# 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:

### 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:

### 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.
• Use Complex Numbers to solve quadratic equations.
• Perform arithmetic (add, subtract, multiply, divide) with Complex Numbers.
• Simplify imaginary numbers.

Module 3 Resources:

### 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:

### 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 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:

### 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:

• Write a trig function to model a periodic behavior requiring a specific amplitude, period, vertical shift, and horizontal shift (phase shift).
• Graph a 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:

### 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:

### 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: