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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.

Overview

Outcomes

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 students’ 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:
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
2.4. Quadratic Equations
2.4.1. Introduction to Quadratic Equations
2.4.2. Solving Perfect Square Quadratic Equations
2.4.3. Perfect Square Quadratic Equations with One or No Solutions
2.4.4. The Zero Product Rule for Solving Quadratic Equations
2.4.5. Solving Quadratic Equations Using a Difference of Squares
2.4.6. Solving Quadratic Equations with No Constant Term
2.4.7. Solving Quadratic Equations by Factoring
2.4.8. Solving Quadratic Equations with Leading Coefficients by Factoring
2.4.9. Completing the Square
2.4.10. Completing the Square With Odd Linear Terms
2.4.11. Completing the Square With Leading Coefficients
2.4.12. Solving Quadratic Equations by Completing the Square
2.4.13. Solving Quadratic Equations With Leading Coefficients by Completing the Square
2.4.14. The Quadratic Formula
2.4.15. The Discriminant of a Quadratic Equation
2.4.16. Modeling With Quadratic Equations
3.
Quadratic Functions
17 topics
3.5. Quadratic Functions
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. Modeling With 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.
Radical Equations
2 topics
8.20. Radical Equations
8.20.1. Solving Radical Equations
8.20.2. Solving Advanced Radical Equations
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
56 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. Radians
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. Properties of Transformed Tangent and Cotangent Functions
12.35.3. Properties of Transformed Secant and Cosecant Functions
12.35.4. Interpreting Trigonometric Models
12.35.5. 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