# ALGEBRA 2 RESOURCES

### Module 1: Functions and Their Inverses

In Module 1, students review the ideas of independent variables and dependent variables of functions within context while reviewing the idea of inverses. Students learn the definition of a new kind of function (an inverse function) called a Logarithm, which will be further developed in Module 2.

By the end of this module, students should be able to do the following:

- Find the
**inverse equation**for a given function in equation form. **Graph the inverse**of a given function (or relation) whose graph is given.- Make a
**table for the inverse**of a given function whose graph is given. **Verify**two functions are inverses by simplifying a**composition of functions**.- Write the
**logarithmic form**from the**exponential form**, and visa-versa. - Use correct
**function notation**and**inverse notation**.

Module 1 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Functions (basic function review)
- Helpful videos: Khan Academy – Functions (advanced – composition of functions, inverse functions, etc.)
- Helpful explanations: Cool Math – Inverse Functions
- Helpful explanations: Math is Fun – Inverse Functions
- Helpful explanations: Purple Math – Inverse Functions (more advanced)

### Module 2: Logarithmic Functions

In Module 2, students become more familiar with logarithms first introduced in Module 1 and develop a number of logarithmic properties. Students are introduced to the graph of the logarithmic function. A focus is placed on the inverse relationship to the exponential function.

By the end of this module, students should be able to do the following:

- Write the
**exponential form**from the**logarithmic form**, and visa-versa. - Compare the value of
**logarithmic expressions**(which is greater, less than, or equal to). **Evaluate**basic logarithms without a calculator by considering the exponential form.**Graph**logarithmic functions.- Know and use the
**Product Rule**Property of Logarithms to simplify and solve logarithmic expressions and equations. - Know and use the
**Quotient Rule**Property of Logarithms to simplify and solve logarithmic expressions and equations. - Know and use the
**Power Rule**Property of Logarithms to simplify and solve logarithmic expressions and equations.

Module 2 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Exponentials & Logarithms
- Helpful explanations: Math Is Fun – Introduction to Logarithms

### Module 3: Numbers & Operations

In Module 3, students will develop a number of techniques to solve quadratic equations and other complex equations. They will also further explore ideas of exponents initially learned in 8th grade, but now extend concepts to rational numbers. During this module, students will encounter situations that require a new type of number set, the Complex Numbers, and perform arithmetic with these new numbers.

By the end of this module, students should be able to do the following:

- Use
**rules of exponents**on both integer and rational number exponents. - User
**rational exponents**to solve**radicals**and visa-versa. - Solve a quadratic equation by
**completing the square**and taking a**square root**. - Solve a quadratic equation by using the
**Quadratic Formula**. - Use
**Complex Numbers**to solve quadratic equations. - Perform arithmetic (add, subtract, multiply, divide) with
**Complex Numbers**. - Simplify
**imaginary numbers**.

Module 3 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Quadratics
- Virtual Algebra Tiles and Manipulative: CPM Technology Tools – Algebra Tiles

### Module 4: Polynomial Functions

In Module 4, students briefly review 1st degree polynomials (linear) and 2nd degree polynomials (quadratic) and then begin investigating properties of higher degree polynomials. Properties include graph shape, x-intercept(s), y-intercept, end behavior, local max and min values, growth type, degree, factored form, and more. A focus is placed on the relationship between the different representations of polynomials (graph, equation, table). Students will encounter the Math 2 concept of imaginary numbers while working with quadratics and higher degree polynomials.

By the end of this module, students should be able to do the following:

- Determine if an equation or expression is a
**polynomial**. - Identify the
**degree of a polynomial**. - Determine the
**end behavior**of a polynomial. - Identify the
**x-intercept(s)**or**roots**of a polynomial from a graph, or table, or equation. - Identify the
**y-intercept**of a polynomial from a graph, or table, or equation. - Identify any
**local max**or**min**values of a polynomial. - Describe the type of growth of a polynomial (
**constant growth**,**linear growth**,**quadratic growth**, etc.). - Write a polynomial in
**standard form**and**factored form**. - Use
**Polynomial Long Division**to find remaining factors and roots of polynomial. - Determine if a function is an
**even function**, an**odd function**, or**neither**based on its graph.

Module 4 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Polynomials (level 1)
- Helpful videos: Khan Academy – Polynomials (level 2)
- Helpful explanations: Math Is Fun – Intro to Polynomials
- Helpful explanations: YouTube Video – Polynomial Long Division

### Module 5: Rational Functions & Expressions

In Module 5, students explore an entirely new type of function called Rational Functions. Students make important connections between the equation, table, and graph of a rational function while exploring the advanced and abstract concepts of asymptotes and end behavior. Students also learn to manipulate rational expressions in order to add, subtract, multiply, and divide with rational expressions. Lastly, students will use all their techniques to solve rational equations.

By the end of this module, students should be able to do the following:

- Make a table and graph of the basic
**rational function**y=1/x. - Graph a
**transformation**of the basic**rational function**(shifting left/right, up/down, flipping, or widening/narrowing). - Identify the
**end behavior**of a rational function by looking at its equation or graph. - Identify any
**vertical asymptotes**of a rational function by looking at its equation or graph. - Identify any
**horizontal asymptotes**of a rational function by looking at its equation or graph. - Identify any
**x-intercepts**of a rational function by looking at its equation or graph. - Identify any
**y-intercepts**of a rational function by looking at its equation or graph. **Graph**a complicated rational function without technology by using key features (horizontal and vertical asymptotes, x-intercepts and y-intercepts, etc.)- Determine if a rational expression is
**proper**or**improper**. **Simplify**a rational expression.- Add, subtract, multiply, and divide
**rational expressions**. **Solve**a rational equation.

Module 5 Resources:

MVP Algebra 2 Curriculum: MVP Website

Helpful videos: Khan Academy – Rational Relationships

Helpful explanations: Purple Math – Graphing Rational Functions

Module 6: Modeling Periodic Behavior

In Module 6, students develop the idea of periodic functions through applications involving periodic behavior. Students learn many new vocabulary words regarding angles and extend the definition of trig ratios in order to handle periodic situations. Students also see a new way to measure an angle, namely with radians. Ultimately, students use the Unit Circle, a major tool for trigonometric functions and precalculus.

By the end of this module, students should be able to do the following:

- Use
**right triangle trigonometry**to describe periodic behavior. - Use angles in
**standard position**to evaluate trig ratios in all four quadrants. - Extend the
**definition of the trig ratios**to include all real numbers (angles larger than 90 degrees as well as negative angles). - Graph a
**sine function**. - Convert
**degree angles**into**radian angles**, and visa-versa. - Use the
**Unit Circle**to give the exact value of trig expressions.

Module 6 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Right Triangles & Trigonometry
- Helpful videos: Khan Academy – Intro to Radians
- Helpful videos: Khan Academy – Unit Circle Definition of Trig
- Helpful videos: Khan Academy – Graphs of Trig
- Interactive demo: Math Is Fun – Interactive Unit Circle

### Module 7: Trigonometric Functions, Equations and Identities

In Module 7, students continue the study of trigonometric functions by changing the functions amplitude, period, vertical shift, and horizontal shift (phase shift) in order to model periodic behavior. Students also encounter a need for inverse trig relationships and for a formal definition of the tangent function. Lastly, students see a number of trig identities and the reasoning behind their validity.

By the end of this module, students should be able to do the following:

a*Write***trig function**to model a periodic behavior requiring a specific**amplitude**,**period**,**vertical shift**, and**horizontal shift**(**phase shift**).a*Graph***trig function**to model a periodic behavior requiring a specific**amplitude**,**period**,**vertical shift**, and**horizontal shift**(**phase shift**).- Use
**inverse trig**to solve for the unknown angle in a**trig equation**. - Identify the
**restricted domain**for a trig function so that its**inverse**if also a function. - Evaluate the
**tangent function**for various angles around the**Unit Circle**. - Justify why a
**trig identity**is true. **Solve**trig equations.

Module 7 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Basic Trigonometric Identities
- Helpful videos: Khan Academy – Amplitude, Period, Phase Shift, Midline
- Helpful explanations: Purple Math – Graphing Trigonometric Functions
- Interactive Demo: NCTM – Transforming a Sine Graph

### Module 8: Modeling with Functions

In Module 8, students begin by reviewing all of the Family of Functions study so far throughout their education and how we transform them. Students then begin combining different functions by adding, subtracting, multiplying, and dividing functions as well as by making a composition of functions. Students will work with composition of functions in table form, graph form, and equation form. Lastly, students will decompose functions by breaking them into their smaller “original” parts.

By the end of this module, students should be able to do the following:

- Know the original (“parent”) function and graph for all the
**standard family of functions**– linear, exponential, quadratic, polynomial, rational, absolute value, logarithmic, trigonometric, and radical functions. **Transform**each of the original (“parent”) function from all the standard family of functions.- Combine standard family of functions by
**adding**,**subtracting**,**multiplying**, and**dividing functions**as well as by making a composition of functions (ie build more complex functions out of the standard family of functions). **Decompose**a more complicated function by breaking it into its smaller “original” parts.- Model complex behavior using a complicated
**composition of functions**.

Module 8 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Combining Functions
- Helpful videos: Khan Academy – Shifting Functions
- Helpful explanations: Math is Fun – Composition of Functions

### Module 9: Statistics

In Module 9, students are introduced to normal distributions and the Normal Curve. Students examine data that is displayed graphically and numerically in order to determine if it represents a Normal Distribution. They also make conclusions about normally distributed data. Lastly, students compare and apply various methods of sampling populations to collect data.

By the end of this module, students should be able to do the following:

- Identify whether data displayed on a graph is
**normally distributed**. - Describe the features of data
**normally distributed**. - Use the
**mean**and**standard deviation**of normal data to determine data percentages. - Use the
**68%-95%-99.7% Rule**to describe percentages for data distributed normally. - Use the mean and standard deviation of normal data to determine a
**z-score**for specific data values. - Identify the
**population**, the**sample**, and the**parameter**of interest within a context/application. - Describe appropriate methods to
**sample data**from a population.

Module 9 Resources:

- MVP Algebra 2 Curriculum: MVP Website
- Helpful videos: Khan Academy – Normal Distributions
- Helpful videos: Khan Academy – Study Design
- Helpful explanations: Statistics How To – Normal Distributions