Mathematical Foundations II is the second course in a sequence specially crafted for adults seeking the most direct path to prepare for university math courses. Building on the basic algebra covered in Mathematical Foundations I, students in Mathematical Foundations II generalize their understanding of algebra to more advanced functions (including trigonometry) and become acquainted with limits and derivatives in calculus. Upon completing the course, students will be prepared for Mathematical Foundations III, the final stepping stone in preparing for university-level math.
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.
| 1.1.1. | Introduction to Quadratic Equations | |
| 1.1.2. | Solving Perfect Square Quadratic Equations | |
| 1.1.3. | Perfect Square Quadratic Equations with One 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 |
| 1.2.1. | 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. | The Axis of Symmetry of a Parabola | |
| 1.2.7. | The Average of the Roots Formula | |
| 1.2.8. | The Vertex Form of a Parabola | |
| 1.2.9. | Writing the Equation of a Parabola in Vertex Form | |
| 1.2.10. | Domain and Range of Quadratic Functions | |
| 1.2.11. | Finding Intersections of Lines and Quadratic Functions | |
| 1.2.12. | Inverses of Quadratic Functions | |
| 1.2.13. | Modeling Downwards Vertical Motion |
| 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. | Solving Polynomial Equations Using the GCF | |
| 2.3.6. | Solving Cubic Equations by Grouping | |
| 2.3.7. | Graphing Elementary Cubic Functions | |
| 2.3.8. | End Behavior of Polynomials |
| 2.4.1. | Factoring Polynomials Using the GCF | |
| 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 |
| 2.5.1. | 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 |
| 2.6.1. | 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. | Factoring Quartic Polynomials Using the Factor Theorem | |
| 2.6.5. | Multiplicities of the Roots of Polynomials | |
| 2.6.6. | The Remainder Theorem | |
| 2.6.7. | The Rational Roots Theorem |
| 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 |
| 3.8.1. | The Real Number System | |
| 3.8.2. | Sums and Products of Rational and Irrational Numbers |
| 3.9.1. | Imaginary Numbers | |
| 3.9.2. | 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 |
| 3.10.1. | The Complex Plane | |
| 3.10.2. | The Magnitude of a Complex Number | |
| 3.10.3. | The Argument of a Complex Number |
| 4.11.1. | The Arithmetic of Functions | |
| 4.11.2. | Function Composition | |
| 4.11.3. | Describing Function Composition | |
| 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. | Domain and Range of Inverse Functions | |
| 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 |
| 4.12.1. | 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 | |
| 4.12.11. | Finding Points on Transformed Curves | |
| 4.12.12. | The Domain and Range of Transformed Functions |
| 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 |
| 5.14.1. | 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 |
| 5.15.1. | 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 |
| 5.16.1. | 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 | |
| 5.16.5. | Solving Logarithmic Equations With Logarithms on Both Sides |
| 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 | |
| 5.17.5. | Inverses of Exponential and 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 |
| 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 |
| 6.20.1. | Rational Equations With Three Terms | |
| 6.20.2. | Advanced Rational Equations |
| 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. | Inverses of Reciprocal Functions | |
| 6.21.6. | Finding Intersections of Lines and Reciprocal Functions | |
| 6.21.7. | Roots of Rational Functions | |
| 6.21.8. | Vertical Asymptotes of Rational Functions | |
| 6.21.9. | Horizontal Asymptotes of Rational Functions |
| 6.22.1. | Solving Radical Equations | |
| 6.22.2. | Inverses of Radical Functions | |
| 6.22.3. | Graphing the Square Root Function | |
| 6.22.4. | Graph Transformations of Square Root Functions | |
| 6.22.5. | Graphing the Cube Root Function | |
| 6.22.6. | Properties of Transformed Square Root Functions | |
| 6.22.7. | The Domain of a Transformed Radical Function | |
| 6.22.8. | The Range of a Transformed Radical Function | |
| 6.22.9. | Roots of Transformed Radical Functions |
| 7.23.1. | Introduction to Sequences | |
| 7.23.2. | Recursive Sequences | |
| 7.23.3. | Fibonacci Sequences | |
| 7.23.4. | Sigma Notation | |
| 7.23.5. | Properties of Finite Series |
| 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 | |
| 7.24.6. | Finding the Nth Term of an Arithmetic Sequence Given Two Terms | |
| 7.24.7. | Determining Indexes of Terms in Arithmetic Sequences |
| 7.25.1. | Introduction to Geometric Sequences | |
| 7.25.2. | The Recursive Formula for a Geometric Sequence | |
| 7.25.3. | 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. | Determining Indexes of Terms in Geometric Sequences |
| 8.26.1. | Finding the Equation of a Parallel Line | |
| 8.26.2. | Perpendicular Lines in the Coordinate Plane | |
| 8.26.3. | Finding Equations of Perpendicular Lines | |
| 8.26.4. | Midpoints in the Coordinate Plane | |
| 8.26.5. | The Distance Formula | |
| 8.26.6. | The Shortest Distance Between a Point and a Line | |
| 8.26.7. | The Distance Formula in Three Dimensions |
| 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. | Similarity and Similar Polygons | |
| 8.27.13. | Similarity Transformations | |
| 8.27.14. | Reflective Symmetry | |
| 8.27.15. | Rotational Symmetry |
| 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. | Finding Surface Areas Using Nets | |
| 8.28.5. | Volumes of Cubes | |
| 8.28.6. | Volumes of Rectangular Solids | |
| 8.28.7. | Surface Areas of Cubes | |
| 8.28.8. | Volumes of Spheres | |
| 8.28.9. | Surface Areas of Spheres | |
| 8.28.10. | Volumes of Cylinders | |
| 8.28.11. | Surface Areas of Cylinders | |
| 8.28.12. | Euler's Formula for Polyhedra | |
| 8.28.13. | The Five Platonic Solids |
| 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 |
| 9.30.1. | The Law of Sines | |
| 9.30.2. | The Law of Cosines | |
| 9.30.3. | The Area of a General Triangle |
| 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. | Extending the Trigonometric Ratios Using Angles in Radians | |
| 9.31.8. | Extending the Trigonometric Ratios to Negative Angles | |
| 9.31.9. | Extending the Trigonometric Ratios to Large Angles | |
| 9.31.10. | Using the Pythagorean Identity in the First Quadrant | |
| 9.31.11. | Extending the Pythagorean Identity to All Quadrants |
| 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 |
| 9.33.1. | Graphing Sine and Cosine | |
| 9.33.2. | Graphing Tangent and Cotangent | |
| 9.33.3. | Graphing Secant and Cosecant |
| 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 |
| 9.35.1. | 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 |
| 9.36.1. | Properties of Transformed Sine and Cosine Functions | |
| 9.36.2. | Finding Zeros and Extrema of Transformed Sine and Cosine Functions | |
| 9.36.3. | Properties of Transformed Tangent and Cotangent Functions | |
| 9.36.4. | Properties of Transformed Secant and Cosecant Functions |
| 10.37.1. | Introduction to Vectors | |
| 10.37.2. | The Triangle Law for the Addition of Two Vectors | |
| 10.37.3. | The Magnitude of a Vector | |
| 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 |
| 10.38.1. | 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 |
| 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 |
| 11.40.1. | 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 |
| 11.41.1. | 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. | Limits of Piecewise Functions | |
| 11.41.9. | Calculating Limits of Rational Functions by Factoring |
| 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 |
| 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 |
| 12.44.1. | 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. | Calculating Derivatives From Data and Tables | |
| 12.44.8. | Calculating Derivatives From Graphs | |
| 12.44.9. | Differentiating Reciprocal Trigonometric 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 |
| 12.46.1. | 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 |
| 13.47.1. | The Mean of a Data Set | |
| 13.47.2. | Variance and Standard Deviation | |
| 13.47.3. | Estimating Means and Variances For Grouped Data | |
| 13.47.4. | Covariance | |
| 13.47.5. | Sums of Squares | |
| 13.47.6. | The Z-Score |
| 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 |
| 14.49.1. | Sets | |
| 14.49.2. | Probability From Experimental Data | |
| 14.49.3. | Sample Spaces and Events in Probability | |
| 14.49.4. | Single Events in Probability | |
| 14.49.5. | The Complement of an Event | |
| 14.49.6. | Venn Diagrams in Probability |
| 14.50.1. | The Union of Sets | |
| 14.50.2. | The Intersection of Sets | |
| 14.50.3. | Compound Events in Probability From Experimental Data | |
| 14.50.4. | Computing Probabilities for Compound Events Using Venn Diagrams | |
| 14.50.5. | Computing Probabilities of Events Containing Complements Using Venn Diagrams |
| 14.51.1. | The Rules of Sum and 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 | |
| 14.51.7. | Computing Probabilities Using Combinatorics |
