INSTRUCTORS Carleen Eaton Grant Fraser

Home » Mathematics » Algebra 2 » Exponential Growth and Decay

Dr. Carleen Eaton

Dr. Carleen Eaton

Exponential Growth and Decay

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

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

20m 15s

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

19m 10s

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

17m 31s

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

17m 14s

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

25m

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

32m 5s

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

14m 46s

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

23m 7s

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

23m 5s

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

31m 5s

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

21m 42s

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

17m 13s

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

23m 53s

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

27m 12s

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

28m 53s

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

11m 34s

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

21m 36s

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

29m 36s

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

33m 13s

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

28m 25s

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

22m 25s

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

22m 32s

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

31m 48s

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

27m 3s

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

19m 53s

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

35m 45s

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

27m 11s

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

22m 48s

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

30m 7s

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

27m 5s

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

19m 29s

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

13m 27s

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

31m 11s

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

22m 30s

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

33m 29s

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

21m 10s

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

31m 21s

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

31m 27s

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

31m 16s

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

34m 30s

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

22m 42s

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

30m 4s

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

20m 46s

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

41m 11s

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

30m 45s

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

31m 27s

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

40m 54s

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

55m 4s

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

57m 13s

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

20m 21s

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

55m 14s

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

35m 58s

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

45m 54s

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

28m 43s

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

25m 23s

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

21m 14s

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

24m 30s

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

32m 42s

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

41m 27s

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

21m 3s

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

46m 51s

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

38m 15s

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

18m 43s

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

47m 4s

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

21m 16s

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

21m 36s

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

23m 3s

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

22m 43s

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

18m 32s

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

14m 34s

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

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Lecture Comments (8)

1 answer

Last reply by: Dr Carleen Eaton
Mon May 4, 2015 9:10 PM

Post by David Saver on May 1, 2015

In the Half-life example, where does the answer 7 months come from?
Or is it suppose to be 7 years?

1 answer

Last reply by: Dr Carleen Eaton
Mon Nov 25, 2013 9:41 PM

Post by Mirza Baig on November 25, 2013

What the difference between this formula y=a*p^a and y=a(1+r)^t????
Can we use any two formula given below and find the growth of change ??

1 answer

Last reply by: Dr Carleen Eaton
Mon May 4, 2015 9:10 PM

Post by Adela Bravo on May 30, 2012

At 8:27 shouldn't 7months be 7years, because t=years.

1 answer

Last reply by: Dr Carleen Eaton
Sun Nov 27, 2011 4:11 PM

Post by Vanza Bik on November 26, 2011

I have a question? How can I ask you?

Exponential Growth and Decay

  • If growth or decay is occurring by a fixed percentage during each period of time, use the formula y = a(1 + r)t or y = a(1 – r)t. For scientific applications, use the formula y = aekt or y = ae-kt.

Exponential Growth and Decay

A recent college grade bought a brand new car for $ 22,999.99. The value of the car is expected to
depreciate at a rate of .96% per year. What is it's value after 5 years?
  • Use the formula y = a(1 − r)t where a = initial amount, r is the rate and t is the number of years
  • y = a(1 − r)t
  • y = 22999.99(1 − 0.0996)5
  • y = 22999.99(0.9004)5
  • y = 22999.99(0.5918)
  • y = 13611.47
After 5 years the car will be valued at $ 13,611.47.
A high school student bought a car for for $ 25,000. The value of the car is expected to
depreciate at a rate of % 15.00 per year. What is it's value after 4 years?
  • Use the formula y = a(1 − r)t where a = initial amount, r is the rate and t is the number of years
  • y = a(1 − r)t
  • y = 25000(1 − 0.15)4
  • y = 25000(0.85)4
  • y = 25000(0.522)
  • y = 13,050.16
After 4 years the car will be valued at $ 13,050.16
A small apartment - home is expected to increase in value at the rate of 5% per year.
How long would it take for this small apartment - home to tripple in value?
  • Use the equation y = a(1 + r)t. Find t when y = 3a
  • 3a = a(1 + 0.05)t
  • 3a = a(1.005)t
  • 3 = (1.005)t
  • Take the log of both sides
  • log(3) = log(1.005t)
  • log(3) = tlog(1.005)
  • t = [log(3)/log(1.005)] = 220.27
It will take 220.27 years for the home to tripple in value.
The value of gold is expected to increase in value at a rate of 7.45% per year.
How long would it take for the price of gold to double in value?
  • Use the equation y = a(1 + r)t. Find t when y = 2a
  • 2a = a(1 + 0.0745)t
  • 2a = a(1.0745)t
  • 2 = (1.0745)t
  • Take the log of both sides
  • log(2) = log(1.0745t)
  • log(2) = tlog(1.0745)
  • t = [log(2)/log(1.0745)] = 9.65
It will take 9.65 years for the price of gold to double in value.
A new investment strategy promises potential investors a return of investment of 8.05% per year.
How long will it take for this investment to reach 2.5 times its original value?
  • Use the equation y = a(1 + r)t. Find t when y = 2.5a
  • 2.5a = a(1 + 0.0805)t
  • 2.5a = a(1.0805)t
  • 2.5 = (1.0805)t
  • Take the log of both sides
  • log(2.5) = log(1.0805t)
  • log(2.5) = tlog(1.0805)
  • t = [log(2.5)/log(1.0805)] = 11.83
It will take 11.83 years for the investment to reach 2.5 its original value.
Caffeine has a half - life of 5 hours. If caffeine decays based on the equation y = ae − kt, find the coefficient k.
  • Set up the equation. y = 1/2a
  • y = ae − kt
  • [1/2]a = ae − k*5
  • [1/2] = e − k*5
  • Take the ln of both sides
  • ln([1/2]) = lne − k*5
  • ln(0.5) = − k*5
k = [ln(0.5)/( − 5)] = 0.139
Penicillin has a half - life of 45 minutes. If penicillin decays based on the equation y = ae − kt, find the coefficient k.
  • Set up the equation. y = 1/2a
  • y = ae − kt
  • [1/2]a = ae − k*0.75
  • [1/2] = e − k*0.75
  • Take the ln of both sides
  • ln([1/2]) = lne − k*75
  • ln(0.5) = − k*0.75
k = [ln(0.5)/( − 0.75)] = 0.9242
The bacterial growth in the lab can be modeled by y = aekt. At the beginning of the experiment there
were 250,000 bacterial. Ten hours later, there were 500,000. What is the coefficient k?
  • Set up the equation. y = 2a since the bacterial doubled in size.
  • y = aekt
  • 2a = aek*10
  • 2 = ek*10
  • Take the ln of both sides
  • ln(2) = lnek*10
  • ln(2) = k*10
k = [ln(2)/10] = 0.0693
The bacterial growth in the lab can be modeled by y = aekt. At the beginning of the experiment there
were 250,000 bacterial. Ten hours later, there were 500,000. How long would it take for the
bacterial culture to reach 999,999 bacterial?
  • Set up the equation. use the value of k = 0.0693 from the previous problem
  • y = aekt
  • 999,999 = 250,000e0.0693*t
  • 3.999996 = e0.0693*t
  • Take the ln of both sides
  • ln(3.999996) = lne0.0693*t
  • ln(3.999996) = 0.0693*t
t = [ln(3.999996)/0.0693] = 20.004 hours
The bacterial growth in the lab can be modeled by y = aekt. At the beginning of the experiment there
were 250,000 bacterial. Ten hours later, there were 500,000. How long did it take the bacterial to
grow to 375,000?
  • Set up the equation. use the value of k = 0.0693 from the previous problem
  • y = aekt
  • 375,000 = 250,000e0.0693*t
  • 1.5 = e0.0693*t
  • Take the ln of both sides
  • ln(1.5) = lne0.0693*t
  • ln(1.5) = 0.0693*t
t = [ln(1.5)/0.0693] = 5.85 hours

*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

Exponential Growth and Decay

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
  • Decay 0:17
    • Decreases by Fixed Percentage
    • Rate of Decay
    • Example: Finance
  • Scientific Model of Decay 3:37
    • Exponential Decay
    • Radioactive Decay
    • Example: Half Life
  • Growth 9:06
    • Increases by Fixed Percentage
    • Example: Finance
  • Scientific Model of Growth 11:35
    • Population Growth
    • Example: Growth
  • Example 1: Computer Price 14:00
  • Example 2: Stock Price 15:46
  • Example 3: Medicine Disintegration 19:10
  • Example 4: Population Growth 22:33

Transcription: Exponential Growth and Decay

Welcome to Educator.com.0000

In today's lesson, we are going to go over exponential growth and decay.0002

This topic was introduced briefly in a previous lecture on exponential equations.0006

And now, we are going to look at it in greater depth and work on some sample problems.0011

First, decay: there are actually two formulas that we are going to go over for decay, and two for growth.0017

The first decay formula talks about a situation where you have a quantity that decreases by a fixed percentage each year.0024

The equation for this situation is y = a(1 - r)t.0033

This gives the amount, y, of the quantity after t years; so a is the initial amount that you begin with;0040

r is the percent decrease; and in order for this formula to work correctly, you need to express it as a decimal.0056

y is the amount that you end up with, the quantity after t time; and t is the time that has passed.0072

A typical application for this formula would be in finance.0091

For example, if you have $1,000 in your bank account, and you remove 40% of whatever you have in that account each year,0096

and you are trying to figure out how much you would have left in 3 years, you could use this formula.0105

So, I said you began with $1,000; therefore, a = 1,000 dollars--that is how much money you started out with in your savings account.0111

The percent decrease is going to be 40%, because you are taking that amount out each year (or .4).0122

And we are trying to figure out how much money you are going to have left after 3 years,0132

so the unknown is the amount of money that you have left after 3 years; and the time is 3 years.0138

So, trying this problem out using this formula: we are going to take y =...and y is our unknown;0146

a is 1,000; times 1 minus...and it is .4, expressed as a decimal, raised to the t power (which here is 3).0155

That gives me y = 1000(0.6)3; .62 is .36, times .6 is .216; so this leaves me with y = 1000(.216).0166

And multiplying that out, I would find that I would have 216 dollars left in my account after 3 years of removing 40% of the money in the account.0184

So again, this decay formula is the general formula often used for applications, talking about financial applications, interest...0196

Actually, interest would be growth...but interest payments...decreases; we are talking about decreases.0207

The scientific model of decay is slightly different.0217

And in a lot of applications in science, the formula that is used is exponential decay: y = ae-kt.0221

We talked previously about how base e is often used in scientific applications; we see it popping up here again.0231

And as in the last formula, y is the amount that you end up with after t years, whereas a is the initial amount.0237

This time, instead of having a fixed percent that it decreases by, we use a constant.0245

And the constant depends on the situation.0251

A typical situation would involve radioactive decay; and with radioactive decay, or any type of decay, you will be given a constant.0253

There might be a certain isotope or a certain compound that decays according to one k.0262

And then, you look at a different compound, and it might decay a lot faster; and then the value of k would be different.0269

So, just to review what the graph would look like when we are talking about an exponential decrease:0274

the shape of the graph would be roughly like this, over time.0279

You see here: as time moves forward, the amount that you have left is getting smaller and smaller and smaller and smaller, and approaching 0 eventually.0284

This formula, as I mentioned, is important when working with radioactive substances and thinking about radioactive decay.0298

And you may have heard of the concept of half-life.0305

Half-life is the amount of time it takes for half of the material to decay away.0308

So, let's say I have 10 grams of a radioactive substance, and it takes 3 months for it to decay until there are only 5 grams left.0313

The half-life of that substance would be 3 months, the amount of time it takes for half of it to be gone.0326

And let's look at an example of this: let's say that I have a radioactive compound that decays exponentially according to this formula.0333

And k equals 0.1 for a certain radioactive substance; and it follows exponential decay; and I want to know the half-life.0341

All right, let's look at this formula and think about how this will work.0361

I have y = ae-kt, and I am not given the original amount, which seems like it would be a problem;0364

but it is actually not, and here is why: think about what half-life is saying.0370

It is saying that half of the original amount is left.0375

So here, the amount we are left with, y, equals half the original amount, or 1/2a.0382

Knowing that, I can set this up as follows: 1/2a = ae-kt.0390

Once I have this set up like this, the a's will cancel; I am going to divide both sides (I am trying to get rid of the a on the right) by a.0397

That cancels; this cancels; and I just am left with 1/2 = e-kt.0406

Recall, earlier on, when we were working with exponential equations, we talked about how it is possible to solve these by taking the log of both sides.0414

So, let's go ahead and fill in what we have for k, which is -.1.0426

And what we are looking for is the time--the amount of time it takes for half of the substance to decay.0432

So, we have it down to this point; and what I can do is take a log of both sides.0440

And since I am working with base e, I am going to go ahead and work with the natural log,0444

although you actually could use any log, as long as it has the same base.0448

That is going to give me ln(1/2) = ln(e-.1t).0451

Because these are inverses, the ln(e)...these are essentially going to cancel each other out.0461

And on the right, I am going to be left with -.1t.0468

On the left, I am going to actually rewrite this as ln(2-1); and recall that 2-1 is the same as 1/2.0475

All right, let's go on up here to finish this one out.0484

ln(2-1) = -.1t: recall, from the rules of working with logs, that I can rewrite this as -1ln(2), or just -ln(2), equals -.1t.0486

I am going to divide both sides by -.1; the negatives are going to cancel out, and this is going to give me ln(2)/.1 = t.0508

So, I am rewriting this like this, ln(2) divided by .1.0517

You can use your calculator and the natural log button on there to find out what ln(2) is, divided by .1.0526

And that actually comes out to approximately 7 months; so it would take 7 months for this compound to decay until there is only half left.0533

Therefore, the half-life is 7 months.0541

That was decay; now we are talking about growth.0546

OK, so again, there are two formulas, and they are very similar to the decay formulas.0554

Here, we are talking about growth, so for a quantity that increases by a fixed percentage each year, the equation...0558

this actually should be a plus right there...y = a(1 + r)t gives the amount, y, of the quantity after t years.0571

So, a is the initial amount, just like we talked about with decay.0583

And r is the percent increase, and we need to express that as a decimal.0588

And what we are looking for is the final amount, the amount that we are going to end up with.0592

And this is the general formula; so this is the model that you might see used again when working with money or financial-type problems.0597

Let's look at an example: let's say that you had $500 in your bank account, so a = 500.0607

And you receive 10% interest per year on that.0616

I am going to rewrite that in a decimal form: .1.0620

And what you want to know is how much money you are going to end up with after 3 years.0624

So, rewriting this as y = a(1 + r)t, this is going to give me y = 500 (that is what I started out with);0632

and this is 1 + .1, raised to the t power; and in this case, t is actually 3--let's go ahead and make that 3.0646

y = 500(1.1)30662

And you could use your calculator to determine that 1.1 raised to the third power is equal to 1.331.0669

Multiplying that by 500, you would end up with 665 dollars and 50 cents in your account after 3 years, if you are receiving 10% interest on that $500.0678

So, you can see that this is a really useful formula.0690

The scientific model of growth is analogous to the scientific model of decay that we talked about before.0695

Notice here that now we have a positive exponent; with the decay formula, y = ae...it was -kt; now we have a positive exponent.0700

And the constant k depends on the situation.0711

You might be asked to find it, if they give you what y is, and a, and t; or you may be given the k and then asked to figure out something else.0715

An important application here in science is in population growth.0724

And systems that follow this model of growth increase exponentially, so their graph is going to look approximately like this.0728

So, let's consider an example where you have a population of 1,000 people.0740

There are 1,000 people living in a town, and this town's population is increasing according to this exponential model.0745

And k is 0.2: we want to know the number of people that we are going to end up with in 10 years.0756

So, using this formula, y = aekt, we don't know y; we know that a is 1,000; e; k is .2; and time is 10 years.0770

This gives me y = 1,000e...0.2 times 10--that is just going to give me 2.0790

Now, I can figure this out; I could use my calculator, or you might remember that e is 2.7182.0797

So, if I square that, I will get approximately (for e2) 7.389.0805

Multiply by 1,000: I am going to end up with 7,389 people living in the town 10 years down the road.0817

OK, so that was the four equations that we are working with today: two each for growth and decay.0831

So, let's try some examples: A computer is purchased for $2,000, and it is expected to depreciate at the rate of 20% per year.0840

Depreciate means it is losing value: so you bought your computer, and each year it is worth less,0851

which is frequently the case with objects that you buy.0858

Let's think about what formula we are going to use.0862

Since this is decreasing, we know that we need to use a decay formula.0864

And since it is decreasing at a certain percentage per year, I am going to use the general decay formula, which is y = a(1 - r)t.0868

My initial amount is $2,000; that is what I purchased it for.0880

And the rate of depreciation is 20%; in decimal form, this is .2.0884

I want to know its value after a certain amount of time; and the amount of time is 3 years.0892

With this in mind, I can just go ahead and work this out.0901

y is what I am looking for; a is 2,000; times 1 - .2, raised to the third power.0906

Therefore, y = 2,000...1 minus .2 is .8, to the third power; you can go ahead and work this out on your calculator;0914

and you will find that it comes out to .512, times 2,000, is 1,024 dollars.0926

So, this computer, in three years, at this rate of depreciation, is going to be worth $1,024.0935

Now, we are talking about a stock that is expected to increase in value.0947

Since we are talking about increasing, we are talking about growth.0950

And this is at the rate of 45% per year.0953

Since I am talking about a steady increase that is based on percent per year, I am going to use the general growth formula, not the scientific exponential one.0958

So, I am going to use this formula: y = a(1 + r)t.0968

And I want to know how long it will take for the stock to triple in value.0975

So, you have to think carefully about how to set this up.0979

I know the rate: the rate is given as 45%, which is equal to .45.0982

And I am asked how long it will take for the stock to triple in value; so I want to know0992

when the value I am going to end up with, y, is 3 times the initial value, 3a.0997

So, I have an initial value, a: it doesn't matter what that value is.1003

What I want to know is when it is going to be 3 times whatever a is.1006

So, if you just keep going on this, you see that the a will drop out.1010

So then, I set this up as...let's go ahead and start it over here...for y, I am going to substitute 3a.1013

This equals a; I don't know what a is--I just put a.1021

1 + .45...and I know that what I am looking for, actually, is t: so, I just leave this as t.1024

When I divide, if I want to move this a over here, I am going to divide both sides by a; and the a's will conveniently drop out,1036

leaving us with 1 + .45, raised to the t power: so 3 = 1.45t.1045

Recall that, when you are working with exponential equations with different bases1053

that you can't easily get to become the same base, you can just take the log of both sides.1057

And you can use any base log you want; but two convenient ones are the natural log and the common log.1063

So, I am going to go ahead and take the common log, the base 10 log, of both sides.1070

log(3) = log(1.45t).1075

Using properties of logs, I can rewrite this as t times log(1.45); and I am looking for t.1081

I want to isolate t, so I am going to make this log(3), divided by log(1.45), dividing both sides by log(1.45).1092

And remember that these are just numbers; they have a value--I am going to work with them and move them around, just like I would any other numbers.1103

I am rewriting this so that the t is on the left--a more common form.1112

This is something you can then use your calculator to figure out.1118

And if you take the log of 3, divided by log base 10 of 1.45, you will get approximately equal to 3 years.1121

So, even though I didn't know what my stock was worth to start with, it didn't matter.1134

All I wanted to know is how long it is going to take whatever amount I have to triple; and it is going to take 3 years.1139

Example 3: A medicine disintegrates in the body at a steady rate.1153

We are talking about disintegration, which is a type of decrease; so we are talking about decay.1158

It decays based on the equation y = ae-kt.1163

This is using the scientific model of decay with base e, where k equals 0.125, and t is in hours.1168

Find the half-life of this medicine.1178

We are using this formula; let's rewrite it here; and we want to find the half-life.1182

So, we are looking for the half-life, which is often written this way: t1/2.1186

And I don't know how much medicine we are starting out with, but it doesn't matter.1195

It is similar to the last one we worked out, where it doesn't matter what you are starting out with.1201

You are just looking for a change.1204

I know k; that is given; and I know that what I am going to end up with is half of the initial.1208

Since I am looking for half-life, after this amount of time, y is going to be equal to half of the amount that we started out with, 1/2a.1218

I am coming over here and setting this up, substituting 1/2a for y, equals ae, and then I have -.125 as k; and what I am looking for is t.1226

If I divide both sides by a, the a will drop out.1243

This gives me 1/2 = e-.125t.1246

One way to solve this is to take the log of both sides; I am going to go ahead and take the natural log of both sides; this is ln(1/2) = ln(e-.125t).1255

Because these are inverses, I am going to end up with -.125t.1271

To go further with this, I am going to rewrite this left part, ln(1/2), as ln of 2 raised to the -1 power,1277

just like the example we worked out a few slides ago, equals -.125 times t.1286

I can bring this out in front, which would be -1ln(2), or just -ln(2), equals -.125t.1298

I am going to divide both sides by -.125, and this equals t; so these negatives...a negative divided by a negative1307

is going to become a positive, so I come up here and get ln(2) divided by .125 = t.1320

Now, I can find the natural log of 2 and divide that by .125, using a calculator.1329

And it turns out that this is approximately equal to 5 hours.1335

Therefore, the half-life of this medication, given this constant, is approximately 5 hours, based on using this model for exponential decay.1340

Example 4: The population growth of a city can be modeled exponentially with a constant of k = 0.01.1355

The current population is 100,000; what will it be in 100 years?1363

Since this is exponential growth, and we are given a constant, I am going to use the scientific model for growth, that formula, y = aekt.1368

All right, I have been given a k; this is 0.01; and I have been given the original population, which is 100,000.1382

I have been given a t of 100 years, and I am asked to find y.1395

I want to know what the population is going to end up being after 100 years.1401

Therefore, this becomes y = 100,000 times e; and that would be a k of .01, times a t of 100.1407

Therefore, this is y = 100,000 times e; and this is just going to be .01 times 100, so that is going to give me 1.1418

Recall that e is 2.71828, so instead of e to the first power, it is just e, which I know the value of.1428

Then, I go ahead and multiply this times 100,000 to get that 271,828 will be the population in 100 years.1441

Again, this was simply a problem involving the use of the scientific model for growth.1456

That concludes this lesson on Educator.com on exponential growth and decay; thanks for visiting!1464

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