)1391

to a positive value, up here at 6, I knew that there had to be a zero in between those; and I did find that zero.1399

OK, I am going to approach this, also, by first finding the zeroes.1406

Let's work with the corresponding equation and set this equal to 0.1411

Now, looking at this, there are no common factors to pull out; and I have four terms, so I am going to factor by grouping.1417

Grouping means that we are going to put x3 - x2 together, and add that to -3x + 3.1424

Now, looking at this, I have a common factor of x2; that leaves behind an x and a -1.1434

Here, I have a common factor of -3; I pull that out--that leaves an x behind here, and a -1, because -3 and -1 would give me the 3 back.1442

Now, I see that I have a common binomial factor of x - 1; I am going to pull that out in front.1455

When that is pulled out, that leaves behind x2 - 3.1462

OK, using the zero product property, I know that x - 1 = 0 or x2 - 3 = 0.1467

If either of these expressions is 0, then the equation will hold true; this left side will be equal to 0.1476

So, this is easy: just add 1 to both sides--that gives me x = 1; and then, over here,1485

I am going to add 3 to both sides, and that is going to give me x2 = 3.1490

I am going to take the square root of both sides, and that is going to give me x = ±√3.1496

OK, so I have zeroes at x = 1, x = √3, and x = -√3.1504

Well, the square root of 3 can be estimated at about 1.7; so let's rewrite this as 1.7 and -1.7, so that we can graph it.1517

OK, I have a zero here at 1; I have a zero here at 1.7 (which is about there) and at -1.7 (which is right about there).1528

All right, again, I need to just find a few more points to get a complete graph.1539

I am going to let x equal -2; and if I plug that into here, I will get out a value of a function that is -3.1547

I am going to let x equal -1, and if you do the calculation on that and substitute -1 in here, you are going to find that y is 4.1559

Plotting that out: when x is -2, f(x), or y, is -3, right here.1570

When x is -1, f(x) is 4; and you see the location principle at work--that, since we switched from negative to positive,1580

that means that there is a zero somewhere in here, somewhere between x is -2 and x is -1.1588

And we already know that there is a zero between those two, and it is at -1.7.1597

OK, finding a few more points: let's let x equal 0; f(x) is going to be 3.1605

When x is 2, plugging that in here, we get a value for the function of 1.1612

So, when x is 0, f(x) is 3; when x is 2, f(x) is 1.1621

So now, we have enough points to get some sense of what is happening.1633

And I am going to start here and connect these.1639

Now, using end behavior, I have an odd degree (this is degree 3), which means the two ends are going to go in opposite directions.1653

And I have a positive leading coefficient, meaning that, over here on the right, as x gets very large, y is going to get large.1663

Over here on the left, as x gets very small, f(x), or y, is going to get very small.1670

I am also going to look and see that my degree is 3; so I have, at most, 2 relative maximums and relative minimums.1677

And I have a relative maximum here, and I have a relative minimum here; so I have them both.1690

So again, I am handling this by finding the zeroes to get a few points on the x-axis,1699

finding some more points to find the shape of the rest of this area of the graph,1704

and using end behavior to predict how the graph will look at large and small values.1709

So, let's take this and set it equal to 0.1715

All right, as far as factoring this, this is a trinomial; and we are used to working with these when they are quadratic equations, and the first term is x2.1722

But it is really not that much different with x4, because x2 times x2 is x4.1730

So, I am looking here, and I see that I have a positive term here, and a negative here.1740

And the only way to get that is if I am multiplying a negative times a negative; that gives me a positive.1745

And yet, when I sum up the outer and inner terms after multiplying, I will get a negative term here.1751

Factors of 8 are 1 and 8, and 2 and 4; and I need them to add up to 6.1757

So, if I added -2 and -4, I am going to get -6; so I know that these are the factors that I want to use.1764

OK, I can't do anything more with this, so I am just going to leave this as it is.1774

However, this is the difference of two squares, so I can factor that out a bit more to this.1777

OK, so using the zero product property: x2 - 2 = 0 or x - 2 = 0 or x + 2 = 0.1785

So, starting with these easier ones: x = 2, x = -2; I found two zeroes.1794

Now, I am going to look over here: x2 = 2--taking the square root of both sides gives me x = ±√2.1803

All right, so I have four zeroes; I have zeroes at x = 2, -2, and then √2 is approximately 1.4...1813

so let's rewrite that as 1.4 and -1.4 to help me with the graph.1824

So, plotting those out: I have a zero here, at -2, and then another one at 2.1830

I have a zero here at 1.4 and -1.4.1836

I need a few more points to complete my graph; so right up here, let's find x and f(x).1843

When x is -1, if you work this out, you get 1; you get -6, because this x2 would just be 1, plus 8; so it is going to give you 9 - 6, or 3.1852

When x is 0, that cancels; that cancels; that becomes 0; that leaves me with 8 right here.1866

And then finally, when x is 1, this will be 1, minus 6, plus 8; so again, I will have 3.1873

All right, so when x is -1, f(x) is 3; when x is 0, f(x) is way up here, about here, at 8 (8 would be right about here); and then, when x is 1, f(x) is 3.1882

I am going to connect these points; and I am also going to use my knowledge of end behavior.1905

And since this is an even degree, both ends are either going to point up or down.1909

And since this is a positive leading coefficient, both ends will actually be up.1915

So, when x is very small, y is going to be large; and then, I know that after this point, I am going to go up,1921

because when x is very large, y is going to be very large.1938

After this point, I am going to continue on up, because when x is small, f(x) is going to be very large,1942

because it is an even degree with a positive leading coefficient.1948

Now, I am checking that I have degree 4; since this is degree 4, the greatest number of relative maximums and minimums I am going to have are 3.1951

And I have a relative minimum there; I have a relative maximum here; and I have a relative minimum here.1964

So, I found all three relative maximums and minimums.1973

So again, finding the zeroes, I found 1, 2, 3, 4 zeroes.1976

Then, I plotted a few other points to give me the shape of the graph.1983

I used my knowledge of end behavior to figure out what is going on out here and here.1987

And then, I just verified that I had the generally correct shape by seeing that I would expect, at most, three relative maximums and minimums.1991

And I found that I had all three.1999

That concludes this session of Educator.com on analyzing the graphs of polynomials.2003

And I will see you again next lesson!2008

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