# 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 Precalculus

## Content

Building on the foundations laid in Integrated Math I, students will extend their understanding of functions to encompass function arithmetic (including function composition), local extrema, periodicity, and even and odd functions. Students will carry out detailed explorations of function invertibility and investigate how to perform multi-step graph transformations. They will learn about formal sequence notation and explore recursive sequences.

As part of this course, students will master techniques for solving quadratic equations and explore quadratic functions in-depth, including how they're used to model real-world problems. Students will also deepen their understanding of exponential functions, explore the relationship between exponentials and logarithms, and the laws of logarithms.

This course aims to broaden students' knowledge of geometry, including concepts such as congruence, similarity, circle theorems, and solid geometry. Students will lay robust foundations in trigonometry, including exploring the unit circle, the Pythagorean identity, and how trigonometry is used to solve practical problems,

Finally, students will develop foundations in probability and combinatorics and explore essential tools for analyzing bivariate data.

Upon successful completion of this course, students will have mastered the following:

• Solving quadratic equations by factoring, completing the square, and using the quadratic formula.
• Using the discriminant to determine the number of solutions to a quadratic equation.
• Identifying elementary quadratic functions undergoing shifts and reflections.
• Calculating roots of quadratic functions.
• Computing the axis of symmetry of a parabola by taking the average of the roots, expressing it in vertex form, or using a formula.
• Sketching graphs of general quadratic functions and stating their domain and range.
• Finding intersection points of lines and quadratic functions.
• Using quadratic functions to model real-world applications involving vertical motion, revenue, cost, and profit.

### Functions & Sequences

• Function arithmetic, including function composition.
• Defining and determining local extrema of functions.
• Accurately describing the behavior of a function near a vertical asymptote.
• Classifying functions as periodic, even, odd, one-to-one, and many-to-one, and solving problems using these properties.
• Describing the conditions required for a function to be invertible and how the graph, domain, and range of a function are related to that of its inverse.
• Calculating inverses of linear and quadratic functions
• Calculating the average rate of change of a function over an interval.
• Graphing transformations of functions, including composite and absolute value transformations.
• Calculating a sequence's terms explicitly or recursively and working with Fibonacci sequences.

### Exponentials & Logarithms

• Applying composite transformations to graphs of exponential functions and determining their properties.
• Converting expressions between exponential and logarithmic form.
• Applying the laws of logarithms to simplify expressions.

### Geometry

• Understanding and applying the ASA, AAS, SAS, SSS, and HL congruence criteria.
• Understanding and applying the concept of similarity and solving problems involving similar polygons.
• The AA, SSS, and SAS similarity criteria and the midpoint and proportionality theorems.
• Showing that two figures are similar using similarity transformations.
• Solving circle problems using the inscribed angle theorem, Thales' theorem, properties of inscribed quadrilaterals, tangent lines to circles, and circle similarity.
• Calculating lengths within rectangular solids using the distance formulas.
• Calculating surface areas and volumes of polyhedra and non-polyhedra,
• Describing solids of revolution and computing their volumes.
• Applying Euler's formula for polyhedra.
• Describing the five Platonic solids.

### Trigonometry

• Calculating trigonometric and reciprocal trigonometric ratios in right triangles.
• Using trigonometry to calculate angles and sides in right triangles.
• Deriving and applying the special trigonometric ratios for acute angles.
• Understanding the relationship between cofunctions.
• Solving practical problems using trigonometry, including cases with multiple triangles.
• Understand the concept of radian measure and use radians to calculate arc lengths and areas of circular sectors.
• Describing how the unit circle extends the trigonometric ratios to non-acute and negative angles.
• Applying the Pythagorean identity.

### Probability & Statistics

• Essential concepts from probability, including sample spaces, event complements, compound events, Venn diagrams, unions and intersections of events, and using these ideas to solve real-world problems.
• Carrying out elementary analyses on bivariate data, including interpreting scatter plots, making predictions using trend lines, interpreting residues, interpolation vs. extrapolation, linear correlation, and distinguishing between correlation and causation.
1.
34 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. 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
 1.3.1. Modeling Downwards Vertical Motion 1.3.2. Modeling Upwards Vertical Motion 1.3.3. Vertical Motion 1.3.4. Revenue, Cost, and Profit Functions 1.3.5. Constructing Revenue, Cost, and Profit Functions 1.3.6. Maximizing Profit and Break-Even Points
2.
Functions & Sequences
32 topics
2.4. Functions
 2.4.1. The Arithmetic of Functions 2.4.2. Function Composition 2.4.3. Describing Function Composition 2.4.4. Local Extrema of Functions 2.4.5. One-To-One Functions 2.4.6. Introduction to Inverse Functions 2.4.7. Calculating the Inverse of a Function 2.4.8. Inverses of Quadratic Functions 2.4.9. Graphs of Inverse Functions 2.4.10. Domain and Range of Inverse Functions 2.4.11. Invertible Functions 2.4.12. Plotting X as a Function of Y 2.4.13. Periodic Functions 2.4.14. Even and Odd Functions 2.4.15. Unbounded Behavior of Functions Near a Point 2.4.16. The Average Rate of Change of a Function
2.5. Graph Transformations of Functions
 2.5.1. Vertical Translations of Functions 2.5.2. Horizontal Translations of Functions 2.5.3. Vertical Stretches of Functions 2.5.4. Horizontal Stretches of Functions 2.5.5. Vertical Reflections of Functions 2.5.6. Horizontal Reflections of Functions 2.5.7. Combining Graph Transformations: Two Operations 2.5.8. Combining Graph Transformations: Three or More Operations 2.5.9. Constructing Functions Using Transformations 2.5.10. Combining Reflections With Other Graph Transformations 2.5.11. Finding Points on Transformed Curves 2.5.12. The Domain and Range of Transformed Functions 2.5.13. Absolute Value Graph Transformations
2.6. Sequences
 2.6.1. Introduction to Sequences 2.6.2. Recursive Sequences 2.6.3. Fibonacci Sequences
3.
Exponentials & Logarithms
16 topics
3.7. Graphs of Exponential Functions
 3.7.1. Vertical Translations of Exponential Growth Functions 3.7.2. Vertical Translations of Exponential Decay Functions 3.7.3. Interpreting Graphs of Exponential Functions 3.7.4. Combining Graph Transformations of Exponential Functions 3.7.5. Properties of Transformed Exponential Functions
3.8. Introduction to Logarithms
 3.8.1. Converting From Exponential to Logarithmic Form 3.8.2. Converting From Logarithmic to Exponential Form 3.8.3. Evaluating Logarithms 3.8.4. The Natural Logarithm 3.8.5. The Common Logarithm 3.8.6. Simplifying Logarithmic Expressions
3.9. The Laws of Logarithms
 3.9.1. The Product Rule for Logarithms 3.9.2. The Quotient Rule for Logarithms 3.9.3. The Power Rule for Logarithms 3.9.4. Combining the Laws of Logarithms 3.9.5. The Change of Base Formula for Logarithms
4.
Geometry
48 topics
4.10. Congruence
 4.10.1. Rigid Motions and Congruence 4.10.2. The ASA Congruence Criterion 4.10.3. The AAS Congruence Criterion 4.10.4. The SAS Congruence Criterion 4.10.5. The SSS Congruence Criterion 4.10.6. The HL Congruence Criterion
4.11. Similarity
 4.11.1. Similarity and Similar Polygons 4.11.2. Side Lengths and Angle Measures of Similar Polygons 4.11.3. Areas of Similar Polygons 4.11.4. Working With Areas of Similar Polygons 4.11.5. Similarity Transformations 4.11.6. The AA Similarity Criterion 4.11.7. The SSS Similarity Criterion 4.11.8. The SAS Similarity Criterion 4.11.9. The Midpoint Theorem 4.11.10. The Triangle Proportionality Theorem
4.12. Circles
 4.12.1. Inscribed Angles 4.12.2. Problem-Solving With Inscribed Angles 4.12.3. Thales' Theorem 4.12.4. Angles in Inscribed Right Triangles 4.12.5. Inscribed Quadrilaterals 4.12.6. Tangent Lines to Circles 4.12.7. Circle Similarity
4.13. Introduction to Solid Geometry
 4.13.1. Identifying Three-Dimensional Shapes 4.13.2. Faces, Vertices, and Edges of Polyhedrons 4.13.3. Nets of Polyhedrons 4.13.4. Finding Surface Areas Using Nets 4.13.5. The Distance Formula in Three Dimensions 4.13.6. Euler's Formula for Polyhedra 4.13.7. The Five Platonic Solids
4.14. Rectangular Solids and Pyramids
 4.14.1. Volumes of Cubes 4.14.2. Surface Areas of Cubes 4.14.3. Face Diagonals of Cubes 4.14.4. Diagonals of Cubes 4.14.5. Volumes of Rectangular Solids 4.14.6. Surface Areas of Rectangular Solids 4.14.7. Diagonals of Rectangular Solids 4.14.8. Volumes of Pyramids 4.14.9. Surface Areas of Pyramids
4.15. Non-Polyhedrons
 4.15.1. Volumes of Cylinders 4.15.2. Surface Areas of Cylinders 4.15.3. Volumes of Right Cones 4.15.4. Slant Heights of Right Cones 4.15.5. Surface Areas of Right Cones 4.15.6. Volumes of Spheres 4.15.7. Surface Areas of Spheres 4.15.8. Conical Frustums 4.15.9. Volumes of Revolution
5.
Trigonometry
32 topics
5.16. Introduction to Trigonometry
 5.16.1. Angles and Sides in Right Triangles 5.16.2. The Trigonometric Ratios 5.16.3. Calculating Trigonometric Ratios Using the Pythagorean Theorem 5.16.4. Calculating Side Lengths of Right Triangles Using Trigonometry 5.16.5. Calculating Angles in Right Triangles Using Trigonometry 5.16.6. Modeling With Trigonometry 5.16.7. The Reciprocal Trigonometric Ratios 5.16.8. Trigonometric Ratios in Similar Right Triangles 5.16.9. Trigonometric Functions of Complementary Angles 5.16.10. Special Trigonometric Ratios 5.16.11. Calculating the Area of a Right Triangle Using Trigonometry 5.16.12. Solving Multiple Right Triangles Using Trigonometry