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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Graphing Quadratic Functions

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Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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
VII. 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
VIII. 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
IX. 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
X. 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
XI. 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 (13)

0 answers

Post by DJ Sai on September 3 at 11:49:00 AM

What happened to the graph at 17:11

2 answers

Last reply by: Cole Lovin
Tue Sep 20, 2016 9:08 PM

Post by Francisco Ramirez Cruz on July 12, 2014

i need help with  2/x + 6/x-2 =-5/2

1 answer

Last reply by: Dr Carleen Eaton
Mon Mar 19, 2012 6:54 PM

Post by Karen Shipp on March 18, 2012

How did you find the Y-intercept? (Karen is my mothers name. I am Gabriel)

1 answer

Last reply by: Dr Carleen Eaton
Mon Nov 14, 2011 11:17 PM

Post by Mary Moss on November 14, 2011

How do you find the x-int.?

4 answers

Last reply by: julius mogyorossy
Tue Sep 23, 2014 2:11 PM

Post by Jeff Mitchell on March 12, 2011

At about 26:40 into lecture in example 3, you said f(6) = -62 + 12(6)+8 = 44 because -6^2 = -36 but doesn't a negative number squared equal a +36? and therefore the answer would be +36+72+8 = 116 ?

Jeff

Graphing Quadratic Functions

  • The graph of a quadratic function is a parabola.
  • Use the axis of symmetry to help you graph a parabola. Graph the right or left half and then reflect the graph across the axis of symmetry.
  • The maximum or minimum value of the function occurs at the vertex. Use the formula for the vertex to find the maximum or minimum.

Graphing Quadratic Functions

Find the equation of the axis of symmetry, the coordinates of the vertex and graph f(x) = x2 − 4
  • Axis of Symmetry:
    x=−[b/2a]
    a = 1, b = 0,
    x = − [0/2] = 0
    Draw vertical, dashed line through x = 0
  • Vertex Coordinates:
    (− [b/2a],f( − [b/2a] )
    a = 1, b = 0,
    (0,f(0))
    f(0) = − 4
    (0, − 4)
  • Points to Graph: Choose 3 points after the axis of symmetry. Since axis is located at x = 0, choose three points after the x = 0 mark. You may also choose 3 points before

  • x
    1
    2
    3
    y = f(x) = x2 − 4
    − 3
    0
    5
  • You will notice that only half the parabola is shown with points x = 1, 2, 3. To Draw the other half
  • use the fact that Quadratics are Symmetric about their axis of Symmetry.
  • Point Symmetric with A will be 1 unit on the left of x = 0 at ( − 1, − 3)
  • Point Symmetric with B will b 2 units on the left of x = 0, at ( − 2,0)
  • Point Symmetric with C will be 3 units on the left of x = 0, at ( − 3,5)
  • Plot and draw a smooth curve.
Find the equation of the axis of symmetry, the coordinates of the vertex and graph f(x) = x2 − 6x + 4
  • Axis of Symmetry:
    x = − [b/2a]
    a = 1, b = − 6,
    x = − [( − 6)/2] = 3
    Draw vertical, dashed line through x = 3
  • Vertex Coordinates:
    ( − [b/2a],f( − [b/2a] )
    a = 1, b = − 6,
    (3,f(3))
    f(3) = 32 − 6(3) + 4
    = 9 − 18 + 4 = − 5
    (3, − 5)
  • Points to Graph: Choose 3 points after the axis of symmetry. Since axis is located at x = 3, choose three points after the x = 3 mark. You may also choose 3 points before

  • x
    4
    5
    6
    y = f(x) = x2 − 6x + 4
    − 4
    − 1
    4
  • You will notice that only half the parabola is shown with points x = 4, 5, 6. To Draw the other half
  • use the fact that Quadratics are Symmetric about their axis of Symmetry.
  • Point Symmetric with A will be 1 unit on the left of x = 3 at (2, − 4)
  • Point Symmetric with B will b 2 units on the left of x = 3, at (1, − 1)
  • Point Symmetric with C will be 3 units on the left of x = 3, at (0,4)
  • Plot and draw a smooth curve.
Find the equation of the axis of symmetry, the coordinates of the vertex and graph f(x) = 2x2 − 8x + 8
  • Axis of Symmetry:
    x = − [b/2a]
    a = 2, b = − 8,
    x = − [( − 8)/2(2)] = [8/4] = 2
    Draw vertical, dashed line through x = 2
  • Vertex Coordinates:
    ( − [b/2a],f( − [b/2a] )
    a = 2, b = − 8,
    (2,f(2))
    f(2) = 2(2)2 − 8(2) + 8
    = 8 − 16 + 8 = 0
    (2,0)
  • Points to Graph: Choose 3 points after the axis of symmetry. Since axis is located at x = 2, choose three points after the x = 2 mark. You may also choose 3 points before

  • x
    3
    4
    5
    f(x) = 2x2 − 8x + 8
    2
    8
    18
  • You will notice that only half the parabola is shown with points x = 3, 4, 5. To Draw the other half
  • use the fact that Quadratics are Symmetric about their axis of Symmetry.
  • Point Symmetric with A will be 1 unit on the left of x = 2 at (1,2)
  • Point Symmetric with B will b 2 units on the left of x = 2, at (0,8)
  • Point Symmetric with C will be 3 units on the left of x = 2, at ( − 1,18)
  • Plot and draw a smooth curve.
Find the equation of the axis of symmetry, the coordinates of the vertex and graph f(x) = x2 + 6x + 8
  • Axis of Symmetry:
    x = − [b/2a]
    a = 1, b = 6,
    x = − [6/2] = − 3
    Draw vertical, dashed line through x=-3
  • Vertex Coordinates:
    ( − [b/2a],f( − [b/2a] )
    ( − 3,f( − 3))
    f( − 3) = ( − 3)2 + 6( − 3) + 8
    = 9 − 18 + 8 = − 1
    ( − 3, − 1)
  • Points to Graph: Choose 3 points after the axis of symmetry. Since axis is located at x = 3, choose three points after the x = 3 mark. You may also choose 3 points before

  • x
    − 2
    − 1
    0
    y = f(x) = x2 + 6x + 8
    0
    3
    8
  • You will notice that only half the parabola is shown with points x = − 2, − 1, 0. To Draw the other half
  • use the fact that Quadratics are Symmetric about their axis of Symmetry.
  • Point Symmetric with A will be 1 unit on the left of x = − 3 at ( − 4,0)
  • Point Symmetric with B will b 2 units on the left of x = − 3, at ( − 5,3)
  • Point Symmetric with C will be 3 units on the left of x = − 3, at ( − 6,8)
  • Plot and draw a smooth curve.
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = − x2 + 10x − 2
  • Max or Min:
    a = -1, b = 10, c = -2
    Because ä" is negative, this quadratice is a maximum
  • Find Max:
    a = -1, b = 10, c = -2
    x=−[b/2a]
    x = − [b/2a] = − [10/(2( − 1))] = 5
    To find the maximum, plug in x = 5 into the quadratic
    f(x) = − x2 + 10x − 2
    f(5) = − (5)2 + 10(5) − 2
    f(5) = − 25 + 50 − 2 = 23
  • Domain: All Real Numbers
  • Range: Given that this quadratic is a maxium, the values taken up by ÿ" cannont exceed the maximum value, therefore,
  • Range: All Real Numbers for y ≤ 23
The Quadratic is a Maximum;
Max = 23;
Domain = All Real Numbers;
Range = All Real Numbers for y ≤ 23
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = x2 − 2x − 1
  • Max or Min:
    a = 1, b = -2, c = -1
    Because ä" is positive, this quadratice is a minimum
  • Find Min:
    a = 1, b = -2, c = -1
    x=−[b/2a]
    x = − [b/2a] = − [(−2)/2] = 1
    To find the minimum, plug x=1 into the quadratic
    f(x) = − x2 + 10x − 2
    f(1) = (1)2 − 2(1) − 1
    f(1) = 1 − 2 − 1 = − 2
  • Domain: All Real Numbers
  • Range: Given that this qudratic is a minimum, the value taken up by ÿ" cannont be below the minimum value, therefore,
  • Range = All Real Numbers for y ≥ −2
This quadratice is a minimum;
Min = -2;
Domain = All Real Numbers;
Range = All Real Numbers for y ≥ −2
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = − 3x2 − 12x − 1
  • Max or Min:
    a = -3, b = -12, c = -1
    Because ä" is negative, this quadratice is a maximum
  • Find Max:
    a = -3, b = -12, c = -1
    x=−[b/2a]
    x = − [b/2a] = − [(−12)/(2(−3))] = −2
    To find the maximum, plug x=-2 into the quadratic
    f(x) = − 3x2 − 12x − 1
    f( − 2) = − 3( − 2)2 − 12( − 2) − 1
    f( − 2) = − 12 + 24 − 1 = 11
  • Domain: All Real Numbers
  • Range: Given that this quadratic is a maxium, the values taken up by ÿ" cannont exceed the maximum value, therefore,
  • Range: All Real Numbers for y ≤ 11
This quadratice is a maximum; Max = 11;
Domain = All Real Numbers;
Range = All Real Numbers for y ≤ 11
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = − 3x2 − 6x
  • Max or Min:
    a = -3, b = -6, c = 0
    Because ä" is negative, this quadratice is a maximum
  • Find Max:
    a = -3, b = -6, c = 0
    x=−[b/2a]
    x = − [b/2a] = − [(−6)/(2(−3))] = −1
    To find the maximum, plug x=-1 into the quadratic
    f(x) = − 3x2 − 6x
    f( − 1) = − 3( − 1)2 − 6( − 1)
    f( − 1) = − 3 + 6 = 3
  • Domain: All Real Numbers
  • Range: Given that this quadratic is a maxium, the values taken up by ÿ" cannont exceed the maximum value, therefore,
  • Range: All Real Numbers for y ≤ 3
This quadratice is a maximum; Max = 3;
Domain = All Real Numbers;
Range = All Real Numbers for y ≤ 3
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = 2x2 − 8x + 4
  • Max or Min:
    a = 2, b = -8, c = 4
    Because ä" is positive, this quadratice is a minimum
  • Find Min:
    a = 2, b = -8, c = 4
    x=−[b/2a]
    x = − [b/2a] = − [(−8)/2(2)] = 2
    To find the minimum, plug x=2 into the quadratic
    f(x) = 2x2 − 8x + 4
    f(2) = 2(2)2 − 8(2) + 4
    f(2) = 8 − 16 + 4 = − 4
  • Domain: All Real Numbers
  • Range: Given that this qudratic is a minimum, the value taken up by ÿ" cannont be below the minimum value, therefore,
  • Range = All Real Numbers for y ≥ −4
This quadratice is a minimum;
Min = -4;
Domain = All Real Numbers;
Range = All Real Numbers for y ≥ −4
Determine whether the function has a maximum or a minimum.
Find the max or min, and state the domain and range.
f(x) = 5x2 + 20x + 4
  • Max or Min:
    a = 5, b = 20, c = 4
    Because ä" is positive, this quadratice is a minimum
  • Find Min:
    a = 5, b = 20, c = 4
    x=−[b/2a]
    x = − [b/2a] = − [20/2(5)] = −2
    To find the minimum, plug x=-2 into the quadratic
    f(x) = 5x2 + 20x + 4
    f( − 2) = 5( − 2)2 + 20( − 2) + 4
    f( − 2) = 20 − 40 + 4 = − 16
  • Domain: All Real Numbers
  • Range: Given that this qudratic is a minimum, the value taken up by ÿ" cannont be below the minimum value, therefore,
  • Range = All Real Numbers for y ≥ −16
This quadratice is a minimum;
Min = -16;
Domain = All Real Numbers;
Range = All Real Numbers for y ≥ −16

*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

Graphing Quadratic Functions

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

  • Intro 0:00
  • Quadratic Functions 0:12
    • A is Zero
    • Example: Parabola
  • Properties of Parabolas 2:08
    • Axis of Symmetry
    • Vertex
    • Example: Parabola
  • Minimum and Maximum Values 9:02
    • Positive or Negative
    • Upward or Downward
    • Example: Minimum
    • Example: Maximum
  • 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

Transcription: Graphing Quadratic Functions

Welcome to Educator.com.0000

Today, we are going to start talking about quadratic equations and inequalities.0002

And the first thing we are going to review is graphing quadratic functions.0006

Recall from Algebra I that a quadratic function has the form f(x) = ax2 + bx + c.0012

So, that is the quadratic function in standard form, where a does not equal 0.0022

And if a were to equal 0, this section would drop out; and this is actually what makes it a quadratic function--the x2.0027

Otherwise, you would just end up with a linear function, which we have described earlier on in the course.0036

So, let's first talk about an example of a quadratic function.0045

It could be something such as f(x) = x2 + 4x - 3, or f(x) = -x2 + 1.0053

And here, in this second example, b is actually equal to 0; and that is allowed, so this drops out.0069

But as I just discussed, a cannot equal 0.0076

We are going to be working with the graphs of these functions today, and first just talking about the general shape of these curves.0080

The curve of a graph of a quadratic function is a shape called a parabola, and they may open upward;0087

a parabola can open upward; it could open downward; and you will see varying shapes, such as wider parabolas and narrower parabolas.0097

And as we get deeper into discussing this topic, you will be able to tell, just from looking at the quadratic function,0116

roughly what the shape is going to be and which direction it is going to curve in.0123

So, getting started with some properties of parabolas: a parabola has an axis of symmetry, and that axis is described by the equation x = -b/2a.0128

And what the axis of symmetry is: it is a vertical line that divides the parabola into two symmetric halves.0142

And the parabola intersects this axis of symmetry at what is called the vertex.0153

Recall that the x-coordinate of the vertex is given by x = -b/2a.0160

To illustrate this with an example: if I were looking at the function f(x) = -x2 - 2x + 1,0168

then first let's look for the axis of symmetry; and the axis of symmetry is up at -b/2a.0184

Well, here a is equal to -1, because recall that standard form is ax2 + bx + c, so a is -1, b is -2, and c is 1.0192

So, if I want -b/2a, that is going to give me -(-2)/2(-1).0211

So, looking at that, that will be a negative and a negative, to give me a positive, over -2, which equals -1.0220

Therefore, the axis of symmetry is going to be a vertical line right here at -1; so, this is the axis of symmetry.0230

Now, this point where x equals -1 is the x-coordinate of the vertex.0249

However, I need to find the y-coordinate; so in order to do that, I am going to go back and look for f(-1).0258

So, I am replacing my x terms with -1; that is going to give me -1 squared, is 1, so that is -1.0267

And then, that gives me -2 times -1, which will become a positive, so that is + 2, + 1.0284

So, this is 3 minus 1; so that equals 2.0291

What I did is found the x-coordinate for the vertex, substituted that in for my x-value, and found that f(-1) = 2.0296

So, the y-value for the vertex is going to be 2; the vertex is going to be at the point (-1,2).0304

So, that is (-1,2) right here.0312

OK, now, to further graph this, let's go ahead and find some points.0316

And it is best to pick some points around the vertex, because if I pick points way out here, my graph is not going to be as accurate.0323

I want to get a general shape of the curve in the area of the vertex.0329

OK, so my vertex is right here; pick some x-values, and then let's go ahead and determine what the y-values are, based on that.0333

When x is -2 (you can go ahead and work this out for yourself), if you worked the whole thing out, you would find that y is 1.0348

OK, when x is 0, you can see that these two are going to drop out, and y is going to be 1.0359

When x is 1, that is going to give me -1, minus 2, plus 1; that is going to come out to -2.0368

So, one more point, maybe: when x is 2, working this out, this is going to give me 2 squared; that is -4.0391

And then, 2 times 2, with a negative, is also -4; plus 1--that is going to equal -7; so here we get -7.0403

Go ahead and plot out the rest of these points.0415

When x is -2, y is 1; when x is 0, y is 1; when x is 1, y is -2.0417

And then, this one is way down here, so we will leave that off.0431

But looking at how the axis of symmetry can help you with graphing:0434

here I have one point on the left, and I am going...I have a couple points on the right.0438

Because I know that the axis of symmetry divides this into two symmetric halves, I can actually add another point right over here.0445

So, because this is symmetrical, I have a point here on one side of the vertex; I have another one here.0455

Looking at the axis of symmetry, if I have a point right here, which is an x-value that is 1, 2 away from the vertex,0462

so on this side it is 1, 2 away from the vertex at -3, and then the y is 1, 2, 3, 4 down;0474

here I would have it, again, compared to the vertex, 1, 2, 3, 4 down.0482

So, that is going to reflect it right across here.0488

So, using the axis of symmetry, I can find the mirror image point.0491

Even if I have just graphed this half, if I have the axis of symmetry, I can reflect across to the other half.0499

Just reviewing: an axis of symmetry for a parabola is given by the equation right here; and I found my axis of symmetry.0504

And that is a vertical line; the parabola actually intersects that axis at the vertex.0515

The x-value of the vertex is also given by this equation; and once I have found that (which was -1),0521

I can substitute that in and find the function's value for y at that point (which turned out to be 2).0530

The axis of symmetry can help you with graphing.0538

Now, we talked a little bit about the vertex; but the vertex can give you either a maximum or minimum value of the function,0544

depending on the shape of the curve--whether it opens upward or downward.0554

That tells you if you have a maximum value or a minimum value.0561

So, investigating this a little bit further: if the coefficient of the x2 term is positive, the graph will open upward,0564

whereas if the coefficient of the x2 term is negative, the graph opens downward.0577

So, recall that the standard form is ax2 + bx + c, so that is the standard form of the function f(x).0585

And if I look at a, if a is greater than 0 (that means that the x2 term is positive), the parabola opens upward.0599

If this is a negative value--if a is less than 0--the parabola is going to open downward.0613

Just by looking at the equation I am given, I will have a general sense of the shape--whether it opens upward or downward.0621

Now, if this is positive, and it opens upward, let's give an example.0631

If I have a parabola like this, and it opens upward, this tells me that a is greater than 0.0638

The vertex is going to be a minimum; and thinking about what that means--0645

that means the minimum value that y can have--the minimum value of the function.0654

If the coefficient of x2 is positive, the graph opens upward, and the function has a minimum value.0659

Right here at the vertex, that is as small a value as you will find for y.0666

y only gets bigger from there, no matter what x-value you put in.0672

Conversely, if I have a parabola that opens downward (here a is less than 0--I have a parabola that opens downward),0676

in this case, the vertex is going to give me a maximum value.0687

So, right here, y is approximately equal to 1; that is as large as y is going to get.0698

No matter what value I plug in for x, y is just going to be smaller than that.0705

So again, this gives us more information about the graph, just by looking at what this coefficient is right here-- the coefficient of x2.0711

So, in order to graph, as I said, once you find the vertex, you look at the equation and say, "OK, it is going to open [upward or downward]."0722

And then, look for some points around the vertex to get the shape of the graph.0730

Does it open very wide? Is it a very shallow graph, or is it a very steep graph?0735

And we will talk more about that later on--about the shape.0745

But for now, just focus on the vertex--whether the parabola opens upward or downward,0749

and then on finding some x and y values in order to help you make the graph.0755

OK, let's practice these with some examples.0762

Find the equation of the axis of symmetry, the coordinates of the vertex, and the graph of this function.0765

OK, first recall that the axis of symmetry is given by the vertical line at x = -b/2a.0773

And then, always just review what standard form is in order to find this: it is ax2 + bx + c.0784

So here, I have x =, and then my a value is 1; there is no bx term, therefore b must be 0.0793

There is no term with a coefficient and then just x; that implies that b is 0.0805

And then, here, c is -9; but I really don't need to worry about that right now.0811

So, I am going to go ahead and substitute these in; and as soon as I see that b is 0,0815

it honestly doesn't matter what a is, because if you take 0 and divide it by anything, you are going to get 0.0822

Now, this is the equation for the axis of symmetry; it is a vertical line at x = 0, right here.0828

The axis of symmetry is just going to follow this y-axis.0843

The coordinates of the vertex: well, I have my x-coordinate right here, which is 0.0848

That means that this axis of symmetry is going to intersect this parabola at x = 0, and then at some y-coordinate.0855

In order to find the y-coordinate, let's find f(0).0863

f(0) is -9; so, let's make this -2, -4, -6, -8, -10; this is the y-coordinate for the vertex.0876

The x-coordinate is x = 0, so the coordinates of the vertex are (0,-9); (0,-9) is right there.0898

Now, let me go ahead and write that in--that is what that is--that is the vertex, right there.0911

Now, in order to graph this, I need to find some other points; so let's find some values for x, and then figure out what f(x) is at those points.0918

When x is 1, that would be 1 squared minus 9; you are going to get -8 for y.0931

When x is 2, 2 squared is 4, minus 9--that is going to give me 5.0940

Let's see: when x is 3, 3 squared is 9, minus 9 is 0.0948

So, let's try these points; and that is going to give me: when x is 1, y is -8; when x is 2, y is going to be...actually, it should be -5 right here.0954

When x is 3, y is 0; 3 is about here.0974

OK, now recall that the axis of symmetry can help me to graph--it can save me work,0980

because here, I have some points on one side of the axis of symmetry, and that means that on the other side...0986

I can just reflect across to get points that are mirror-image--approximately here, and then here at...this is here,0994

and then here is at (3,0), so I am going to go over here to (-3,0) and put a point.1009

Therefore, I was able to cut down my work by knowing that the axis of symmetry divides this into two symmetrical halves.1017

So, checking, I did find the equation of the axis of symmetry, x = 0.1026

I found the coordinates of the vertex, and that was (0,-9).1032

And I graphed by finding some points, and then reflecting across, using the axis of symmetry.1037

Again, find the equation of the axis of symmetry, the coordinates of the vertex, and the graph.1046

The equation of the axis of symmetry is given by x = -b/2a.1054

And here, I have a = 1, b = -4, because f(x) in standard form is ax2 + bx + c.1066

So, I am going to go ahead and find this; the axis of symmetry is x = -b (that is negative -4), over 2 times a (which is 1).1077

Therefore, x equals...that negative and negative gives me...positive 4 over 2, which equals 2.1090

So, this is the axis of symmetry; I'll go ahead and form an x-y axis.1099

The axis of symmetry is going to be right here at x = 2, and there is going to be a vertical line passing through that point.1132

My next thing that I need to do is find the coordinates of the vertex.1142

And I have the x-coordinate (that is x = 2), but I need to find the y-coordinate.1146

And the y-coordinate is going to be given for f(2), which equals 2 squared, minus 4, times 2, plus 4.1150

So, that gives me 4 minus 8, plus 4, so that is 4 + 4 is 8, minus 8 is 0.1164

So, this equals 0; so the y-coordinate f(2) for the vertex is 0.1172

So, the vertex is at (2,0); that is where the axis of symmetry is going to intersect with the graph.1180

Then, to finish graphing this, I just need to find some points--some x-values and some corresponding y-values.1189

So, let's start out with some points around the vertex.1200

When x is 1, that is going to give me 1 squared (is 1), minus 4, plus 4; so these two are going to cancel, and I am just going to get 1 right here.1205

OK, I already have 2; that is my vertex; how about 3?1219

When you go ahead and plug this in and figure it out, you are going to get 3 times 3 (that is 9), minus 3 times 4 (is 12), plus 4.1225

So, that is 9 minus 12 (is going to give me -3), plus 4: that is going to give me 1.1234

OK, when x is 4, if you work this out, you will find that y is 4; and then another easy point--when x is 0, these drop out; you get 4 here also.1241

When x is 1, y is 1; when x is 3, y is 1; when x is 4, y is 4; when x is 0, y is 4.1252

Now again, I could have used the axis of symmetry; I could have found even fewer points,1266

and then just used symmetry to find the points on the other side of that axis.1269

So, I completed the tasks that I was asked to do, which are to find the axis of symmetry (and that is right here at x = 2);1279

to find the vertex (the vertex is here at (2,0)), and to graph this (which is a parabola opening upward,1288

with a minimum value--the vertex here is a minimum, because this opens upward--at (2,0)).1299

In this next one, we are asked to determine whether the function has a maximum or a minimum,1310

Find the maximum or minimum, and state the domain and range.1315

So, we don't even have to graph this one; but we first need to look at it and determine if it has a maximum or a minimum.1319

And the way we figure that out is by seeing what this coefficient of x2 is--what a is.1326

Here, a equals -1; recall that, when a is less than 0--when a is negative--the parabola opens downward.1337

Now, just sketching this out to think about it: if the parabola opens down--it faces downward--1354

the vertex is going to give me a maximum value; that is as large as y is going to get; y only gets smaller.1365

So here, the vertex is a maximum value--it doesn't get any larger.1380

If this were to open upward, then the vertex would be a minimum value.1393

So, here the parabola opens downward; my vertex is a maximum.1404

Now, I am asked to find that maximum or minimum--in other words, to find the vertex.1409

And recall that the vertex is given by -b/2a.1413

So, I have that my a is -1; here, b is 12; so let's find the vertex.1423

It is going to be a vertical line at x = -b, over 2 times -1.1431

So, that is going to give me x = -12/-2, or x = 12/2, or x = 6.1443

That gives me the equation; there is going to be a line at x = 6--that is the axis of symmetry.1454

So, the axis of symmetry is at x = 6.1463

And that is a maximum value; so the axis of symmetry is x = 6.1468

The next thing I am asked to find is the domain and the range.1475

Well, looking at the domain, I could say that x is really any real number.1480

I could say it is one, a thousand, negative two hundred...so the domain is going to be all real numbers.1489

It may be tempting to say, "Oh, the range is also all real numbers."1498

But if you think about it, it is actually not: now, let's go a little bit deeper and find this maximum point--let's find the vertex.1502

I found that x equals 6, so I know that my x-coordinate is 6; but let's find the y-coordinate, which is going to be at f(6).1512

So, that f(6) equals -6 squared, plus 12, times 6, plus 8, which is going to equal...-6 times -6 is 36; 12 times 6 is 72.1521

6 times 6 is 36, and that is negative, so it is -36, plus 72, plus 8.1541

6 squared; take a negative; plus 12 times 6; plus 8; this is going to give me -36 + 72 + 8, which actually comes out to 44.1551

So, my vertex is actually going to be at the point (6,44); and what that is telling me is that this is the largest y is ever going to get.1562

Now, y can be an infinite number of smaller values; but this is the maximum it is going to get.1574

Therefore, the range is actually all real numbers, where y is less than or equal to 44.1582

So, it might be tempting to just look at this and say "all real numbers" for y, but that, in fact, is not true,1594

because we have a maximum here that y cannot be greater than.1599

Therefore, answering all of this, I determined that I have a vertex that is a maximum.1603

I found what the maximum is, and it was x is 6 and y is 44; so the maximum is y = 44.1609

And the domain and the range--the domain is all real numbers; the range is all real numbers such that y is less than or equal to 44.1621

OK, Example 4: Determine whether the function has a maximum or a minimum.1630

Find the maximum or minimum, and state the domain and range.1636

Recall standard form, and that I need to look at the coefficient of x--I need to look at a; and here, a equals 3.1642

And since that is greater than 0--since a is positive--this tells me that this parabola opens upward.1653

And since it opens upward, just to help visualize it, the vertex right here is going to be a minimum.1660

It is going to be the smallest value that y can achieve for any x.1674

They want me to find that minimum value; so I am going to recall that the x-coordinate of the minimum value is defined by -b/2a.1679

I know what a is; b is -12; so, just figuring that out, that is negative, times -12, over 2 times a (which is 3).1690

That is going to give me...a negative and a negative is a positive 12, divided by 2 times 3, which is 6; so this is going to give me 12/6; that is 2.1708

OK, so the x-coordinate for the minimum is 2; to figure out the y-coordinate, I am going to figure out f(2),1723

which is going to be 3 times 2 squared, minus 12 times 2, plus 7.1737

This is going to give me 3 times 4, minus 12 times 2 (is 24), plus 7; so this is 12 minus 24, plus 7.1746

So, that is going to give me -12 + 7, which is -5.1759

Therefore, the minimum value is going to be y = -5; so this is my minimum.1765

Now, looking at this, in reality, actually, this is going to be at x = 2, y = -5; it is actually going to be more like this.1777

We weren't asked to graph it, but just to help visualize it: -5, let's say, is down here; this is actually going to be down here, in this quadrant.1787

OK, now, I found that I have a minimum; I have found that the minimum occurs at y = -5.1797

What is the domain? Looking, I can make x anything I want--any real number--so the domain is going to be all real numbers.1804

What is the range? Well, there is a limit on what that can be, because I just said that the minimum value for y is -5, right here.1818

That is the minimum value; therefore, y can be greater, but it can never be less than that.1834

So, actually, the range is all real numbers, where y is greater than or equal to -5.1841

The domain is all real numbers; the range is more restricted--it is all real numbers greater than or equal to -5.1853

OK, so we found that we had a minimum, and we know that because a is positive.1866

We found that using the formula -b/2a to get the x-coordinate, which is 2.1872

Then, we found f(x) using 2, so f(2), to tell me that y equals -5.1879

Since y is -5, then there is a limit here on the range.1887

The domain is all real numbers, whereas the range is only those real numbers greater than or equal to -5.1894

That concludes this session of Educator.com, where we covered graphing quadratic functions.1901

I will see you next lesson!1907

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