# Integrated Math II

Our comprehensive, fully accredited, Common Core-aligned Integrated Math II course builds upon the strong foundations acquired in Integrated Math I, delving deeper into algebra, functions, geometry, trigonometry, probability, and statistics. This course will further develop students' mathematical understanding and problem-solving skills, preparing them for success in our Integrated Math III course, the final stepping stone before Calculus.

## Content

In this comprehensive course, students will fully master quadratic equations and functions, achieving fluency in the various methods for solving problems that involve quadratics. They will also explore how quadratic functions can be adeptly used in modeling real-world situations. Students will also explore fundamental concepts related to complex numbers.

Building on foundations laid out in Integrated Math I, students will extend their understanding of functions to encompass function arithmetic, composition, periodicity, even and odd functions, and invertible functions. They will be carefully guided through the process of performing graph transformations of functions and will learn to compute inverse functions seamlessly.

As the course progresses, students will enrich their knowledge of exponential functions. This includes a deep dive into logarithms, examining their graphs and properties. Students will learn to solve both exponential and logarithmic equations and further develop their aptitude for modeling scenarios with exponential functions, such as calculating compound interest.

This course aims to broaden studentsâ€™ knowledge to include crucial geometric concepts such as similarity, fundamental circle theorems, conic sections, and solid geometry. Concurrently, students will lay a robust foundation in trigonometry, delving into the unit circle and its pivotal role in extending trigonometric ratios and graphing trigonometric functions.

Finally, students lay solid foundations in probability and combinatorics and build on previous understandings of correlation and regression, further solidifying their skills in these areas.

Upon successful completion of this course, students will have mastered the following:
• Develop a comprehensive understanding of the hierarchical nature of the number system and the role of complex numbers within it.
• Solve quadratic equations using various methods, including applying the zero product rule and completing the square and the quadratic formula.
• Graph and analyze quadratic functions, calculate the axis of symmetry using various methods and find intersections of lines and quadratic functions.
• Model real-world situations using quadratic functions, including vertical motion, revenue, cost, and profit functions.
• Further develop their understanding of more general polynomials, including finding the least common factor and least common multiple of two polynomials and describing numerical relationships using polynomial identities.
• Explore and develop a basic comprehension of complex numbers, including addition, subtraction, and multiplication. They will solve quadratic equations with complex roots and understand the cyclic property of the imaginary unit.
• Extend their knowledge of functions to incorporate function composition, describe one-to-one and many-to-one functions, local extrema, periodicity, even and odd functions, and vertical asymptotes of functions.
• Understand the concept of an invertible function, including how graphs of functions and their inverses are related, their domains and ranges, and compute inverses of linear, quadratic, exponential, and logarithmic functions.
• Strengthen their understanding of transformations in the plane to include graph transformations of functions, including vertical and horizontal translations, stretches, and reflections, composite transformations with multiple operations, and absolute value transformations.
• Work with absolute value functions and their associated graphs, describe their properties and further develop their understanding of absolute value equations to more complex cases involving extraneous solutions.
• Further explore exponentials and logarithms, including evaluating logarithms, applying the laws of logarithms, graphing exponential and logarithmic functions, and solving exponential and logarithmic equations.
• Model real-world situations using exponential functions, such as those involving compound interest, including continuously compounded interest.
• Understand geometric sequences, including finding a given geometric sequence's nth term and common ratio, and work with recursive formulas for geometric sequences.
• Utilize sigma notation, calculate the sum of arithmetic series, apply the binomial theorem, and explain how it relates to Pascal's Triangle.
• Solve radical equations, including cases with no real solutions.
• Understand and apply the concept of similarity, including utilizing similarity criteria for triangles, working with areas of similar polygons, and similarity transformations.
• Explore properties of circles, such as inscribed angles, Thales' theorem, and tangent lines.
• Investigate circles and parabolas as conic sections.
• Learn to identify and work with three-dimensional shapes, calculate surface areas and volumes of polyhedra and non-polyhedra, and explore Euler's formula and Platonic solids.
• Develop a strong foundation in trigonometry, including understanding and applying trigonometric ratios, working with radians, and utilizing the unit circle.
• Master graphing and analyzing trigonometric functions, including applying graph transformations and describing their resulting properties.
• Learn the basics of probability and combinatorics, including Venn diagrams, compound events, factorials, permutations, and combinations.
• Analyze data using statistical methods, such as computing a data set's mean, variance, standard deviation, z-scores, and covariance.
• Further explore correlation and regression, including calculating linear correlation coefficients, carrying out linear regression, and analyzing residuals.
1.
Number Systems
10 topics
1.1. The Number System
 1.1.1. The Real Number System 1.1.2. Writing Repeating Decimals as Fractions 1.1.3. Sums and Products of Rational and Irrational Numbers
1.2. Introduction to Complex Numbers
 1.2.1. Imaginary Numbers 1.2.2. Quadratic Equations with Purely Imaginary Solutions 1.2.3. Complex Numbers 1.2.4. Adding and Subtracting Complex Numbers 1.2.5. Multiplying Complex Numbers 1.2.6. Solving Quadratic Equations With Complex Roots 1.2.7. The Cyclic Property of the Imaginary Unit
2.
Polynomials
19 topics
2.3. Polynomials
 2.3.1. The Least Common Multiple of Two Monomials 2.3.2. The Least Common Multiple of Two Polynomials 2.3.3. Describing Numerical Relationships Using Polynomial Identities
3.
17 topics
 3.5.1. Graphing Elementary Quadratic Functions 3.5.2. Vertical Reflections of Quadratic Functions 3.5.3. Graphs of General Quadratic Functions 3.5.4. Roots of Quadratic Functions 3.5.5. The Discriminant of a Quadratic Function 3.5.6. The Axis of Symmetry of a Parabola 3.5.7. The Average of the Roots Formula 3.5.8. The Vertex Form of a Parabola 3.5.9. Writing the Equation of a Parabola in Vertex Form 3.5.10. Domain and Range of Quadratic Functions 3.5.11. Finding Intersections of Lines and Quadratic Functions
 3.6.1. Modeling Downwards Vertical Motion 3.6.2. Modeling Upwards Vertical Motion 3.6.3. Vertical Motion 3.6.4. Revenue, Cost, and Profit Functions 3.6.5. Constructing Revenue, Cost, and Profit Functions 3.6.6. Maximizing Profit and Break-Even Points
4.
Functions
29 topics
4.7. Functions
 4.7.1. The Arithmetic of Functions 4.7.2. Function Composition 4.7.3. Describing Function Composition 4.7.4. Local Extrema of Functions 4.7.5. One-To-One Functions 4.7.6. Introduction to Inverse Functions 4.7.7. Calculating the Inverse of a Function 4.7.8. Inverses of Quadratic Functions 4.7.9. Graphs of Inverse Functions 4.7.10. Domain and Range of Inverse Functions 4.7.11. Invertible Functions 4.7.12. Plotting X as a Function of Y 4.7.13. Periodic Functions 4.7.14. Even and Odd Functions 4.7.15. Unbounded Behavior of Functions Near a Point 4.7.16. The Average Rate of Change of a Function
4.8. Graph Transformations of Functions
 4.8.1. Vertical Translations of Functions 4.8.2. Horizontal Translations of Functions 4.8.3. Vertical Stretches of Functions 4.8.4. Horizontal Stretches of Functions 4.8.5. Vertical Reflections of Functions 4.8.6. Horizontal Reflections of Functions 4.8.7. Combining Graph Transformations: Two Operations 4.8.8. Combining Graph Transformations: Three or More Operations 4.8.9. Constructing Functions Using Transformations 4.8.10. Combining Reflections With Other Graph Transformations 4.8.11. Finding Points on Transformed Curves 4.8.12. The Domain and Range of Transformed Functions 4.8.13. Absolute Value Graph Transformations
5.
Absolute Value Functions
8 topics
5.9. Absolute Value Functions
 5.9.1. Absolute Value Graphs 5.9.2. Vertical Reflections of Absolute Value Graphs 5.9.3. Stretches of Absolute Value Graphs 5.9.4. Combining Transformations of Absolute Value Graphs 5.9.5. Domain and Range of Absolute Value Functions 5.9.6. Roots of Absolute Value Functions 5.9.7. Equations Connecting Absolute Value and Linear Functions 5.9.8. Absolute Value Equations With Extraneous Solutions
6.
Exponentials & Logarithms
34 topics
6.10. Introduction to Logarithms
 6.10.1. Converting From Exponential to Logarithmic Form 6.10.2. Converting From Logarithmic to Exponential Form 6.10.3. Evaluating Logarithms 6.10.4. The Natural Logarithm 6.10.5. The Common Logarithm 6.10.6. Simplifying Logarithmic Expressions
6.11. The Laws of Logarithms
 6.11.1. The Product Rule for Logarithms 6.11.2. The Quotient Rule for Logarithms 6.11.3. The Power Rule for Logarithms 6.11.4. Combining the Laws of Logarithms 6.11.5. The Change of Base Formula for Logarithms
6.12. Graphs of Exponential Functions
 6.12.1. Vertical Translations of Exponential Growth Functions 6.12.2. Vertical Translations of Exponential Decay Functions 6.12.3. Interpreting Graphs of Exponential Functions 6.12.4. Combining Graph Transformations of Exponential Functions 6.12.5. Properties of Transformed Exponential Functions
6.13. Graphs of Logarithmic Functions
 6.13.1. Graphing Logarithmic Functions 6.13.2. Combining Graph Transformations of Logarithmic Functions 6.13.3. Properties of Transformed Logarithmic Functions 6.13.4. Inverses of Exponential and Logarithmic Functions
6.14. Exponential Equations
 6.14.1. Solving Exponential Equations Using Logarithms 6.14.2. Solving Equations Containing the Exponential Function 6.14.3. Solving Exponential Equations With Different Bases 6.14.4. Solving Exponential Equations With Different Bases Using Logarithms 6.14.5. Solving Exponential Equations Using the Zero-Product Property
6.15. Logarithmic Equations
 6.15.1. Solving Logarithmic Equations 6.15.2. Solving Logarithmic Equations Containing the Natural Logarithm 6.15.3. Solving Logarithmic Equations Using the Laws of Logarithms 6.15.4. Solving Logarithmic Equations by Combining the Laws of Logarithms 6.15.5. Solving Logarithmic Equations With Logarithms on Both Sides 6.15.6. Solving Logarithmic Equations Using the Zero-Product Property
6.16. Modeling with Exponential Functions
 6.16.1. Modeling With Compound Interest 6.16.2. Continuously Compounded Interest 6.16.3. Converting Between Exponents
7.
Sequences & Series
17 topics
7.17. Geometric Sequences
 7.17.1. Introduction to Geometric Sequences 7.17.2. The Recursive Formula for a Geometric Sequence 7.17.3. The Nth Term of a Geometric Sequence 7.17.4. Translating Between Explicit and Recursive Formulas for Geometric Sequences 7.17.5. Finding the Common Ratio of a Geometric Sequence Given Two Terms 7.17.6. Determining Indexes of Terms in Geometric Sequences
7.18. Arithmetic Series
 7.18.1. Sigma Notation 7.18.2. Properties of Finite Series 7.18.3. Expressing an Arithmetic Series in Sigma Notation 7.18.4. Finding the Sum of an Arithmetic Series 7.18.5. Finding the First Term of an Arithmetic Series 7.18.6. Calculating the Number of Terms in an Arithmetic Series 7.18.7. Modeling With Arithmetic Series
7.19. The Binomial Theorem
 7.19.1. Pascal's Triangle and the Binomial Coefficients 7.19.2. Expanding a Binomial Using Binomial Coefficients 7.19.3. The Special Case of the Binomial Theorem 7.19.4. Approximating Values Using the Binomial Theorem
8.
2 topics
9.
Geometry
17 topics
9.21. Similarity
 9.21.1. Similarity and Similar Polygons 9.21.2. Side Lengths and Angle Measures of Similar Polygons 9.21.3. Areas of Similar Polygons 9.21.4. Working With Areas of Similar Polygons 9.21.5. Similarity Transformations 9.21.6. The Angle-Angle (AA) Criterion for Similar Triangles 9.21.7. The Side-Side-Side (SSS) Criterion for Similar Triangles 9.21.8. The Side-Angle-Side (SAS) Criterion for Similar Triangles 9.21.9. The Midpoint Theorem 9.21.10. The Triangle Proportionality Theorem
9.22. Circles
 9.22.1. The Inscribed Angle Theorem 9.22.2. Problem Solving Using the Inscribed Angle Theorem 9.22.3. Thales' Theorem 9.22.4. Angles in Inscribed Right Triangles 9.22.5. Inscribed Quadrilaterals 9.22.6. Tangent Lines to Circles 9.22.7. Circle Similarity
10.
Conics
15 topics
10.23. Circles as Conic Sections
 10.23.1. The Center and Radius of a Circle in the Coordinate Plane 10.23.2. Equations of Circles Centered at the Origin 10.23.3. Equations of Circles Centered at a General Point 10.23.4. Finding the Center and Radius of a Circle by Completing the Square 10.23.5. Calculating Intercepts of Circles 10.23.6. Intersections of Circles with Lines
10.24. Parabolas as Conic Sections
 10.24.1. Upward and Downward Opening Parabolas 10.24.2. Left and Right Opening Parabolas 10.24.3. The Vertex of a Parabola 10.24.4. Calculating the Vertex of a Parabola by Completing the Square 10.24.5. The Focus-Directrix Property of a Parabola 10.24.6. Calculating the Focus of a Parabola 10.24.7. Calculating the Directrix of a Parabola 10.24.8. Calculating Intercepts of Parabolas 10.24.9. Intersections of Parabolas With Lines
11.
Solid Geometry
25 topics
11.25. Introduction to Solid Geometry
 11.25.1. Identifying Three-Dimensional Shapes 11.25.2. Faces, Vertices, and Edges of Polyhedrons 11.25.3. Nets of Polyhedrons 11.25.4. Finding Surface Areas Using Nets 11.25.5. The Distance Formula in Three Dimensions 11.25.6. Euler's Formula for Polyhedra 11.25.7. The Five Platonic Solids
11.26. Rectangular Solids and Pyramids
 11.26.1. Volumes of Cubes 11.26.2. Surface Areas of Cubes 11.26.3. Face Diagonals of Cubes 11.26.4. Diagonals of Cubes 11.26.5. Volumes of Rectangular Solids 11.26.6. Surface Areas of Rectangular Solids 11.26.7. Diagonals of Rectangular Solids 11.26.8. Volumes of Pyramids 11.26.9. Surface Areas of Pyramids
11.27. Non-Polyhedrons
 11.27.1. Volumes of Cylinders 11.27.2. Surface Areas of Cylinders 11.27.3. Volumes of Right Cones 11.27.4. Slant Heights of Right Cones 11.27.5. Surface Areas of Right Cones 11.27.6. Volumes of Spheres 11.27.7. Surface Areas of Spheres 11.27.8. Conical Frustums 11.27.9. Volumes of Revolution
12.
Trigonometry
57 topics
12.28. Introduction to Trigonometry
 12.28.1. Angles and Sides in Right Triangles 12.28.2. The Trigonometric Ratios 12.28.3. Calculating Trigonometric Ratios Using the Pythagorean Theorem 12.28.4. Calculating Side Lengths of Right Triangles Using Trigonometry 12.28.5. Calculating Angles in Right Triangles Using Trigonometry 12.28.6. Modeling With Trigonometry 12.28.7. The Reciprocal Trigonometric Ratios 12.28.8. Trigonometric Ratios in Similar Right Triangles 12.28.9. Trigonometric Functions of Complementary Angles 12.28.10. Special Trigonometric Ratios 12.28.11. Calculating the Area of a Right Triangle Using Trigonometry 12.28.12. Solving Multiple Right Triangles Using Trigonometry
 12.29.1. Introduction to Radians 12.29.2. Calculating Arc Length Using Angles in Radians 12.29.3. Calculating Areas of Sectors Using Angles in Radians 12.29.4. Trigonometric Ratios With Radians
12.30. The Unit Circle
 12.30.1. Angles in the Coordinate Plane 12.30.2. Negative Angles in the Coordinate Plane 12.30.3. Coterminal Angles 12.30.4. Calculating Reference Angles 12.30.5. Properties of the Unit Circle in the First Quadrant 12.30.6. Extending the Trigonometric Ratios Using the Unit Circle 12.30.7. Extending the Trigonometric Ratios Using Angles in Radians 12.30.8. Extending the Trigonometric Ratios to Negative Angles 12.30.9. Extending the Trigonometric Ratios to Large Angles 12.30.10. Using the Pythagorean Identity in the First Quadrant 12.30.11. Extending the Pythagorean Identity to All Quadrants
12.31. Special Trigonometric Ratios
 12.31.1. Finding Trigonometric Ratios of Quadrantal Angles 12.31.2. Trigonometric Ratios of Quadrantal Angles Outside the Standard Range 12.31.3. Finding Trigonometric Ratios of Special Angles Using the Unit Circle 12.31.4. Evaluating Trigonometric Expressions 12.31.5. Further Extensions of the Special Trigonometric Ratios
12.32. Graphing Trigonometric Functions
 12.32.1. Graphing Sine and Cosine 12.32.2. Graphing Tangent and Cotangent 12.32.3. Graphing Secant and Cosecant
12.33. Properties of Trigonometric Functions
 12.33.1. Describing Properties of the Sine Function 12.33.2. Describing Properties of the Cosine Function 12.33.3. Describing Properties of the Tangent Function 12.33.4. Describing Properties of the Secant Function 12.33.5. Describing Properties of the Cosecant Function 12.33.6. Describing Properties of the Cotangent Function
12.34. Graph Transformations of Trigonometric Functions
 12.34.1. Vertical Translations of Trigonometric Functions 12.34.2. Vertical Stretches of Trigonometric Functions 12.34.3. Horizontal Translations of Trigonometric Functions 12.34.4. Horizontal Stretches of Trigonometric Functions 12.34.5. Combining Graph Transformations of Sine and Cosine 12.34.6. Graph Transformations of Tangent and Cotangent 12.34.7. Combining Graph Transformations of Tangent and Cotangent 12.34.8. Combining Graph Transformations of Secant and Cosecant 12.34.9. Graphing Reflections of Trigonometric Functions 12.34.10. Graphing Reflections of Trigonometric Functions: Three or More Transformations
12.35. Properties of Transformed Trigonometric Functions
 12.35.1. Properties of Transformed Sine and Cosine Functions 12.35.2. Finding Zeros and Extrema of Transformed Sine and Cosine Functions 12.35.3. Properties of Transformed Tangent and Cotangent Functions 12.35.4. Properties of Transformed Secant and Cosecant Functions 12.35.5. Interpreting Trigonometric Models 12.35.6. Modeling With Trigonometric Functions
13.
Probability & Combinatorics
23 topics
13.36. Introduction to Probability
 13.36.1. Sets 13.36.2. Probability From Experimental Data 13.36.3. Sample Spaces and Events in Probability 13.36.4. Single Events in Probability 13.36.5. The Complement of an Event 13.36.6. Venn Diagrams in Probability 13.36.7. Geometric Probability
13.37. Compound Events in Probability
 13.37.1. The Union of Sets 13.37.2. The Intersection of Sets 13.37.3. Compound Events in Probability From Experimental Data 13.37.4. Computing Probabilities for Compound Events Using Venn Diagrams 13.37.5. Computing Probabilities of Events Containing Complements Using Venn Diagrams 13.37.6. Computing Probabilities for Three Events Using Venn Diagrams 13.37.7. The Addition Law of Probability 13.37.8. Applying the Addition Law With Event Complements 13.37.9. Mutually Exclusive Events
13.38. Combinatorics
 13.38.1. The Rule of Sum and the Rule of Product 13.38.2. Factorials 13.38.3. Factorials in Variable Expressions 13.38.4. Ordering Objects 13.38.5. Permutations 13.38.6. Combinations 13.38.7. Computing Probabilities Using Combinatorics
14.
Statistics
8 topics
14.39. Analyzing Data
 14.39.1. Sampling 14.39.2. The Mean of a Data Set 14.39.3. Variance and Standard Deviation 14.39.4. Covariance 14.39.5. The Z-Score
14.40. Correlation and Regression
 14.40.1. The Linear Correlation Coefficient 14.40.2. Linear Regression 14.40.3. Residuals and Residual Plots