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# Precalculus

Learn advanced trigonometry and core concepts in probability and statistics. Encounter objects from higher math including complex numbers, vectors, and matrices.

## Content

### Rational Functions

• Sketch graphs of rational functions by identifying asymptotes.
• Leverage factoring and sign tables as a strategy to solve equations and inequalities involving polynomials and rational functions.

• Apply trigonometric identities to simplify trigonometric expressions.
• Solve trigonometric equations by leveraging properties of trigonometric functions and extending previous knowledge of linear and quadratic equations.

### Conic Sections

• Draw connections between algebraic and geometric properties of ellipses and hyperbolas.

### Parametric and Polar Curves

• Extend prior knowledge of rectangular curves to parametric and polar curves.

### Complex Numbers

• Manipulate complex numbers algebraically, visualize them geometrically in the complex plane, and connect the algebraic and geometric interpretations through Euler’s formula.

### Vectors and Matrices

• Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
• Interpret matrices as linear transformations of points in the coordinate plane.

### Probability

• Define probability using the formal language of sets.
• Apply combinatorial techniques to compute probabilities in real-world modeling contexts.
• Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
• Define probability mass functions and apply binomial and geometric distributions in modeling contexts.

### Statistics

• Understand the utility of statistical sampling and be able to identify sources of bias.
• Compute statistical measures of the center and spread of a data set.
• Fit a linear regression to a data set and interpret the coefficients.
• Construct an appropriate regression model for a data set and use it to make predictions.
1.
Sequences and Series
18 topics
1.1. Arithmetic Series
 1.1.1. Introduction to Sigma Notation 1.1.2. Properties of Finite Series 1.1.3. Expressing an Arithmetic Series in Sigma Notation 1.1.4. Finding the Sum of an Arithmetic Series 1.1.5. Finding the First Term of an Arithmetic Series 1.1.6. Calculating the Number of Terms in an Arithmetic Series 1.1.7. Modeling With Arithmetic Series
1.2. Finite Geometric Series
 1.2.1. The Sum of a Finite Geometric Series 1.2.2. The Sum of the First N Terms of a Geometric Series 1.2.3. Writing Geometric Series in Sigma Notation 1.2.4. Finding the Sum of a Geometric Series Given in Sigma Notation 1.2.5. Solving Geometric Series Problems Using Exponential Equations and Inequalities 1.2.6. Modeling With Geometric Series 1.2.7. Modeling Financial Problems Using Geometric Series
1.3. The Binomial Theorem
 1.3.1. Pascal's Triangle and the Binomial Coefficients 1.3.2. Expanding a Binomial Using Binomial Coefficients 1.3.3. The Special Case of the Binomial Theorem 1.3.4. Approximating Values Using the Binomial Theorem
2.
Rational Functions
16 topics
2.4. Rational Expressions and Equations
 2.4.1. Closure Properties of Polynomials 2.4.2. Closure Properties of Rational Expressions 2.4.3. Rational Equations With Three Terms 2.4.4. Advanced Rational Equations 2.4.5. Further Advanced Rational Equations
2.5. Rational Functions
 2.5.1. Finding Roots of Rational Functions 2.5.2. Vertical Asymptotes of Rational Functions 2.5.3. Locating Holes in Rational Functions 2.5.4. Horizontal Asymptotes of Rational Functions 2.5.5. End Behavior of Rational Functions 2.5.6. Infinite Limits of Rational Functions 2.5.7. Infinite Limits of Rational Functions: Advanced Cases 2.5.8. The Domain and Range of a Rational Function 2.5.9. Identifying a Rational Function From a Graph 2.5.10. Identifying a Rational Function From a Graph Containing Holes 2.5.11. Identifying the Graph of a Rational Function
3.
Inequalities
18 topics
 3.6.1. Solving Elementary Quadratic Inequalities 3.6.2. Solving Quadratic Inequalities From Graphs 3.6.3. Solving Quadratic Inequalities Using the Graphical Method 3.6.4. Solving Quadratic Inequalities Using the Sign Table Method 3.6.5. Solving Discriminant Problems Using Quadratic Inequalities
3.7. Polynomial Inequalities
 3.7.1. Inequalities Involving Powers of Binomials 3.7.2. Solving Polynomial Inequalities Using a Graphical Method 3.7.3. Solving Polynomial Inequalities Using Special Factoring Techniques and the Graphical Method 3.7.4. Solving Polynomial Inequalities Using the Sign Table Method
3.8. Rational Inequalities
 3.8.1. Solving Rational Inequalities 3.8.2. Further Solving of Rational Inequalities
3.9. Non-Polynomial Inequalities
 3.9.1. Solving Inequalities Involving Radical Functions 3.9.2. Solving Inequalities Involving Exponential Functions 3.9.3. Solving Inequalities Involving Logarithmic Functions 3.9.4. Solving Inequalities Involving Exponential Functions and Polynomials 3.9.5. Solving Inequalities Involving Positive and Negative Factors 3.9.6. Solving Two-Variable Nonlinear Inequalities 3.9.7. Further Solving of Two-Variable Nonlinear Inequalities
4.
Trigonometry
42 topics
4.10. The Inverse Trigonometric Functions
 4.10.1. Graphing the Inverse Sine Function 4.10.2. Graphing the Inverse Cosine Function 4.10.3. Graphing the Inverse Tangent Function 4.10.4. Evaluating Expressions Containing Inverse Trigonometric Functions 4.10.5. Further Evaluating Expressions Containing Inverse Trigonometric Functions
4.11. Trigonometric Identities
 4.11.1. Simplifying Expressions Using Basic Trigonometric Identities 4.11.2. Simplifying Expressions Using the Pythagorean Identity 4.11.3. Alternate Forms of the Pythagorean Identity 4.11.4. Simplifying Expressions Using the Secant-Tangent Identity 4.11.5. Alternate Forms of the Secant-Tangent Identity 4.11.6. Simplifying Trigonometric Expressions Using the Cotangent-Cosecant Identity 4.11.7. Simplifying Trigonometric Expressions Using Cofunction Identities
4.12. The Sum and Difference Formulas
 4.12.1. The Sum and Difference Formulas for Sine 4.12.2. The Sum and Difference Formulas for Cosine 4.12.3. The Sum and Difference Formulas for Tangent 4.12.4. Calculating Trigonometric Ratios Using the Sum Formula for Sine 4.12.5. Calculating Trigonometric Ratios Using the Sum Formula for Cosine 4.12.6. Calculating Trigonometric Ratios Using the Sum Formula for Tangent 4.12.7. Writing Sums of Trigonometric Functions in Amplitude-Phase Form
4.13. The Double-Angle Formulas
 4.13.1. The Double-Angle Formula for Sine 4.13.2. Verifying Trigonometric Identities Using the Double-Angle Formula for Sine 4.13.3. Using the Double-Angle Formula for Sine With the Pythagorean Theorem 4.13.4. The Double-Angle Formula for Cosine 4.13.5. Verifying Trigonometric Identities Using the Double-Angle Formulas for Cosine 4.13.6. Finding Exact Values of Trigonometric Expressions Using the Double-Angle Formulas for Cosine 4.13.7. Simplifying Expressions Using the Double-Angle Formula for Tangent 4.13.8. Verifying Trigonometric Identities Using the Double-Angle Formula for Tangent
4.14. Elementary Trigonometric Equations
 4.14.1. Elementary Trigonometric Equations Containing Sine 4.14.2. Elementary Trigonometric Equations Containing Cosine 4.14.3. Elementary Trigonometric Equations Containing Tangent 4.14.4. Elementary Trigonometric Equations Containing Secant 4.14.5. Elementary Trigonometric Equations Containing Cosecant 4.14.6. Elementary Trigonometric Equations Containing Cotangent 4.14.7. General Solutions of Elementary Trigonometric Equations
4.15. Trigonometric Equations Containing Transformed Functions
 4.15.1. General Solutions of Trigonometric Equations With Transformed Functions 4.15.2. Trigonometric Equations Containing Transformed Sine Functions 4.15.3. Trigonometric Equations Containing Transformed Cosine Functions 4.15.4. Trigonometric Equations Containing Transformed Tangent Functions
 4.16.1. Solving Trigonometric Equations Using the Sin-Cos-Tan Identity 4.16.2. Solving Trigonometric Equations Using the Zero-Product Property 4.16.3. Quadratic Trigonometric Equations Containing Sine or Cosine 4.16.4. Quadratic Trigonometric Equations Containing Tangent or Cotangent
5.
Vectors
29 topics
5.17. Introduction to Vectors
 5.17.1. Representing Given Information as a Vector 5.17.2. The Triangle Law for the Addition and Subtraction of Two Vectors 5.17.3. Calculating the Magnitude of a Vector From Given Information 5.17.4. Problem Solving Using Vector Diagrams 5.17.5. Parallel Vectors 5.17.6. Unit Vectors 5.17.7. Linear Combinations of Vectors and Their Properties 5.17.8. Describing the Position Vector of a Point Using Known Vectors
5.18. Vectors in 2D Cartesian Coordinates
 5.18.1. Two-Dimensional Vectors Expressed in Component Form 5.18.2. Addition and Scalar Multiplication of Cartesian Vectors in 2D 5.18.3. Calculating the Magnitude of Cartesian Vectors in 2D 5.18.4. Calculating the Direction of Cartesian Vectors in 2D 5.18.5. Calculating the Components of Cartesian Vectors in 2D 5.18.6. Velocity and Acceleration for Plane Motion 5.18.7. Calculating Displacement for Plane Motion
5.19. Vectors in 3D Cartesian Coordinates
 5.19.1. Three-Dimensional Vectors in Component Form 5.19.2. Addition and Scalar Multiplication of Cartesian Vectors in 3D 5.19.3. Calculating the Magnitude of Cartesian Vectors in 3D
5.20. The Dot Product
 5.20.1. Calculating the Dot Product Using Angle and Magnitude 5.20.2. Calculating the Dot Product Using Components 5.20.3. The Angle Between Two Vectors 5.20.4. Calculating a Scalar Projection 5.20.5. Calculating a Vector Projection
5.21. The Cross Product
 5.21.1. Calculating the Cross Product of Two Vectors Using the Definition 5.21.2. Calculating the Cross Product Using Determinants 5.21.3. Finding Areas Using the Cross Product 5.21.4. The Scalar Triple Product 5.21.5. Volumes of Parallelepipeds 5.21.6. Finding Volumes of Tetrahedrons and Pyramids Using Vector Products
6.
Matrices
37 topics
6.22. Introduction to Matrices
 6.22.1. Introduction to Matrices 6.22.2. Index Notation for Matrices 6.22.3. Adding and Subtracting Matrices 6.22.4. Properties of Matrix Addition 6.22.5. Scalar Multiplication of Matrices 6.22.6. Zero, Square, Diagonal and Identity Matrices 6.22.7. The Transpose of a Matrix
6.23. Matrix Multiplication
 6.23.1. Multiplying a Matrix by a Column Vector 6.23.2. Multiplying Square Matrices 6.23.3. Conformability for Matrix Multiplication 6.23.4. Multiplying Matrices 6.23.5. Powers of Matrices 6.23.6. Multiplying a Matrix by the Identity Matrix 6.23.7. Properties of Matrix Multiplication 6.23.8. Representing 2x2 Systems of Equations Using a Matrix Product 6.23.9. Representing 3x3 Systems of Equations Using a Matrix Product
6.24. Determinants
 6.24.1. The Determinant of a 2x2 Matrix 6.24.2. The Geometric Interpretation of the 2x2 Determinant 6.24.3. The Minors of a 3x3 Matrix 6.24.4. The Determinant of a 3x3 Matrix
6.25. The Inverse of a Matrix
 6.25.1. Introduction to the Inverse of a Matrix 6.25.2. Inverses of 2x2 Matrices 6.25.3. Calculating the Inverse of a 3x3 Matrix Using the Cofactor Method 6.25.4. Solving 2x2 Systems of Equations Using Inverse Matrices 6.25.5. Solving 3x3 Systems of Equations Using Inverse Matrices
6.26. Linear Transformations
 6.26.1. Introduction to Linear Transformations 6.26.2. The Standard Matrix of a Linear Transformation 6.26.3. Linear Transformations of Points and Lines in the Plane 6.26.4. Linear Transformations of Objects in the Plane 6.26.5. Dilations and Reflections as Linear Transformations 6.26.6. Shear and Stretch as Linear Transformations 6.26.7. Right-Angle Rotations as Linear Transformations 6.26.8. Rotations as Linear Transformations 6.26.9. Combining Linear Transformations Using 2x2 Matrices 6.26.10. Inverting Linear Transformations 6.26.11. Area Scale Factors of Linear Transformations 6.26.12. Singular Linear Transformations in the Plane
7.
Conic Sections
20 topics
7.27. Ellipses as Conic Sections
 7.27.1. Introduction to Ellipses 7.27.2. Equations of Ellipses Centered at the Origin 7.27.3. Equations of Ellipses Centered at a General Point 7.27.4. Finding the Center and Axes of Ellipses by Completing the Square 7.27.5. Finding Intercepts of Ellipses 7.27.6. Finding Intersections of Ellipses and Lines 7.27.7. Foci of Ellipses 7.27.8. Vertices and Eccentricity of Ellipses 7.27.9. Directrices of Ellipses 7.27.10. The Area of an Ellipse
7.28. Hyperbolas as Conic Sections
 7.28.1. Equations of Hyperbolas Centered at the Origin 7.28.2. Equations of Hyperbolas Centered at a General Point 7.28.3. Asymptotes of Hyperbolas Centered at the Origin 7.28.4. Asymptotes of Hyperbolas Centered at a General Point 7.28.5. Finding Intercepts and Intersections of Hyperbolas 7.28.6. Transverse Axes of Hyperbolas 7.28.7. Conjugate Axes of Hyperbolas 7.28.8. Foci of Hyperbolas 7.28.9. Eccentricity and Vertices of Hyperbolas 7.28.10. Directrices of Hyperbolas
8.
Parametric Equations
7 topics
8.29. Parametric Equations
 8.29.1. Graphing Curves Defined Parametrically 8.29.2. Cartesian Equations of Parametric Curves 8.29.3. Finding Intercepts of Curves Defined Parametrically 8.29.4. Finding Intersections of Parametric Curves and Lines 8.29.5. Parametric Equations of Circles 8.29.6. Parametric Equations of Ellipses 8.29.7. Parametric Equations of Parabolas
9.
Polar Equations
6 topics
9.30. Polar Coordinates
 9.30.1. Introduction to Polar Coordinates 9.30.2. Converting from Polar Coordinates to Cartesian Coordinates 9.30.3. Polar Equations of Circles Centered at the Origin 9.30.4. Polar Equations of Radial Lines 9.30.5. Polar Equations of Circles Centered on the Coordinate Axes 9.30.6. Finding Intersections of Polar Curves
10.
Complex Numbers
28 topics
10.31. The Complex Plane
 10.31.1. The Complex Plane 10.31.2. The Magnitude of a Complex Number 10.31.3. The Argument of a Complex Number 10.31.4. Arithmetic in the Complex Plane 10.31.5. Geometry in the Complex Plane
10.32. Further Complex Numbers
 10.32.1. The Complex Conjugate 10.32.2. Special Properties of the Complex Conjugate 10.32.3. The Complex Conjugate and the Roots of a Quadratic Equation 10.32.4. Dividing Complex Numbers 10.32.5. Solving Equations by Equating Real and Imaginary Parts 10.32.6. Extending Polynomial Identities to the Complex Numbers
10.33. Complex Numbers in Polar Form
 10.33.1. The Polar Form of a Complex Number 10.33.2. Products of Complex Numbers Expressed in Polar Form 10.33.3. Quotients of Complex Numbers Expressed in Polar Form 10.33.4. The CIS Notation
10.34. De Moivre's Theorem
 10.34.1. De Moivre's Theorem 10.34.2. Finding Powers of Complex Numbers Using De Moivre's Theorem 10.34.3. The Power-Reducing Formulas for Sine and Cosine 10.34.4. Euler's Formula 10.34.5. Roots of Unity 10.34.6. Properties of Roots of Unity 10.34.7. Square Roots of Complex Numbers 10.34.8. Higher Roots of Complex Numbers
10.35. The Fundamental Theorem of Algebra
 10.35.1. The Fundamental Theorem of Algebra for Quadratic Equations 10.35.2. The Fundamental Theorem of Algebra with Cubic Equations 10.35.3. Solving Cubic Equations With Complex Roots 10.35.4. The Fundamental Theorem of Algebra with Quartic Equations 10.35.5. Solving Quartic Equations With Complex Roots
11.
Probability & Combinatorics
42 topics
11.36. Introduction to Probability
 11.36.1. Probability From Experimental Data 11.36.2. Sample Spaces and Events in Probability 11.36.3. Single Events in Probability 11.36.4. The Complement of an Event 11.36.5. Introduction to Sets 11.36.6. The Union of Sets 11.36.7. The Intersection of Sets 11.36.8. Venn Diagrams in Probability 11.36.9. Geometric Probability
11.37. Compound Events in Probability
 11.37.1. Compound Events in Probability From Experimental Data 11.37.2. Computing Probabilities for Compound Events Using Venn Diagrams 11.37.3. Computing Probabilities for Three Events Using Venn Diagrams 11.37.4. The Addition Law of Probability 11.37.5. Mutually Exclusive Events
11.38. Conditional Probability
 11.38.1. Conditional Probabilities From Venn Diagrams 11.38.2. Conditional Probabilities From Tables 11.38.3. The Multiplication Law for Conditional Probability 11.38.4. The Law of Total Probability 11.38.5. Tree Diagrams for Dependent Events 11.38.6. Tree Diagrams for Dependent Events: Applications 11.38.7. Independent Events 11.38.8. Tree Diagrams for Independent Events
11.39. Discrete Random Variables
 11.39.1. Probability Mass Functions of Discrete Random Variables 11.39.2. Cumulative Distribution Functions for Discrete Random Variables 11.39.3. Expected Values of Discrete Random Variables 11.39.4. The Binomial Distribution 11.39.5. Modeling With the Binomial Distribution 11.39.6. The Geometric Distribution 11.39.7. Modeling With the Geometric Distribution
11.40. The Normal Distribution
 11.40.1. The Standard Normal Distribution 11.40.2. The Normal Distribution 11.40.3. Mean and Variance of the Normal Distribution 11.40.4. Percentage Points of the Standard Normal Distribution 11.40.5. Modeling With the Normal Distribution 11.40.6. The Empirical Rule for the Normal Distribution
11.41. Combinatorics
 11.41.1. The Rule of Sum and the Rule of Product 11.41.2. Factorials 11.41.3. Factorials in Variable Expressions 11.41.4. Ordering Objects 11.41.5. Permutations 11.41.6. Combinations 11.41.7. Computing Probabilities Using Combinatorics
12.
Statistics
18 topics
12.42. Analyzing Data
 12.42.1. Sampling 12.42.2. The Mean of a Data Set 12.42.3. Variance and Standard Deviation 12.42.4. Covariance 12.42.5. The Z-Score
12.43. Correlation
 12.43.1. Scatter Plots 12.43.2. Trend Lines 12.43.3. Making Predictions Using Trend Lines 12.43.4. Interpreting Coefficients of Trend Lines 12.43.5. Linear Correlation 12.43.6. Residuals and Residual Plots 12.43.7. The Linear Correlation Coefficient 12.43.8. Correlation vs. Causation
12.44. Regression
 12.44.1. Selecting a Regression Model 12.44.2. Linear Regression 12.44.3. Quadratic Regression 12.44.4. Semi-Log Scatter Plots 12.44.5. Exponential Regression