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 |
| 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 |
| 2.4.1. | 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 |
| 2.5.1. | 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. | 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. | 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 |
| 8.27.1. | 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 |
| 8.28.1. | 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 |
| 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. | Properties of Transformed Tangent and Cotangent Functions | |
| 9.36.3. | 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. | 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 |
| 10.38.1. | 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. | 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. | Covariance | |
| 13.47.4. | 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 | |
| 14.50.6. | The Addition Law of Probability | |
| 14.50.7. | Applying the Addition Law With Event Complements | |
| 14.50.8. | Mutually Exclusive Events |
| 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 |
