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Mathematical Foundations II

Master the algebra of advanced functions including quadratics, logarithms, and trigonometry. Dive deep into the theory of polynomials, learn the basics of limits, derivatives, and integrals from calculus, and explore a variety of concepts from higher math including complex numbers, vectors, probability, and statistics.





Nonlinear Equations and Graphing

Polynomial Division and Factoring

Graphing Polynomials and Rational Functions

Limits, Derivatives, and Integrals


Complex Numbers and Vectors

Probability and Statistics

21 topics
1.1. Quadratic Equations
1.1.1. Introduction to Quadratic Equations
1.1.2. Solving Perfect Square Quadratic Equations
1.1.3. Perfect Square Quadratic Equations with One Solution or No Solutions
1.1.4. The Zero Product Rule for Solving Quadratic Equations
1.1.5. Solving Quadratic Equations Using a Difference of Squares
1.1.6. Solving Quadratic Equations with No Constant Term
1.1.7. Solving Quadratic Equations by Factoring
1.1.8. Solving Quadratic Equations with Leading Coefficients by Factoring
1.1.9. Completing the Square
1.1.10. Completing the Square With Odd Linear Terms
1.1.11. Completing the Square With Leading Coefficients
1.1.12. Solving Quadratic Equations by Completing the Square
1.1.13. Solving Quadratic Equations With Leading Coefficients by Completing the Square
1.1.14. The Quadratic Formula
1.1.15. The Discriminant of a Quadratic Equation
1.2. Quadratic Functions
1.2.1. Graphing Elementary Quadratic Functions
1.2.2. Vertical Reflections of Quadratic Functions
1.2.3. Graphs of General Quadratic Functions
1.2.4. Roots of Quadratic Functions
1.2.5. The Discriminant of a Quadratic Function
1.2.6. Finding Intersections of Lines and Quadratic Functions
25 topics
2.3. Polynomials
2.3.1. Closure Properties of Polynomials
2.3.2. The Least Common Multiple of Two Monomials
2.3.3. The Least Common Multiple of Two Polynomials
2.3.4. Determining the Roots of Polynomials
2.3.5. Graphing Elementary Cubic Functions
2.3.6. End Behavior of Polynomials
2.4. Factoring Polynomials
2.4.1. Factoring Polynomials Using the Greatest Common Factor
2.4.2. Factoring Higher-Order Polynomials as a Difference of Squares
2.4.3. Factoring Cubic Expressions by Grouping
2.4.4. Factoring Sums and Differences of Cubes
2.4.5. Factoring Biquadratic Expressions
2.5. Dividing Polynomials
2.5.1. Dividing Polynomials Using Synthetic Division
2.5.2. Dividing Polynomials by Linear Binomials Using Long Division
2.5.3. Dividing Polynomials Using Long Division
2.5.4. Dividing Polynomials by Manipulating Rational Expressions
2.6. Polynomial Theorems
2.6.1. The Factor Theorem
2.6.2. Determining Polynomial Coefficients Using the Factor Theorem
2.6.3. Factoring Cubic Polynomials Using the Factor Theorem
2.6.4. Multiplicities of the Roots of Polynomials
2.6.5. The Remainder Theorem
2.6.6. The Rational Roots Theorem
2.7. Graphs of Polynomials
2.7.1. Graphing Cubic Curves Containing Three Distinct Real Roots
2.7.2. Graphing Cubic Curves Containing a Double Root
2.7.3. Graphing Cubic Curves Containing One Distinct Real Root
2.7.4. Graphing General Polynomials
Number Systems
13 topics
3.8. The Real Number System
3.8.1. Natural Numbers, Integers, and Rational Numbers
3.8.2. The Real Number System
3.8.3. Sums and Products of Rational and Irrational Numbers
3.9. Introduction to Complex Numbers
3.9.1. Imaginary Numbers
3.9.2. Solving Quadratic Equations with Purely Imaginary Solutions
3.9.3. Complex Numbers
3.9.4. Adding and Subtracting Complex Numbers
3.9.5. Multiplying Complex Numbers
3.9.6. Solving Quadratic Equations With Complex Roots
3.9.7. The Cyclic Property of the Imaginary Unit
3.10. The Complex Plane
3.10.1. The Complex Plane
3.10.2. The Magnitude of a Complex Number
3.10.3. The Argument of a Complex Number
24 topics
4.11. Functions
4.11.1. The Arithmetic of Functions
4.11.2. Composition of Functions
4.11.3. Finding Algebraic Expressions of Composite Functions
4.11.4. Local Extrema of Functions
4.11.5. One-To-One Functions
4.11.6. Introduction to Inverse Functions
4.11.7. Calculating the Inverse of a Function
4.11.8. Graphs of Inverse Functions
4.11.9. The Domain and Range of an Inverse Function
4.11.10. Invertible Functions
4.11.11. Plotting X as a Function of Y
4.11.12. Periodic Functions
4.11.13. Even and Odd Functions
4.11.14. Unbounded Behavior of Functions Near a Point
4.12. Graph Transformations of Functions
4.12.1. Vertical Translations of Functions
4.12.2. Horizontal Translations of Functions
4.12.3. Vertical Stretches of Functions
4.12.4. Horizontal Stretches of Functions
4.12.5. Vertical Reflections of Functions
4.12.6. Horizontal Reflections of Functions
4.12.7. Combining Graph Transformations: Two Operations
4.12.8. Combining Graph Transformations: Three or More Operations
4.12.9. Constructing Functions Using Transformations
4.12.10. Combining Reflections With Other Graph Transformations
Exponentials & Logarithms
27 topics
5.13. Introduction to Logarithms
5.13.1. Converting From Exponential to Logarithmic Form
5.13.2. Converting From Logarithmic to Exponential Form
5.13.3. Evaluating Logarithms
5.13.4. The Natural Logarithm
5.13.5. The Common Logarithm
5.13.6. Simplifying Logarithmic Expressions
5.14. The Laws of Logarithms
5.14.1. The Product Rule for Logarithms
5.14.2. The Quotient Rule for Logarithms
5.14.3. The Power Rule for Logarithms
5.14.4. Combining the Laws of Logarithms
5.14.5. The Change of Base Formula for Logarithms
5.15. Exponential Equations
5.15.1. Solving Exponential Equations Using Logarithms
5.15.2. Solving Equations Containing the Exponential Function
5.15.3. Solving Exponential Equations With Different Bases
5.15.4. Solving Exponential Equations With Different Bases Using Logarithms
5.15.5. Solving Exponential Equations Using the Zero-Product Property
5.16. Logarithmic Equations
5.16.1. Solving Logarithmic Equations
5.16.2. Solving Logarithmic Equations Containing the Natural Logarithm
5.16.3. Solving Logarithmic Equations Using the Laws of Logarithms
5.16.4. Solving Logarithmic Equations by Combining the Laws of Logarithms
5.17. Graphs of Exponential Functions
5.17.1. Vertical Translations of Exponential Growth Functions
5.17.2. Vertical Translations of Exponential Decay Functions
5.17.3. Combining Graph Transformations of Exponential Functions
5.17.4. Properties of Transformed Exponential Functions
5.18. Graphs of Logarithmic Functions
5.18.1. Graphing Logarithmic Functions
5.18.2. Combining Graph Transformations of Logarithmic Functions
5.18.3. Properties of Transformed Logarithmic Functions
Radical & Rational Functions
23 topics
6.19. Rational Expressions
6.19.1. Simplifying Rational Expressions Using Polynomial Factorization
6.19.2. Adding and Subtracting Rational Expressions
6.19.3. Adding Rational Expressions With No Common Factors in the Denominator
6.19.4. Multiplying Rational Expressions
6.19.5. Dividing Rational Expressions
6.19.6. Expressing Rational Functions as Sums of Partial Fractions
6.20. Rational Equations
6.20.1. Rational Equations With Three Terms
6.20.2. Advanced Rational Equations
6.21. Rational Functions
6.21.1. Graphing Reciprocal Functions
6.21.2. Graph Transformations of Reciprocal Functions
6.21.3. Combining Graph Transformations of Reciprocal Functions
6.21.4. Domain and Range of Transformed Reciprocal Functions
6.21.5. Finding Intersections of Lines and Reciprocal Functions
6.21.6. Finding Roots of Rational Functions
6.21.7. Horizontal Asymptotes of Rational Functions
6.22. Radical Functions
6.22.1. Solving Radical Equations
6.22.2. Graphing the Square Root Function
6.22.3. Graph Transformations of Square Root Functions
6.22.4. Graphing the Cube Root Function
6.22.5. Domain, Range, and Roots of Transformed Square Root Functions
6.22.6. The Domain of a Transformed Radical Function
6.22.7. The Range of a Transformed Radical Function
6.22.8. Roots of Transformed Radical Functions
17 topics
7.23. Introduction to Sequences
7.23.1. Introduction to Sequences
7.23.2. Recursive Sequences
7.23.3. Fibonacci Sequences
7.23.4. Introduction to Sigma Notation
7.24. Arithmetic Sequences
7.24.1. Arithmetic Sequences
7.24.2. Recursive Formulas for Arithmetic Sequences
7.24.3. The Nth Term of an Arithmetic Sequence
7.24.4. Translating Between Explicit and Recursive Formulas for Arithmetic Sequences
7.24.5. Finding the Common Difference of an Arithmetic Sequence Given Two Terms
7.24.6. Finding the Nth Term of an Arithmetic Sequence Given Two Terms
7.24.7. Determining Indexes of Terms in Arithmetic Sequences
7.25. Geometric Sequences
7.25.1. Introduction to Geometric Sequences
7.25.2. The Recursive Formula for a Geometric Sequence
7.25.3. Finding the Nth Term of a Geometric Sequence
7.25.4. Translating Between Explicit and Recursive Formulas for Geometric Sequences
7.25.5. Finding the Common Ratio of a Geometric Sequence Given Two Terms
7.25.6. Finding the Index of a Term in a Geometric Sequence
30 topics
8.26. Coordinate Geometry
8.26.1. Parallel Lines in the Coordinate Plane
8.26.2. Finding the Equation of a Parallel Line
8.26.3. Perpendicular Lines in the Coordinate Plane
8.26.4. Finding Equations of Perpendicular Lines
8.26.5. Midpoints in the Coordinate Plane
8.26.6. The Distance Formula
8.26.7. The Distance Formula in Three Dimensions
8.27. Geometric Transformations
8.27.1. Translations of Geometric Figures
8.27.2. Rotations of Geometric Figures
8.27.3. Rotating Objects in the Coordinate Plane Using Functions
8.27.4. Reflections of Geometric Figures in the Cartesian Plane
8.27.5. Reflections of Figures Across Arbitrary Lines
8.27.6. Dilations of Geometric Figures
8.27.7. Dilations of Figures in the Coordinate Plane
8.27.8. Stretches of Geometric Figures
8.27.9. Combining Stretches of Geometric Figures
8.27.10. Combining Geometric Transformations
8.27.11. Rigid Motions and Congruence
8.27.12. Reflective Symmetry
8.27.13. Rotational Symmetry
8.28. Solid Geometry
8.28.1. Identifying Three-Dimensional Shapes
8.28.2. Faces, Vertices, and Edges of Polyhedrons
8.28.3. Nets of Polyhedrons
8.28.4. Volumes of Cubes
8.28.5. Volumes of Rectangular Solids
8.28.6. Surface Areas of Cubes
8.28.7. Volumes of Spheres
8.28.8. Surface Areas of Spheres
8.28.9. Euler's Formula for Polyhedra
8.28.10. The Five Platonic Solids
49 topics
9.29. Introduction to Trigonometry
9.29.1. Angles and Sides in Right Triangles
9.29.2. The Trigonometric Ratios
9.29.3. Calculating Trigonometric Ratios Using the Pythagorean Theorem
9.29.4. Calculating Side Lengths of Right Triangles Using Trigonometry
9.29.5. Calculating Angles in Right Triangles Using Trigonometry
9.29.6. The Reciprocal Trigonometric Ratios
9.29.7. Special Trigonometric Ratios
9.29.8. Calculating the Area of a Right Triangle Using Trigonometry
9.29.9. Introduction to Radians
9.29.10. Trigonometric Ratios With Radians
9.30. Trigonometry with General Triangles
9.30.1. The Law of Sines
9.30.2. The Law of Cosines
9.30.3. The Area of a General Triangle
9.31. The Unit Circle
9.31.1. Angles in the Coordinate Plane
9.31.2. Negative Angles in the Coordinate Plane
9.31.3. Coterminal Angles
9.31.4. Calculating Reference Angles
9.31.5. Properties of the Unit Circle in the First Quadrant
9.31.6. Extending the Trigonometric Ratios Using the Unit Circle
9.31.7. Expressing Negative Angles in Terms of Reference Angles
9.31.8. Expressing Large Angles in Terms of Reference Angles
9.31.9. Using the Pythagorean Identity in the First Quadrant
9.31.10. Extending the Pythagorean Identity to All Quadrants
9.32. Special Trigonometric Ratios
9.32.1. Finding Trigonometric Ratios of Quadrantal Angles
9.32.2. Trigonometric Ratios of Quadrantal Angles Outside the Standard Range
9.32.3. Finding Trigonometric Ratios of Special Angles Using the Unit Circle
9.32.4. Evaluating Trigonometric Expressions
9.32.5. Further Extensions of the Special Trigonometric Ratios
9.33. Graphing Trigonometric Functions
9.33.1. Graphing Sine and Cosine
9.33.2. Graphing Tangent and Cotangent
9.33.3. Graphing Secant and Cosecant
9.34. Properties of Trigonometric Functions
9.34.1. Describing Properties of the Sine Function
9.34.2. Describing Properties of the Cosine Function
9.34.3. Describing Properties of the Tangent Function
9.34.4. Describing Properties of the Secant Function
9.34.5. Describing Properties of the Cosecant Function
9.34.6. Describing Properties of the Cotangent Function
9.35. Graph Transformations of Trigonometric Functions
9.35.1. Vertical Translations of Trigonometric Functions
9.35.2. Vertical Stretches of Trigonometric Functions
9.35.3. Horizontal Translations of Trigonometric Functions
9.35.4. Horizontal Stretches of Trigonometric Functions
9.35.5. Combining Graph Transformations of Sine and Cosine
9.35.6. Graph Transformations of Tangent and Cotangent
9.35.7. Combining Graph Transformations of Tangent and Cotangent
9.35.8. Combining Graph Transformations of Secant and Cosecant
9.35.9. Graphing Reflections of Trigonometric Functions
9.36. Properties of Transformed Trigonometric Functions
9.36.1. Properties of Transformed Sine and Cosine Functions
9.36.2. Properties of Transformed Tangent and Cotangent Functions
9.36.3. Properties of Transformed Secant and Cosecant Functions
11 topics
10.37. Introduction to Vectors
10.37.1. Representing Given Information as a Vector
10.37.2. The Triangle Law for the Addition and Subtraction of Two Vectors
10.37.3. Calculating the Magnitude of a Vector From Given Information
10.37.4. Problem Solving Using Vector Diagrams
10.37.5. Parallel Vectors
10.37.6. Unit Vectors
10.37.7. Linear Combinations of Vectors and Their Properties
10.37.8. Describing the Position Vector of a Point Using Known Vectors
10.38. Vectors in 2D Cartesian Coordinates
10.38.1. Two-Dimensional Vectors Expressed in Component Form
10.38.2. Addition and Scalar Multiplication of Cartesian Vectors in 2D
10.38.3. Calculating the Magnitude of Cartesian Vectors in 2D
Limits & Continuity
21 topics
11.39. Estimating Limits from Graphs
11.39.1. The Finite Limit of a Function
11.39.2. The Left and Right-Sided Limits of a Function
11.39.3. Finding the Existence of a Limit Using One-Sided Limits
11.39.4. Limits at Infinity from Graphs
11.39.5. Infinite Limits from Graphs
11.40. The Algebra of Limits
11.40.1. Limits of Power Functions, and the Constant Rule for Limits
11.40.2. The Sum Rule for Limits
11.40.3. The Product and Quotient Rules for Limits
11.40.4. The Power and Root Rules for Limits
11.41. Limits of Functions
11.41.1. Limits at Infinity of Polynomials
11.41.2. Limits of Reciprocal Functions
11.41.3. Limits of Exponential Functions
11.41.4. Limits of Logarithmic Functions
11.41.5. Limits of Radical Functions
11.41.6. Limits of Trigonometric Functions
11.41.7. Limits of Reciprocal Trigonometric Functions
11.41.8. Calculating Limits of Rational Functions by Factoring
11.42. Continuity
11.42.1. Determining Continuity from Graphs
11.42.2. Defining Continuity at a Point
11.42.3. Left and Right Continuity
11.42.4. Continuity Over an Interval
Introduction to Calculus
28 topics
12.43. Introduction to Differentiation
12.43.1. The Average Rate of Change of a Function
12.43.2. The Average Rate of Change of a Function over a Varying Interval
12.43.3. The Instantaneous Rate of Change of a Function at a Point
12.43.4. Defining the Derivative Using Derivative Notation
12.43.5. The Power Rule for Differentiation
12.43.6. The Sum and Constant Multiple Rules for Differentiation
12.43.7. Calculating the Slope of a Tangent Line Using Differentiation
12.43.8. Calculating the Equation of a Tangent Line Using Differentiation
12.43.9. Calculating the Equation of a Normal Line Using Differentiation
12.43.10. Interpreting the Meaning of the Derivative in Context
12.44. Derivatives of Functions and the Rules of Differentiation
12.44.1. Differentiating Exponential Functions
12.44.2. Differentiating Logarithmic Functions
12.44.3. Differentiating Trigonometric Functions
12.44.4. Second and Higher Order Derivatives
12.44.5. The Product Rule for Differentiation
12.44.6. The Quotient Rule for Differentiation
12.44.7. Differentiating Reciprocal Trigonometric Functions
12.45. Differentiating Composite Functions
12.45.1. The Chain Rule for Differentiation
12.45.2. The Chain Rule With Exponential Functions
12.45.3. The Chain Rule With Logarithmic Functions
12.45.4. The Chain Rule With Trigonometric Functions
12.45.5. Selecting Procedures for Calculating Derivatives
12.46. Indefinite Integrals
12.46.1. The Antiderivative
12.46.2. The Constant Multiple Rule for Indefinite Integrals
12.46.3. The Sum Rule for Indefinite Integrals
12.46.4. Integrating the Reciprocal Function
12.46.5. Integrating Exponential Functions
12.46.6. Integrating Trigonometric Functions
12 topics
13.47. Analyzing Data
13.47.1. The Mean of a Data Set
13.47.2. Variance and Standard Deviation of Population Data
13.47.3. Covariance of Population Data
13.47.4. The Z-Score
13.48. Correlation and Regression
13.48.1. Scatter Plots
13.48.2. Trend Lines
13.48.3. Making Predictions Using Trend Lines
13.48.4. Linear Correlation
13.48.5. The Linear Correlation Coefficient
13.48.6. Linear Regression
13.48.7. Residuals and Residual Plots
13.48.8. Selecting a Regression Model
Probability & Combinatorics
18 topics
14.49. Introduction to Probability
14.49.1. Probability From Experimental Data
14.49.2. Sample Spaces and Events in Probability
14.49.3. Single Events in Probability
14.49.4. The Complement of an Event
14.49.5. Introduction to Sets
14.49.6. The Union of Sets
14.49.7. The Intersection of Sets
14.49.8. Venn Diagrams in Probability
14.50. Compound Events in Probability
14.50.1. Compound Events in Probability From Experimental Data
14.50.2. Computing Probabilities for Compound Events Using Venn Diagrams
14.50.3. The Addition Law of Probability
14.50.4. Mutually Exclusive Events
14.51. Combinatorics
14.51.1. The Rule of Sum and the Rule of Product
14.51.2. Factorials
14.51.3. Factorials in Variable Expressions
14.51.4. Ordering Objects
14.51.5. Permutations
14.51.6. Combinations