Algebra II

Curriculum Materials- Algebra II Envision Math

Course Overview

This course requires that students have a solid background in Algebra I. Students will be expected to have a solid comprehension solving multi-step equations, writing equations of lines and graphing linear functions. New topics include imaginary numbers, quadratic equations, polynomial and rational functions, and logarithms. Emphasis is placed on the understanding of the process and theory of various techniques, as well as on problem solving and modeling data. The necessity to understand the concepts is much stronger in Algebra II than students have experienced in the more skill driven Algebra I.

Topic 1 - Linear Functions and Systems

Learning Targets

Lesson 1-1 - Key Features of Functions- The key features of a graph— including the domain, range, and intercepts—reveal the relationship between two quantities.
Lesson 1-2 - Transformations of Functions -A function of the form f(x)=a∙f[b(x−h)]+k is transformed by changing the values of a, b, h, or k. Changing the value of h or k results in a horizontal or vertical translation, changing the sign of a or b results in a reflection across one of the axes, and changing the value of a or b results in a horizontal or vertical stretch or compression.
Lesson 1-3- Piecewise-Defined Functions- A piecewise-defined function is used to model situations in which there are different rules for different parts of the domain of the function.
Lesson 1-6-Linear Systems- The solution of a system of linear equations or inequalities is the set of ordered pairs that satisfy all the equations or inequalities in the system.

Essential Question

What are the ways in which functions can be used to represent and solve problems involving quantities?

Instructional Snapshot

Students will:

Identify key features of a graph of a function, including the intercepts, positive and negative intervals, and areas where the function is increasing or decreasing.
Calculate and interpret the average rate of change of a function over a specified interval.
Write the domain and range of functions using set-builder and interval notations.
Graph a transformed function by identifying the effect of on the graph of replacing

f(x) by f(x)+k, kf(x), f(kx), and f(x+k) or specific values of k.

Write an equation of a transformed function.
Relate the domain of a function to its graph and the real-world situation it describes.
Create and graph piecewise-defined functions, including absolute value functions and step functions.
Create and use a piecewise-defined function from real-world data.
Write a piecewise-defined rule from a graph.
Solve linear systems graphically and algebraically.
Identify regions that satisfy systems of inequalities.

Topic 2 - Quadratic Functions and Equations

Learning Targets

Lesson 2-1 -Vertex Form of a Quadratic Function - All quadratic functions are transformations of the parent function f(x)=x^2. The vertex form of a quadratic function highlights the key features of the function’s graph and shows how the graph of the parent function can be transformed.
Lesson 2-2- Standard Form of a Quadratic Function - A quadratic function in vertex form can be rewritten in standard form to highlight different features of the function’s graph. The key features are used to interpret values in context.
Lesson 2-3 - Factored Form of a Quadratic Function -The factored form of a quadratic function is used to find the zeros of the function by identifying the values that make one or both of the factors equal to zero.
Lesson 2-4 -Complex Numbers and Operations- A complex number contains both real and imaginary parts. The four basic operations can be applied to complex numbers.
Lesson 2-5- Completing the Square- A quadratic equation can be solved by completing the square to transform the equation to an equivalent equation, (x−p)^2=q.
Lesson 2-6- The Quadratic Formula-The Quadratic Formula can be used to solve any quadratic equation, including those with complex solutions.
Lesson 2-7- Linear-Quadratic Systems-A linear-quadratic system consists of a linear equation and a quadratic equation. The points of intersection are the solutions.

Essential Question

How do you use quadratic functions to model situations and solve problems?

Instructional Snapshot

Students will:

Create quadratic functions in vertex form to represent relationships between variables as shown in their graphs.
Graph functions on coordinate axes using their key features.
Interpret key features of the graph of a quadratic function.
Create quadratic functions written in standard form.
Identify key features of quadratic functions and graph a quadratic function written in standard form.
Write a quadratic equation in factored form and use it to identify the zeros of the function it defines.
Determine the intervals over which a quadratic function is positive or negative.
Add, subtract, and multiply complex numbers using the properties of operations and the relation i^2=−1
Use complex numbers to represent numbers that are not on the real number line.
Transform a quadratic equation into the form (x−p)^2=q by completing the square.
Complete the square to reveal the minimum or maximum value of a quadratic expression.
Use the Quadratic Formula to solve quadratic equations that have complex solutions.
Use algebra to solve a linear-quadratic system.
Solve a linear-quadratic system using graphing and explain why the points of intersection are the solutions.

Topic 3 - Polynomial Functions

Learning Targets

3-1- Graphing Polynomial Functions-A polynomial function is a function whose rule is either a monomial or a sum of monomials. The key features of the graph of a polynomial function—such as its end behavior, intercepts, and turning points—can be used to sketch a graph of the function.
3-2- Adding, Subtracting, and Multiplying Polynomial Functions-Just as with real numbers, the properties of operations can be used to add, subtract, and multiply polynomials. Polynomial functions can be used to represent and compare real-world situations.
3-3- Polynomial Identities- Polynomial identities and the Binomial Theorem are helpful tools for efficiently rewriting expressions and describing mathematical relationships.
3-4- Dividing Polynomials- Polynomial expressions can be divided by linear factors using long division or synthetic division. The Remainder Theorem is used to determine the remainder of a division problem.
3-5- Zeros of Polynomial Functions-The zeros of a polynomial function can be determined using factoring or synthetic division. The zeros of a function can be used to sketch its graph.
3-6- Theorems about Roots of Polynomial Functions- Theorems such as the Rational Root Theorem, the Fundamental Theorem of Algebra, and the Conjugate Root Theorems are helpful tools for determining the roots of a polynomial function.
3.7- Transformations of Polynomials Functions- Polynomial functions are categorized as even, odd, or neither. Even functions are symmetric about the y-axis, and for all x in the domain, f(x)=f(−x) Odd functions are symmetric about the origin, and for all x in the domain f(−x)=−f(x).

Essential Question

What can the equation of the polynomial function tell about the graph and what can the graph about the polynomial function tell about the solutions?

Instructional Snapshot

Students will:

Graph polynomial functions and show the key features of the graph.
Predict the end behavior of polynomial functions by interpreting the leading coefficients and degrees.
Sketch graphs showing key features, given a verbal description.
Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations.
Compare a polynomial function represented algebraically with one represented graphically.
Prove polynomial identities and use them to multiply and factor polynomials.
Expand binomials using the Binomial Theorem and coefficients determined by Pascal’s Triangle.
Divide polynomial expressions using long division.
Use synthetic division to rewrite rational expressions.
Identify the zeros of a function by factoring or using synthetic division.
Use the zeros of a polynomial function to sketch its graph.
Extend polynomial theorems and identities to find the real and complex solutions of polynomial equations.
Write polynomial functions using conjugates.
Recognize even and odd functions from their graphs and algebraic equations.
Identify the effect on the graphs of cubic and quartic functions of replacing f(x) with f(x)+k, kf(x), f(kx),and f(x+k).
Given a graph, determine the equation as it is related to its parent cubic function or quartic function.

Topic 4- Rational Functions

Learning Targets

Lesson 4-1-Inverse Variation and Reciprocal Function- The reciprocal function is used to model inverse variation, which is a proportional relationship between two variables such that when one variable increases, the other decreases.
Lesson 4-2- Graphing Rational Functions- A rational function is any function R(x)=P(x)/Q(x) where P(x) and Q(x) are polynomial functions. The domain of a rational function is all real numbers except any x-values for which Q(x) equals zero. The graph of a rational function has one or more asymptotes, which guide the end behavior of the graph.
Lesson 4-3- Multiplying and Dividing Rational Functions- Rational expressions form a system similar to the system of rational numbers and can be multiplied and divided by applying the properties of operations as they apply to rational expressions.
Lesson 4-4- Adding and Subtracting Rational Expressions -The properties of operations used to add and subtract rational numbers can be applied to adding and subtracting rational expressions.
Lesson 4-5- Solving Rational Equations - Rational equations contain a rational expression and can be solved by multiplying each side of the equation by a common denominator to eliminate the fractions. Any solution that is excluded from the domain of the original equation is extraneous.

Essential Question

What are rational functions and what are the key features of their graphs?

Instructional Snapshot

Students will:

Use inverse variation to write and graph the reciprocal function.
Identify the effect of transformations on the graph of the reciprocal function and define the effects of h and k on the function f(x) = 1 /x −h +k
Graph rational functions by identifying asymptotes and end behavior.
Rewrite simple rational expressions in different forms using long division.
Use the structure of rational expressions to rewrite simple rational expressions in different forms.
Understand that rational expressions form a system analogous to the system of rational numbers and use that understanding to multiply and divide rational expressions.
Understand that rational expressions form a system analogous to the system of rational numbers and use that understanding to add and subtract rational expressions.
Solve rational equations in one variable.
Identify extraneous solutions to rational equations and give examples of how they arise.

Topic 5- Function Operations

Learning Targets

5-1- nth Roots, Radicals, and Rational Exponents-A rational exponent has an equivalent radical expression and any radical expression may be written with a rational exponent. This foundation is necessary to simplify expressions and solve equations.
5-2- Properties of Exponents and Radicals- The properties of integer exponents can be applied to terms with rational exponents, as well as to radicals. The properties of exponents and radicals can be used to rewrite radical expressions. When rewriting radical expressions, like radicals, which have the same index, can be added and subtracted.
5-3- Graphing radical functions - The function g(x)=a n√x−h+k represents the transformation of the parent radical function f(x)= n√x, where a stretches compresses the graph vertically, h translates the graph horizontally, and k translates the graph vertically.
5-4- Solving Radical Equations- Solving equations that include radicals or rational exponents is like solving rational equations. You can use the Power of a Power Property to eliminate radicals and rational exponents.
5-5 -Function Operations- Functions can be combined by operations (+, -, ×, ÷) and by composition. The result of the operation or composition can be described as a single function. The domain of the result may be different from the domains of the original functions.
5-6- Inverse Relations and Functions-The inverse of a function is found by exchanging the roles of the independent and dependent variables. Composition can be used to verify that two functions are inverses.

Essential Questions

How are rational exponents and radical equations used to solve real-world problems?

Instructional Snapshot

Students will:

Find all real nth roots of a number.
Evaluate expressions with rational exponents.
Use nth roots to solve equations by rewriting expressions using the properties of exponents.
Use the properties of exponents and radicals to identify ways to rewrite radical expressions
Interpret radical expressions that represent a quantity in terms of its context.
Graph radical functions, including square root and cube root functions.
Identify the effect of transformations on the key features of the graphs of radical functions.
Solve radical equations in one variable.
Explain how extraneous solutions may arise when solving radical equations.
Solve radical inequalities and apply the solution within a real-world context.
Combine functions by addition, subtraction, multiplication, or division, and identify the domain of the result.
Compose functions, specifying the order in which the functions are applied and describing the domain of the composite function.
Use tables, graphs, and equations to represent the inverse of a relation.
Write an equation for the inverse of a function by restricting the domain.
Verify that one function is the inverse of another using composition.

Topic 6- Exponential and Logarithmic Functions

Learning Targets

6-1- Key features of Exponential Functions- The rate of exponential growth or decay is the ratio between two consecutive output values in an exponential function.
6-2- Exponential Models- Exponential models are useful for representing situations in which the rate increases by the same percent for each period of time and for interpreting problems that involve compound interest. Exponential regression can be used to generate exponential models for real-world contexts.
6-3 Logarithms- A logarithmic function is the inverse of an exponential function. Logarithms are found by determining the exponent that must be applied to a base to yield a given result.
6-4- Logarithmic Functions- The inverse relationship between exponential and logarithmic functions reveals key features of the graphs of both functions. Logarithmic functions can be used to model several real-world situations.
6-5 Properties of Logarithms- Properties of Logarithms can be used to rewrite logarithmic expressions and to evaluate logarithms by changing the base.
6-6- Exponential and Logarithmic Equations- Some exponential equations can be solved by rewriting both sides with a common base. For others, rewriting the equation using logarithms and applying properties of logarithms is a more efficient method.

Essential Questions

How do you use exponential and logarithmic functions to model situations and solve problems?

Instructional Snapshot

Students will:

Interpret key features of exponential functions represented by graphs, tables, and equations.
Graph transformations of exponential functions showing intercepts and end behavior.
Model quantities that increase or decrease by a fixed percent each time period using exponential functions.
Understand the inverse relationship between exponents and logarithms.
Use logarithms to solve exponential models.
Evaluate logarithms using technology.
Graph logarithmic functions, and interpret their key features.
Write and interpret the inverses of exponential and logarithmic functions.
Use Properties of Logarithms to rewrite logarithmic expressions.
Use the Change of Base Formula to evaluate logarithmic expressions and solve equations.
Use logarithms to express the solutions to exponential models.
Solve exponential and logarithmic equations.

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