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INSTRUCTORS Carleen Eaton Grant Fraser
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Dr. Carleen Eaton

Dr. Carleen Eaton

Square Root Functions and Inequalities

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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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Square Root Functions and Inequalities

  • Exclude all values that make the radicand negative.
  • The graph of a square root function is half of a parabola.
  • The initial point of the graph can be translated horizontally or vertically from the origin, depending on the additive constants in the function.

Square Root Functions and Inequalities

Graph y = √{2x}
  • Step 1: Determine the domain
  • Since you cannot have a negative inside the radical sign, your domain is limited to values greater or equal to zero inside the radical.
  • 2x > 0
  • x > 0
  • Your values of x can be greater or equal to zero
  • Step 2 - Create a table of values
  • xy=√{2x}
    2 
    8 
    18 
    32 
  • xy=√{2x}
    2 = √{2(2)} = √4 = 2
    8 = √{2(8)} = √{16} = 4
    18 = √{2(18)} = √{36} = 6
    32 = √{2(32)} = √{64} = 8
  • Step 3 - Draw a smooth curve
Graph y = [1/4]√{2x}
  • Step 1: Determine the domain
  • Since you cannot have a negative inside the radical sign, your domain is limited to values greater or equal to zero inside the radical.
  • 2x > 0
  • x > 0
  • Your values of x can be greater or equal to zero
  • Step 2 - Create a table of values
  • xy = [1/4]√{2x}
    2 
    8 
    18 
    32 
  • xy = [1/4]√{2x}
    2 = [1/4]√{2(2)} = [1/4]√4 = [1/2]
    8 = [1/4]√{2(8)} = [1/4]√{16} = 1
    18 = [1/4]√{2(18)} = [1/4]√{36} = [6/4] = [3/2]
    32 = [1/4]√{2(32)} = [1/4]√{64} = 2
  • Step 3 - Draw a smooth curve
Graph y = 6√{3x}
  • Step 1: Determine the domain
  • Since you cannot have a negative inside the radical sign, your domain is limited to values greater or equal to zero inside the radical.
  • 3x > 0
  • x > 0
  • Your values of x can be greater or equal to zero
  • Step 2 - Create a table of values
  • xy = 6√{3x}
    3 
    12 
    27 
    81 
  • xy = 6√{3x}
    3 = 6√{3(3)} = 6√9 = 6*3 = 18
    12 = 6√{3x} = 6√{3*12} = 6√{36} = 6*6 = 36
    27 = 6√{3*27} = 6√{81} = 6*9 = 54
    81 = 6√{3*48} = 6√{144} = 6*12 = 72
  • Step 3 - Draw a smooth curve
Graph y = √{x − 2} + 3
  • Step 1: Determine the domain
  • Since you cannot have a negative inside the radical sign, your domain is limited to values greater or equal to zero inside the radical.
  • x − 2 ≥ 0
  • x ≥ 2
  • Your values of x can be greater or equal to 2
  • Step 2 - Create a table of values
  • xy = √{x − 2} + 3
    2 
    6 
    11 
    18 
  • xy = √{x − 2} + 3
    2 = √{2 − 2} + 3 = 0 + 3 = 3
    6 = √{6 − 2} + 3 = √4 + 3 = 2 + 3 = 5
    11 = √{11 − 2} + 3 = √9 + 3 = 3 + 3 = 6
    18 = √{18 − 2} + 3 = √{16} + 3 = 4 + 3 = 7
  • Step 3 - Draw a smooth curve
Graph y = √{2x − 6} + 5
  • Step 1: Determine the domain
  • Since you cannot have a negative inside the radical sign, your domain is limited to values greater or equal to zero inside the radical.
  • 2x − 6 ≥ 0
  • 2x ≥ 6
  • x ≥ 3
  • Your values of x can be greater or equal to 3
  • Step 2 - Create a table of values
  • xy = √{2x − 6} + 5
    3 
    5 
    11 
    21 
  • xy = √{2x − 6} + 5
    3 = √{2(3) − 6} + 5 = √{6 − 6} + 5 = 5
    5 = √{2(5) − 6} + 5 = √{10 − 6} + 5 = √4 + 5 = 7
    11√{2(11) − 6} + 5 = √{22 − 6} + 5 = √{16} + 5 = 9
    21√{2(21) − 6} + 5 = √{42 − 6} + 5 = √{36} + 5 = 11
  • Step 3 - Draw a smooth curve
Graph y ≥ √{x − 1} + 2
  • In order to graph an a square root function, you must first determine the domain. You cannot have
  • negatives inside the square root.
  • x − 1 ≥ 0
  • x ≥ 1
  • Your domain is restricted to values greater or equal to 1.
  • Step 1: Create a table of values to graph the square root function.
  • xy = √{x − 1} + 2
    1 
    5 
    10 
    17 
    26 
  • xy = √{x − 1} + 2
    1 = √{1 − 1} + 2 = 0 + 2 = 2
    5 = √{5 − 1} + 2 = √4 + 2 = 2 + 2 = 4
    10√{10 − 1} + 2 = √9 + 2 = 3 + 2 = 5
    17√{17 − 1} + 2 = √{16} + 2 = 4 + 2 = 6
    26√{26 − 1} + 2 = √{25} + 2 = 5 + 2 = 7
    item Step 2 - Graph the square root function using a solid line for the boundary line.
  • Step 3 - Using a test point, check to see which way to shade along the boundary line
  • Test (5,0)
    y ≥ √{x − 1} + 20 ≥ √{5 − 1} + 2
     0 ≥ √4 + 2
     0 ≥ 4
     Not True
  • Shade outside the boundary line not including (5,0). Be carefull when shading to the left of the imaginary line
  • passing through x = 1, nothing should be shaded in that region.
Graph y√{x + 4} + 2
  • In order to graph an a square root function, you must first determine the domain. You cannot have
  • negatives inside the square root.
  • x + 4 ≥ 0
  • x ≥ − 4
  • Your domain is restricted to values greater or equal to − 4.
  • Step 1: Create a table of values to graph the square root function.
  • xy = √{x + 4} + 2
    -4 
    0 
    5 
    12 
    21 
  • xy = √{x + 4} + 2
    -4 = √{ − 4 + 4} + 2 = 0 + 2 = 2
    0 = √{0 + 4} + 2 = √4 + 2 = 2 + 2 = 4
    5√{5 + 4} + 2 = √9 + 2 = 3 + 2 = 5
    12√{12 + 4} + 2 = √{16} + 2 = 4 + 2 = 6
    21√{21 + 4} + 2 = √{25} + 2 = 5 + 2 = 7
  • Step 2 - Graph the square root function using a solid line for the boundary line.
  • Step 3 - Using a test point, check to see which way to shade along the boundary line
  • Test (5,0)
    y ≥ √{x + 4} + 20 ≥ √{0 + 4} + 2
     0 ≥ √4 + 2
     0 ≥ 4
     Not True
  • Shade outside the boundary line not including (0,0). Be carefull when shading to the left of the imaginary line
  • passing through x = − 4, nothing should be shaded in that region.
Graph y <√{x + 4} − 6
  • In order to graph an a square root function, you must first determine the domain. You cannot have
  • negatives inside the square root.
  • x + 4 ≥ 0
  • x ≥ − 4
  • Your domain is restricted to values greater or equal to − 4.
  • Step 1: Create a table of values to graph the square root function.
  • xy = √{x + 4} − 6
    -4 
    0 
    5 
    12 
    21 
  • xy = √{x + 4} − 6
    -4 = √{ − 4 + 4} − 6 = 0 − 6 = − 6
    0 = √{0 + 4} − 6 = √4 − 6 = 2 − 6 = − 4
    5√{5 + 4} − 6 = √9 − 6 = 3 − 6 = − 3
    12√{12 + 4} − 6 = √{16} − 6 = 4 − 6 = − 2
    21√{21 + 4} − 6 = √{25} − 6 = 5 − 6 = − 1
  • Step 2 - Graph the square root function using a dashed line for the boundary line.
  • Step 3 - Using a test point, check to see which way to shade along the boundary line
  • Test (0,-6)
    y <√{x + 4} − 6 − 6 <√{0 + 4} − 6
      − 6 <√4 − 6
      − 6 <− 4
     True
  • Shade inside the boundary line including (0, − 6). Be carefull when shading to the left of the imaginary line
  • passing through x = − 4, nothing should be shaded in that region.
Graph y <√x + 9
  • In order to graph an a square root function, you must first determine the domain. You cannot have
  • negatives inside the square root.
  • x ≥ 0
  • Your domain is restricted to values greater or equal to 0.
  • Step 1: Create a table of values to graph the square root function.
  • xy = √x + 9
    0 
    4 
    9 
    16 
    25 
  • xy = √x + 9
    0 = √0 + 9 = 0 + 9 = 9
    4 = √4 + 9 = 2 + 9 = 11
    9√9 + 9 = 3 + 9 = 12
    16√{16} + 9 = 4 + 9 = 13
    25√{25} + 9 = 5 + 9 = 14
  • Step 2 - Graph the square root function using a dashed line for the boundary line.
  • Step 3 - Using a test point, check to see which way to shade along the boundary line
  • Test (4,2)
    y <√x + 92 <√4 + 9
     2 < 2 + 9
     2 < 11
     True
  • Shade inside the boundary line including (4,2). Be carefull when shading to the left of the imaginary line
  • passing through x = 0, nothing should be shaded in that region.

*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

Square Root Functions and Inequalities

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
  • Square Root Functions 0:07
    • Examples: Square Root Function
    • Example: Not Square Root Function
    • Radicand
    • Example: Restriction
  • Graphing Square Root Functions 3:42
    • Example: Graphing
  • Square Root Inequalities 8:47
    • Same Technique
    • Example: Square Root Inequality
  • 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

Transcription: Square Root Functions and Inequalities

Welcome to Educator.com.0000

Today, we will be talking about square root functions and inequalities.0002

First, defining what a square root function is: a square root function contains a square root involving a variable.0007

So, let's look at some examples before we go on to talk more about these.0016

A square root function could be something like f(x) = √(2x + 1), or g(x) = √(x2 + 4) - 2x.0022

Notice that it says "a square root involving a variable"; so there is a variable under the square root sign.0039

If I had a function--say h(x) =...let's say, instead of x2 + 4, I said √3, minus 2x.0046

This is not a square root function; and it is not a square root function because there is no variable in the radicand.0057

So, recall that whatever is under the square root sign here is the radicand.0072

And recall that terminology, because here it says that the radicand must be non-negative.0080

So, the domain is restricted to values that make the radicand non-negative.0085

Let's look at this first one, f(x) = √(2x + 1): if this expression were to become negative,0092

then what I would end up with here is...let's say I ended up with something like...this was a value like -4:0110

So, let's let x equal -4; then, what I would end up with is 2 times -4, plus 1; and that would give me -8 + 1, and that would be √-7.0119

The problem is that that is not a real number; and although we have talked about complex numbers0133

and imaginary numbers, we are going to restrict our discussion of square root functions to only functions that result in real numbers.0142

We are not going to allow values such as this.0151

Instead, when we look at a square root function, the first thing we are going to do is determine what the domain will be.0154

And we can do that by saying that this domain, this radicand, needs to be greater than or equal to 0.0160

So, we need to find values of x that will allow the radicand to be greater than or equal to 0, and our domain will be restricted to those values.0167

If I have f(x) = √(2x + 1), I am saying that I want 2x + 1 to be greater than or equal to 0.0177

Therefore, subtracting 1 from both sides gives me that 2x must be greater than or equal to -1.0185

Dividing both sides by 2 gives me x ≥ -1/2.0193

So, the domain is restricted to values of x that are greater than or equal to -1/2, for this function.0199

This function is not defined; we are not defining the function for values of x that are less than -1/2.0211

OK, in graphing a square root function, we are going to exclude values of x that make the radicand negative.0222

So, I just discussed how to find what the domain is; and values outside the domain are restricted.0227

And we won't even include those values on the graph.0233

For example, looking at the function f(x) = √(x + 3) - 2: what I want to do is make sure that x + 3 ≥ 0,0239

so that I don't end up with a negative radicand, and then an imaginary or complex number.0259

I am going to subtract 3 from both sides; and this is telling me that I must restrict my domain to x ≥ -3.0265

That is going to be the domain; I am not going to define this function for values of x less than -3; those are excluded values.0274

Values of x that are less than -3 are excluded from the domain.0282

OK, so with that in mind, we can pick some values for x that are part of the domain, and then evaluate the function for those values.0302

So, if I let x be -3 (because it says "greater than or equal to," so -3 is included), then I am going to get a 0 under here.0313

The square root of that is 0; minus 2 is going to give me -2 for y.0321

OK, when x is -2, the radicand will be 1; the square root of that is 1; minus 2 gives me -1.0329

Let's let x be -1: when x is -1, what I am going to end up with is the square root of 2.0341

Well, the square root of 2 is approximately 1.4; so that is going to give me 1.4 - 2, which is going to give me -0.6.0348

OK, when x is 1, this becomes 4; the square root of that is 2; minus 2 is 0.0362

Now, it is good to pick values that are easy to work with, so I am going to think about perfect squares.0370

How can I get perfect squares under here (like 1 + 3 gave me 4--that was a perfect square; -3 gave me 0--I can get a square root of that easily)?0376

Another nice number to work with is 9: pick values for x strategically--if I make x 6, this becomes 9, and the square root of that is 3; minus 2 gives me 1.0386

Another perfect square is 16: I just think, "OK, 16 - 3 is 13; 13 would be a good value to work with."0399

13 + 3 is 16; the square root of that is 4; subtract 2 from that, and I have 2.0408

Now, you will notice, I picked a large value here; and it was a convenient value;0416

but also, if you look at the coordinate axis I am using here, it is short but long,0421

because, to get a good graph, you are going to need to pick some big x-values,0427

because often, these graphs are not very tall, but they are wide--the slope is not very great.0433

For each change in x, I don't get a big change in y.0440

And then, if I don't pick enough values, I am not going to be able to find the shape of my graph.0444

Here, when x is -1, 2...-3 is right here...y is -2; that is (-3,-2), right here.0449

When x is -2, y is -1; -1 puts me right up here; -.6 is going to be about there; (1,0)--when x is 1, y is 0.0460

When x is 6, y is 1--not much change in the graph there--not much of a slope.0476

When x is 13, y is 2; and that is why I picked some values out here--so I could get a better sense of what this graph looks like.0484

So, just keep that in mind.0493

Now, one thing to also keep in mind is that the graph starts here, or ends here, because this function is not defined for values of x beyond this.0496

Beyond this is not defined, so I can't graph the function past x = -3, for values of x less than -3.0515

This is graphing square root functions: now, talking about graphing square root inequalities:0528

when we are graphing an inequality involving square roots, we are going to use0533

the same techniques that we used to graph linear and quadratic inequalities.0536

And recall that, when we handled these, what we did is...let's say quadratic inequalities:0540

the first thing we would do is graph the corresponding quadratic equation.0545

That equation formed the boundary line for our solution set; and then we used a test point to find on which side of the boundary the solution set lay.0550

And we are going to use those same techniques here, except now we are working with square root inequalities.0561

For example, if I am given the inequality y ≥ √2x, the one extra thing I do have to do is find the excluded values that we talked about.0566

I am going to need to graph this corresponding equation, which is going to be y = √2x.0585

But I have to make sure that I don't end up with some excluded value.0592

So, I know that, when I look at the radicand, I need 2x (the radicand) to be greater than or equal to 0.0597

So, that means that, if I divide both sides by 2, I am going to get that x must be greater than or equal to 0.0609

So, this is the domain; for this inequality, I am only going to graph the corresponding equation for a domain where x is greater than or equal to 0.0616

First, graph the corresponding equation; and in this case, the corresponding equation here is y = √2x.0627

And I can plot points, now that I have figured out what my domain is.0643

And the smallest value for my domain is 0; so when x is 0, y is 0.0647

When x is 1, that gives me the square root of 2, which is about 1.4.0654

When x is 2, that gives me a perfect square, 4; the square root of that is 2.0660

I like to work with perfect squares; I know that (even though it seems like a strange number to pick)--if I pick 4.5 and multiply it by 2, I will get 9.0665

And the square root of 9 is 3, so that makes it easy on that end of things.0676

And I will pick one more value; and again, I want to look for perfect squares, if I can, to make my life easy.0681

So, we used 4; we ended up with 4 when we made x 2.0688

When we made x 4.5, we ended up with 9; if I make x 8, I am going to get 8, times 2 is 16;0692

that is a perfect square, and the square root of that is 4.0699

I have enough points to plot; and when x is 0, y is 0; when x is 1, y is 1.4--about there; when x is 2, y is 2.0703

When x is 4.5, y is going to be 3; and then, when x is 8, y will be 4, right here.0715

Now, this is where the graph starts--or ends, depending on how you want to look at it.0733

I cannot define values of this function--evaluate the function--for any value smaller than x; the domain is restricted.0740

So, this is what the graph is going to look like.0751

Now, something else to point out: we are working with an inequality, and I did make this a solid line.0754

As previously, when we are working with inequalities, if it is ≥ or ≤, that means that the boundary line is part of the solution set.0761

If this had been a strict inequality, if it was greater than or just less than, I would have made this a dashed line0771

to indicate that the boundary line is not part of the solution set.0779

So, I graphed the corresponding equation; my second step is to use a test point to determine which side of the boundary the solution set is on.0782

Now, often, we use (0,0), the origin, as the test point.0794

But in this case, that is on the boundary line; and you want to find a test point that is away from the boundary line.0799

So, I am going to choose (right here) (3,0) as a test point.0805

And recall that, to use a test point, we go back to the inequality, y ≥ √2x, and we insert these values.0811

We substitute these values: so, I am going to let y be 0, and I am going to let x be 3.0824

That is going to give me 2 times 3; and this is going to end up being 0 ≥ √6.0833

And even if you don't know the exact value of the square root of 6 (it is about 2.5, but), we know that it is a positive number.0844

And we know that the square root of 4 is 2, and we know that the square root of 6 is going to be greater than that.0851

So, I know that this is not true: so this is not valid; therefore, the test point is not part of the solution set.0856

When I inserted these values, I came up with something not valid; so it is not part of the solution set.0872

So, the solution set is actually up here, not down here.0876

And I am going to shade in this region, and the boundary line is actually going to be included in my solution set.0880

But I need to keep in mind that this inequality is not going to be defined for any values of x smaller than 0.0886

So, I am not going to shade over here: the graph ends here; my shading ends there.0895

OK, so that was inequalities with square roots.0899

And with square root inequalities, you graph the corresponding equation, find the boundary line,0902

look at the inequality to determine if you should use a dashed line or a solid line,0909

and then use a test point to find if the solution set is above or below the boundary line.0914

The first example is asking me to graph y = √4x.0921

First, we are going to define the domain; and the domain has to be such that 4x is greater than or equal to 0.0928

I am dividing both sides by 4, so x has to be greater than or equal to 0.0935

All values for x that are less than 0 are excluded.0940

Now, I can plot points, starting with 0: 0 times 4 is 0; the square root of that is 0.0948

1 times 4 is 4; the square root of that is 2; 2 times 4 is 8; the square root of 8--you can use a calculator,0957

and figure out that it is approximately 2.8; or you can just estimate by knowing that0969

it is going to be greater than a square root you know of, such as the square root of 4,0975

but less than another square root that you are familiar with.0980

4 times 4 is 16; and so, the square root of that is going to be 4; and then the square root of 6 times 4...0985

that is 24, and the square root of 24 is approximately equal to 5--a little bit less than 5, actually...4.9.0995

We know that the square root of 25 is 5, so the square root of 24 is going to be slightly less than that; we will say about 4.9--close enough for this graph.1013

When x is 0, y is 0; when x is 1, y is 2; when x is 2, y is a little bit below 3...2.8...around there.1024

When x is...let's move this over just a bit; it is going to be right about there...and when x is 4, y is 4.1036

And then, one more point here: we are going to have x be 6, and y is going to be a little bit below 5, just to give me one more point to work with.1051

Now, again, the graph begins right here; that is just going to be a point, and then this is going to be an arrow, continuing on out.1067

So, we are beginning the graphing by finding the domain, x ≥ 0, and then plotting some points.1074

OK, here y is less than -√6x; so I am working with a strict inequality.1084

And I first need to think about excluded values, because I want to graph the corresponding equation, y = -√6x.1093

And I know that the radicand, 6x, must be greater than or equal to 0.1106

Dividing both sides by 6 tells me that x has to be greater than or equal to 0; so this is the domain.1112

So, as I plot points, I am only going to choose values for x that are at least 0.1120

Starting out with 0: 6 times 0 is 0; the square root of that is 0; -0 is still 0.1129

So, when x is 1, this is going to give me the square root of 6, and that is going to be about 2.5.1138

The square root of 6 is about 2.5, and I need to make that negative--about -2.5.1149

2 times 6 is 12; the square root of 12 is around 3.5; I am making that negative: -3.5.1156

4 times 6 is 24; and the square root of that, we said, is a little bit less than 5; and making that negative, it is about -4.9.1166

And then, 5 times 6 is 30; the square root of that is about 5.5, so it is going to give me -5.5.1178

So, this is enough to plot; and I am going to use a dashed line, because it is a strict inequality,1186

meaning that the boundary line is not part of the solution set.1192

So, when x is 0, y is 0; when x is 1, y is going to be -2.5; when x is 2, y is going to be -3.5...about right there.1196

When x is 4 (that is 1, 2, 3, 4, 5...) y is going to be a little bit more than -5...right about there.1225

And then, I am going to put a -6 down here; and when x is 5, this is going to give me -5.5.1241

So, this is curving like this.1251

Now, I am going to use a dashed line for this, since it is a strict inequality; and the graph begins right there.1256

I graphed the boundary line; now I need to find a test point; and again, I am not going to use (0,0) as a test point, because it is on the boundary line.1270

Instead, I am going to go ahead and use (3,0), so my test point is (3,0).1278

I am going to look at this: y < -√6x; let y be 0; let x be 3.1290

Is 0 less than -√18? Well, the square root of 18 is about 4.2, and so this is not true.1302

Even if you didn't know what the square root of 18 is, you know that it is going to be greater than √16, which is 4.1318

So, it is going to be some number greater than 4; the square root of 18 will be a little bit more than 4.1324

And when we make that negative, it is going to be negative 4 point something, and you know that 0 is not less than a negative number.1330

This is not true; so the test point is not part of the solution set, which tells me that the solution set is down here.1337

And again, I am not going to shade in past x = 0, because this graph--this function--is not defined for values of x less than 0.1344

Those are excluded values.1356

OK, we are supposed to graph this square root function, y = √(x + 2) - 3.1362

Let's find the excluded values: x + 2 must be greater than or equal to 0, so x must be greater than or equal to -2.1371

This is my domain, so I am not going to attempt to plot points where x is less than -2.1380

Let's start out with -2: when x is -2, the radicand is 0; the square root of 0 is 0, minus 3 gives me -3.1390

When x is -1, -1 + 2 is 1; the square root of 1 is 1; minus 3 is -2.1403

When x is 2, 2 + 2 is 4; the square root of that is 2; minus 3 is -1.1412

When x is 3, this gives me 5; the square root of 5 is about 2.2; subtracting 3 from that...2.2 - 3 is -0.8.1420

Again, I like to look for perfect squares in the radicand to make things easy.1437

And 9 is a perfect square, so I am going to let x be 7; this will become 9; the square root of 9 is 3; minus 3 is 0.1440

And I am going to also let x be 14, because that will give me 16; 14 + 2 is 16; the square root of that is 4; minus 3 is 1.1449

OK, so when x is -2, y is -1, 2, 3, so right here; when x is -1, y is -2; when x is 2, y is -1.1462

When x is 3, y is -0.8, right about there; when x is 7, y is 0; and when x is 14, y is 1.1481

OK, so I am going to go ahead and draw a line through these points, and alter that just a little bit so it goes through there...1498

And this continues on: again, we have one of these short, very wide graphs, where right here, we don't have a lot of slope.1508

So, I needed to find some large values of x for x.1517

And this finishes out the graph; the graph starts right there at x = -2 (I can't graph anything past there) and continues on.1522

Here I have an inequality; and I am going to start out...what I want to do is graph the corresponding equation, y = √(x - 4) + 2.1538

But I know that I need to find excluded values; and I know that x - 4 must be greater than or equal to 0,1549

since I don't want to have a negative radicand; I need to have values in the radicand that are 0 or greater.1556

If I add 4 to both sides, I find that the domain is that x must be greater than or equal to 4.1563

So, I am going to plot points accordingly; and I know that x can be 4--that is the smallest value I can give it.1572

4 - 4 is 0; the square root of that is 0; plus 2 gives me 2.1581

OK, I am looking for perfect squares: a perfect square...if I have 1, I can easily get the square root of that.1589

So, if I let x be 5, and I subtract 4 from that, I am going to get 1; the square root is 1; plus 2 gives me 3.1596

Let's see, another perfect square is 4; so I am going to let x equal 8; 8 - 4 is 4; the square root of that is 2; plus 2 is 4.1610

Another perfect square would be 9, so I am going to let x equal 13, because 13 - 4 is 9; the square root of that is 3; add 2 to that--that gives me 5.1620

So, I have some values that I can plot out.1631

And I am going to use a solid line, because this inequality is greater than or equal to; it is not a strict inequality.1636

So, when x is 4, y is 2; and that is the start of my graph--I can't define this function for values less than 4.1643

OK, when x is 5, y is 3; when x is 8, then y is 4; and when x is 13, y is 5...right about there.1655

OK, and this is a solid line that I am going to use; and the graph begins here.1678

Now, this is an inequality, and I need to figure out which side of this boundary line the solution set is.1686

And I know that I am not going to worry about values over here--just ones where x is greater than or equal to 4.1696

So, I am going to go ahead and...I don't want to pick a point on the boundary line, and (0,0) is not even part of this graph at all;1703

so, I am going to go ahead and pick (5,0) as a test point.1709

Test point (5,0) for y ≥ √(x - 4) + 2: when y is 0, let's see if this holds true.1716

5 - 4 + 2: is 0 greater than the square root of 1, plus 2? Is 0 greater than or equal to 1 + 2?1730

Is 0 greater than or equal to 3? No, it is not, so the test point is not part of the solution set.1740

This is not part of the solution set; so what I need to do is shade in this region of the graph1754

above the boundary line, being careful not to go over here, because that is not part of the graph; those are not defined areas.1766

To find the graph of this inequality, I graphed the corresponding equation, first finding excluded values1777

(that values that were less than 4 are excluded), then using values that are allowed to plot points,1784

and then using a test point and finding that my test point is not part of the solution set down here.1791

So, I went ahead and shaded in up there.1797

That concludes this lecture for Educator.com; and thanks for visiting!1801

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