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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.

## Content

### Logarithms

• Understand the relationship between logarithms and exponents and use this understanding to evaluate logarithms.

### Trigonometry

• Apply trigonometric functions to solve for unknown sides and angles in right triangles.
• Leverage the unit circle as a conceptual framework for evaluating special trigonometric ratios.
• Graph and describe properties of trigonometric functions by relating them to the unit circle.
• Use the law of sines and the law of cosines to solve for unknown angles and sides in general triangles.

### Nonlinear Equations and Graphing

• Leverage factoring and the quadratic formula as complementary techniques for solving quadratic equations and graphing quadratic functions.
• Extend previous knowledge of algebraic techniques to solve nonlinear equations involving polynomials, radicals, exponents, and logarithms.
• Understand how transformations affect the shape of a function's graph and use this understanding to graph transformed quadratic, absolute value, exponential, radical, logarithmic, and trigonometric functions.

### Polynomial Division and Factoring

• Understand the relationship between zeros and factors of polynomials.
• Divide polynomials using synthetic and long division.
• Understand the relationship between the value of a polynomial at a given input and the remainder obtained when dividing the polynomial by the corresponding binomial.
• Leverage the rational roots theorem as a strategy to factor polynomials.

### Graphing Polynomials and Rational Functions

• Understand how the multiplicity of a root of a polynomial relates to the shape of the graph near the root.
• Sketch graphs of polynomial functions by identifying end behavior, roots, and behavior near roots.
• Sketch graphs of rational functions by identifying asymptotes.

### Limits, Derivatives, and Integrals

• Estimate limits from graphs and compute limits using algebraic manipulation.
• Identify continuous functions from graphs and define continuity in terms of limits.
• Interpret the derivative as the instantaneous rate of change or the slope of the tangent line.
• Apply rules to compute derivatives of a variety of algebraic functions, including higher-order derivatives.
• Evaluate indefinite integrals by finding antiderivatives.

### Sequences

• Evaluate terms of a sequence and compute sums given in sigma notation.
• Determine the formula of an arithmetic or geometric sequence.

### Complex Numbers and Vectors

• Manipulate complex numbers algebraically and visualize them geometrically in the complex plane.
• Generalize prior intuitions about arithmetic to vectors.

### Probability and Statistics

• 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.
• Compute statistical measures of the center and spread of a data set.
• Fit a linear regression to a data set and use it to make predictions.
1.
27 topics
 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. Calculating 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.
Polynomials
26 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. 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. 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
3.
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
4.
Functions
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. 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. 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
5.
Exponentials & Logarithms
29 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 5.16.5. Solving Logarithmic Equations With Logarithms on Both Sides
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 5.17.5. Inverses of Exponential and Logarithmic 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
6.
25 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. 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.
Sequences
18 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. Sigma Notation 7.23.5. Properties of Finite Series
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
8.
Geometry
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
9.
Trigonometry
50 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. 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. 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
10.
Vectors
11 topics
10.37. Introduction to Vectors
 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. 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
11.
Limits & Continuity
22 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. Limits of Piecewise Functions 11.41.9. 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
12.
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
13.
Statistics
12 topics
13.47. Analyzing Data
 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. 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
14.
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