Algebra Two Lessons

Course Lesson List

Below are the lessons for Algebra Two. View All Lessons » to see all lessons from all courses.

Algebra Two

Functions - Function Basics

Intro to Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions

Priority: VIP Knowledge

This is the first proper Mister Math lesson on functions. We'll see what they are, how they work, and how to determine whether or not a relationship can be called a function.

  • Define and understand what a relation is
  • Define and understand what a function is
  • Determine whether or not a relationship is a function by looking at sets of input and output
Function Notation

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions

Priority: VIP Knowledge

In this short VIP Knowledge lesson, we seek to understand the standard function notation that mathematicians use.

  • Read and understand standard function notation
  • Interpret worded math instruction and write it using function notation
Evaluating Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions
  • Algebra Manipulation

Priority: VIP Knowledge

Functions are fancy, special relationships, but they aren't very helpful in math unless we can use them. This lesson first shows us how to evaluate a function at a given input value, and then shows us how to figure out what the input value was if all we know is the output.

  • Understand how functions behave like input/output processes.
  • Write and interpret input / output lists
  • Evaluate functions at a specific value
  • Solve for an unknown input given the output
  • Input an algebraic expression into a function
Properties of Graphs of Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions
  • Graphing
  • Function Analysis
  • Calculator Skills

Priority: High

Every function has a graph, though some are simple and some are complex. This lesson helps us understand the why and how of graphing functions, what it means to say a function is continuous, and what asymptotes are and how to graph them. We can also use a graph to determine whether or not we have a function or a relation, using what we call "The Vertical Line Test".

  • Know how and why functions can be graphed on the coordinate plane
  • Learn important characteristics that a function's graph can have
  • See what asymptotes are and how to denote them on a graph
  • Learn what the vertical line test is and how it works
  • Understand why the vertical line test is a fail-safe method
  • Become a champ at getting a function's graph with a graphing calculator
Piecewise Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions
  • Graphing
  • Function Analysis

Priority: High

While we most commonly work with one function definition for any input value, it is possible to use several function definitions one at a time in different intervals. Here we will see how that is done and how to work with functions of this type.

  • Define what a piecewise function is and the notation we use for them
  • Learn how to work with, evaluate, and graph piecewise functions
  • Determine an unknown piecewise function based on its graph
Defining Domain and Range

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions
  • Graphing
  • Function Analysis

Priority: VIP Knowledge

In this first lesson about domain and range, we will learn some basic techniques for finding the domain and range of functions based on clues from either the function definition or the function graph.

  • Learn what each domain and range means
  • Practice finding domain and range of a function by inspecting the function definition
  • Practice finding domain and range of a function by inspecting the graph of the function
Basic Function Transformations - Translations

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Function Basics

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Tags

  • Functions
  • Graphing
  • Function Analysis

Priority: High

If a function is translated, its original shape remains intact but its graph shifts up, down, left, or right. This lesson shows us how to draw relationships between two functions that are otherwise the same, but translated, or shifted over from one another. We will continue using translations throughout the entirety of Pre-Calculus, including trigonometry.

  • Identify translations based on graphs only
  • Use function notation to understand how to translate a function up, down, left, or right
  • Algebraically find the new translated function from the original function, given the desired translation

Functions - Linear Functions

Straight Lines as Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Linear Functions

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Tags

  • Functions
  • Graphing
  • Linear Equations

Priority: Normal

Already we've seen various function definitions, with all kinds of curvy and crazy graphs. This lesson focuses on the function type that graphs as a straight line, putting together past knowledge you have about two variable linear equations and newer knowledge you've learned about functions.

  • Know how to identify whether or not a function is linear
  • Understand why slope-intercept form is the only easy way to translate between the perspective of a two variable linear equation and the perspective of a function
  • See why this simple function case exhibits similar behavior to lines in the place that we studied in Algebra One, both algebraically and graphically
Modeling with Linear Functions

Algebra Two  $\rightarrow$  Functions  $\rightarrow$  Linear Functions

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Tags

  • Functions
  • Word Problems
  • Linear Equations

Priority: Normal

Any situation in which two quantities are related by a simple slope is linear. This lesson focuses on how to create and use linear functions that are specific to a given situation. We will be able to not only solve word problems, but also interpret results in context of the situation.

  • Generate a linear function based on tables, graphs, or descriptions
  • Interpret a linear function in context of a real-life scenario, including reasonable limits on domain
  • Solve word problems using linear functions, where the function was either given to us or created by us

Complex Numbers - The Imaginary Number

Defining The Imaginary Number

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  The Imaginary Number

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Tags

  • Imaginary Numbers
  • Square Roots
  • Simplifying

Priority: High

The commonly used imaginary number completes the picture in many advanced math topics, even though it doesn't exist. Here we seek to understand what the imaginary number represents, and the basics of working with it.

  • Define the imaginary number
  • Learn how to simplify powers of i
Simplifying Negative Square Roots

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  The Imaginary Number

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Tags

  • Imaginary Numbers
  • Square Roots
  • Simplifying

Priority: High

Now that we understand what the imaginary number is and how it works, we can simplify the square root of negative numbers in a very similar way to how we simplify square roots of positive ones.

  • Learn how to simplify square roots of negative numbers
  • Apply what we already know about reducing square roots of positive integers

Complex Numbers - Using Complex Numbers

Defining Complex Numbers

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  Using Complex Numbers

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Tags

  • Imaginary Numbers

Priority: Normal

Real numbers and imaginary numbers are not like terms and cannot be combined, but their sum comprises a type of number we call a "Complex Number". These are very useful for polynomial analysis and other situations where imaginary numbers are meaningful, such as advanced Physics applications.

  • Define what complex numbers are
  • Understand that each real numbers and imaginary numbers are subsets of the set of complex numbers
  • Know the concepts of norm and conjugate as applicable to complex numbers
Arithmetic with Complex Numbers

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  Using Complex Numbers

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Tags

  • Imaginary Numbers
  • Adding
  • Subtracting
  • Multiplying

Priority: Normal

We will combine what we know about arithmetic with imaginary numbers with what we know about binomial arithmetic to understand the proper way to add, subtract, and multiply complex numbers, including simplifying the final result.

  • Add and subtract complex numbers by focusing on like terms
  • Multiply complex numbers using FOIL and simplify final answer to be one complex number
Rationalizing Complex Expressions

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  Using Complex Numbers

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Tags

  • Imaginary Numbers
  • Rationalizing
  • Simplifying
  • Dividing

Priority: Normal

Dividing by a complex number, or equivalently, working with a fraction that has a complex number denominator, requires rationalization for the final answer. This lesson covers the nuances of dividing by a complex number via rationalization and simplification.

  • Rationalize expressions that contain a complex number in the denominator
  • Learn how to divide by a complex number
Plotting Complex Numbers

Algebra Two  $\rightarrow$  Complex Numbers  $\rightarrow$  Using Complex Numbers

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Tags

  • Imaginary Numbers
  • Coordinate Plane
  • Graphing

Priority: Optional

While we often use the Cartesian Coordinate Plane to plot relationships between two real variables, we can use a modified version to plot complex numbers as points. We will see how to do that here, and the useful properties of complex numbers that we are able to visualize when we do so.

  • See and understand how the Complex Number Plane works
  • Learn how to plot a complex number on the plane
  • Visualize the norm and conjugate of a complex number by using the complex plane

Using Quadratic Relationships - Quadratic Functions

Factoring Quadratics Overview

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Quadratic Functions

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Tags

  • Review
  • Quadratics
  • Factoring

Priority: Optional

This pure review lesson revisits all the techniques for factoring quadratics that we learned in Algebra One, including Trinomial Factoring (aka Reverse FOIL), GCF factoring, Difference of Squares, and Perfect Square Trinomials. With the opportunity to look at them all in one place, we can make sure we're solid on each technique before moving forward with new quadratics concepts in the following lessons.

  • Review factoring techniques from Algebra One
  • Recall factoring techniques including Trinomial Factoring ("reverse FOIL"), GCF factoring, difference of squares, and perfect square trinomials
Properties of Quadratic Functions

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Quadratic Functions

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Tags

  • Quadratics
  • Functions
  • Graphing
  • Coordinate Plane

Priority: VIP Knowledge

In Algebra One we learned a little bit about factoring and solving quadratic equations. In Algebra Two we're going to study quadratics in detail. To start, this lesson will define quadratics in perspective of functions, and review some common characteristics all quadratics share, both graphically and algebraically. Some important concepts we'll go over include the meaning of the intercepts on the graph, how to find the coordinates of the vertex, and how to tell if a given number is a root of the quadratic.

  • Define a quadratic function
  • Know and recognize basic characteristics common to all quadratics
  • See how and why the shape of graphs of quadratic functions is a parabola
  • Find the coordinates of the parabola vertex and the y-intercept
  • Find how many roots a quadratic has and what kind based on the graph
  • Verify that a given number is a root of a quadratic function
Forms of Expressing a Quadratic Function

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Quadratic Functions

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Tags

  • Quadratics
  • Functions

Priority: Normal

Though we most commonly are presented with quadratics in what we call "standard form" ($ax^2 + bx + c$), there are two other forms that we can manipulate a quadratic into. This is very useful in the right situation, as each form has special properties. This lesson familiarizes us with the three forms (Note: a later lesson is dedicated to changing back and forth between the three forms).

  • Recognize the three forms that a quadratic might be arranged in
  • Know the advantages each form has
Graphing Quadratic Relationships

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Quadratic Functions

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Tags

  • Quadratics
  • Graphing

Priority: High

When asked to graph a quadratic from scratch, we can use the common properties that all quadratics share that we recently learned to make the process structured. We will zone in on how to graph quadratic functions in three ways - one for each of the three quadratic forms we now know.

  • Learn the common ideas to graphing a quadratic
  • Know how to graph a quadratic function that is presented in standard form
  • Know how to graph a quadratic function that is presented in factored form
  • Know how to graph a quadratic function that is presented in vertex form
Changing Quadratic Forms

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Quadratic Functions

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Tags

  • Quadratics
  • Simplifying
  • Algebra Manipulation

Priority: Normal

Toward the beginning of this section on quadratics we needed to know the three common forms that quadratic equations come in. Now that we know more about quadratics, we can practice changing back and forth among the three forms - which we are sometimes required to do.

  • Recall what the three quadratic forms are and why each is useful
  • Learn how to change a quadratic back and forth among standard form, factored form, and vertex form

Using Quadratic Relationships - Applications of Quadratics

Problem Solving with Quadratic Factoring

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Factoring
  • Word Problems

Priority: Normal

We will continue what we learned in the last lesson to solve equations that require quadratic factoring. Here, however, we will need to give answers in context of a real-world problem, and sometimes we will have to build the equation to solve instead of being supplied with the equation to solve.

  • Review techniques for solving word problems
  • Setup equations for number based, geometry, and real world word problems, and then solve the problem
Quadratic Solutions via Graphing Calculator

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Graphing
  • Calculator Skills

Priority: Normal

Sometimes it will be sufficient to solve a quadratic equation with a graphing calculator, even though we usually only get answers as decimal approximations. This lesson will show you the basics of using the TI-83/84 functionality, as well as a few other common graphing calculator interfaces such as the Casio graphing calculator.

  • Learn how to solve any quadratic equation quickly using a graphing calculator
  • Learn how to find the vertex of any quadratic using a graphing calculator
Problem Solving with Quadratic Functions

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Solving Equations
  • Word Problems

Priority: Normal

Now that you're an expert on solving quadratic equations, let's look at real world situations that are modeled with quadratic behavior. You will practice setting up your own equation and then solving it.

  • Set up a quadratic equation from a word problem
  • Know to restrict the domain subjectively for realistic possible values of $x$ in context of the problem
  • Become familiar with common situations that have quadratic behavior
Equations in Quadratic Form

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Solving Equations

Priority: Normal

Looks at how to solve equations that are of the general quadratic form (variable squared + variable + constant) but not necessarily $ax^2 + bx + c$.

  • Recognize a general quadratic form, not just the typical $ax^2 + bx + c$
  • Apply factoring techniques that we already know to these special quadratics
  • Solve general quadratic form equations using the quadratic formula.
Systems of One Variable Quadratic Equations

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Systems of Equations
  • Solving Equations

Priority: Optional

In Algebra One we learned how to solve systems of linear equations, which was conceptually equivalent to finding the place that two straight lines intersect. Here we will do the same thing but with quadratics, so that we are conceptually looking for the places where two parabolas intersect.

  • Solve systems of quadratic equations visually
  • Solve systems of quadratics equations algebraically
  • Know when we do and do not need a calculator for this
Quadratic Inequalities

Algebra Two  $\rightarrow$  Using Quadratic Relationships  $\rightarrow$  Applications of Quadratics

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Tags

  • Quadratics
  • Inequalities
  • Graphing

Priority: Normal

When you have an inequality that involves a quadratic expression, we can still use our quadratic knowledge but we have to modify our thinking slightly. We'll see how to methodically get the solutions to quadratic inequalities.

  • Learn how to solve quadratic inequalities algebraically
  • Learn how to graph quadratic inequalities as a two-variable relationship
  • Learn how to solve quadratic inequalities with a graph

Polynomial Functions - Analyzing Polynomials

Polynomial Properties Overview

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Review
  • Quadratics

Priority: Normal

This lesson starts with a quick overview of everything we know about polynomials up to this point, including what they are, what they are not, and important vocab that we should know. We'll also re-visit the GCF and Factor-by-Grouping factoring techniques.

  • Be able to identify whether a given object is or is not a polynomial
  • Re-visit important vocal terms such as degree, coefficient, leading term, etc.
  • Discern the degree of a polynomial for single variable and multi-variable polynomials
  • Review the GCF factoring and Factoring by Grouping techniques
Sums and Differences of Cubes

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Factoring
  • Algebra Rules

Priority: Normal

Similar to the Difference of Squares formula we already know, the Sum and Difference of Cubes formula provides a successful factoring approach to expressions of the form $a^3 \pm b^3$.

  • Know and (probably) memorize the sum of cubes and difference of cubes formulas
  • Practice using the formulas on simple cases of sums and difference of cubes
  • Use GCF factoring and other prior techniques for complex expressions of sums and differences that have one or more variables
Solving Polynomials with Roots and the Factor Theorem

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Factoring
  • Solving Equations
  • Algebra Rules
  • Polynomials

Priority: High

Factoring and using the zero product property is one of the most common ways we solved quadratic equations in the past. We'll extend our prior knowledge to learn how we can use the same property to solve general polynomials, and understand the connection between the roots of a polynomial (the numbers that make the polynomial zero) and the factors of a polynomial (the expressions that divide evenly into the polynomial).

  • Recall what the zero-product property of equations is and how it works
  • Understand what factors of polynomials are by definition
  • Use the zero-product property to solve polynomials that are already factored
  • Understand and use the Factor Theorem for polynomials
  • Define what a root of a polynomial is and what root multiplicity means
  • Use basic factoring techniques to factor polynomials of degree 3 and higher, and then solve the equation using the zero-product property
Solving Polynomial Inequalities

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Polynomials
  • Inequalities

Priority: Optional

This lesson expands on a basic principle we used when we looked at quadratic inequalities. We will use the zeros of the polynomial and sign analysis to find the solution ranges of polynomial inequalities.

  • Use polynomial zeroes and sign analysis to solve polynomial inequalities
Polynomial Long and Synthetic Division

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Polynomials
  • Dividing

Priority: Normal

First we'll look at long division - that's right, a throwback to grade school, compadre. But instead of numbers, we will divide polynomials by polynomials. We will also learn a shortcut method of polynomial division called "synthetic division" that saves a ton of time and writing, and understand when we can and cannot use it.

  • Remember how to divide polynomials by monomials
  • Use the long division technique for division of any polynomial by any other polynomial
  • Learn how to setup polynomial division using shorthand notation that we call "Synthetic Long Division"
  • Practice correctly using this method as an alternative to long-hand polynomial long division notation
  • Know that long-form polynomial long division is required when dividing by non-linear factors
The Remainder Theorem

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Analyzing Polynomials

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Tags

  • Polynomials
  • Dividing

Priority: Normal

There exists an important connection between the result we get when we plug a specific value $c$ in for $x$ in any polynomial $P(x)$, and the result we get when we divide the polynomial by $(x-c)$. The Remainder Theorem tells us what that connection is, and in this lesson, we'll not only see when this information is useful, but also how teachers typically structure quiz questions on this topic.

  • Understand and use the Remainder Theorem for polynomials
  • Decide when long division is an easier way to solve a problem than plugging in values, and vice versa

Polynomial Functions - Advanced Polynomial Properties

Polynomial Graphs and End Behavior

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Graphing

Priority: High

Here we will look more closely at the relationship between characteristics of polynomials and what those characteristics mean when the polynomial is graphed, including degree, leading coefficient, and root multiplicity. We will also discuss "end behavior" and what we can know about it with and without a graph to look at.

  • Understand common properties that all polynomial graphs share
  • See how properties that are specific to the degree of the polynomial affect the shape of its graph
  • Understand how we identify and describe "end behavior" of polynomials
  • Understand the effect of root multiplicity on the graph of the polynomial
  • Be able to sketch a function based on its algebraic function form, either factored or not factored
Theorems for Roots of Polynomials

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Algebra Rules

Priority: Normal

There are several important theorems about polynomials that help us find all the roots of a polynomial, including the ones that are easy to miss. This lesson shows us the most important theorems, and we'll see a few more in the next lesson.

  • Use the Fundamental Theorem of Algebra to determine the number of roots of a polynomial
  • Understand and use the rational roots (aka rational zeroes) theorem
  • Specify a minimum number of real roots for odd degree polynomials
Advanced Theorems for Roots of Polynomials

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Algebra Rules
  • Solving Equations

Priority: Normal

Though they are used infrequently, there are a few theorems about properties of polynomial roots that we are expected to know. This lesson looks at a handful of these theorems, and how they can be used.

  • Learn two theorems about roots that always come in conjugate pairs
  • Understand Descartes's Rule of Signs as well as when and how to use it
  • Discern the pattern between the coefficients of the polynomial and either the sum or product of its roots
Solving Advanced Polynomial Equations

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Factoring
  • Solving Equations

Priority: Normal

This lesson combines the knowledge from the last two lessons, and reviews a few facts that we already knew about polynomials so that we can solve higher degree polynomial equations by hand if needed.

  • Use the theorems from the prior two lessons to solve higher degree polynomials by hand
  • Break a polynomial of degree n down completely so that all $n$ roots are identified
Finding a Polynomial from Its Roots

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Algebra Rules

Priority: Normal

Working backward to find a polynomial solely based on knowing its roots is a great way to recap the wealth of information you now have regarding working with polynomials. Many courses test this skill explicitly as well.

  • Understand why knowing only the roots of a polynomial is not sufficient enough to know exactly what the polynomial is
  • Utilize the pattern between the coefficients of the polynomial and either the sum or product of its roots
  • Leverage relationships among the roots to specify the exact polynomial
Using Technology with Polynomials

Algebra Two  $\rightarrow$  Polynomial Functions  $\rightarrow$  Advanced Polynomial Properties

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Tags

  • Polynomials
  • Calculator Skills
  • Graphing

Priority: Optional

Some of the common things we study about polynomials can be handled in a flash with your graphing calculator. Sometimes the calculator cannot get you the exact, correct answer you need, but rather a decimal approximation - but sometimes that's all we need. This lesson shows you how to accomplish several tasks related to our study of polynomials, including solving a polynomial equation, finding the roots of a polynomial, and finding places where the polynomial has a relative max or min value.

  • Learn how to use your TI or graphing calculator to solve a polynomial equation
  • Learn how to use your TI or graphing calculator to find the roots of a polynomial
  • Learn how to use your TI or graphing calculator to find a relative max or min point of the polynomials

Radical and Rational Relationships - Advanced Radicals and Roots

The Complete List of Exponent Rules

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Advanced Radicals and Roots

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Tags

  • Review
  • Exponents

Priority: VIP Knowledge

This VIP Knowledge lesson starts with the exponent rules that we've learned about so far, and adds a couple more on to the list. We must be absolute exponent rule masters, both for the entirety of this Mr. Math Section, and for almost every part of our future math career!

  • Pull together every exponent rule we've ever learned
Square Root Overview for Advanced Algebra

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Advanced Radicals and Roots

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Tags

  • Review
  • Square Roots
  • Integers
  • Monomials
  • Variable Expressions

Priority: Normal

Before we directly relate square root operations with exponents, we must review and re-master taking square roots, of both integers and variable expressions, as well as simplifying down in both cases.

  • Remaster everything we know about taking square roots and what square roots are
  • Practice simplifying square root expressions involving integers only
  • Practice simplifying square root expression involving variables and general monomials
Integer n-th Roots

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Advanced Radicals and Roots

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Tags

  • Square Roots
  • Radical Expressions
  • n-th Roots
  • Simplifying

Priority: VIP Knowledge

Although, we're already familiar with square roots and how to work with them, we need to be able to take other roots - like third roots, fourth roots, or n-th roots, where n could be anything. We'll introduce these types of roots, understand how they work, and then practice simplifying them for roots with only integers underneath (we'll look at n-th roots of variables in the next lesson).

  • Define the n-th root of a number, where $n$ is an integer
  • Learn how to take the n-th root of whole numbers, when $n$ is an integer
  • Learn how to simplify or reduce an n-th root of an integer to its simplest form, similar to how we learned to reduce square roots
  • Know how to solve simple n-th root equations involving integers
  • Correctly identify positive and negative answers depending on whether you are simplifying an expression or solving an equation
Simplifying n-th Root Variable Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Advanced Radicals and Roots

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Tags

  • Square Roots
  • Radical Expressions
  • n-th Roots
  • Simplifying

Priority: High

Taking n-th roots of variable expressions requires that we deal with two types of objects: the coefficient and its variables. While we will practice doing both at once, the focus of this lesson is on n-th roots of variables, since the n-th root of the coefficient is something we can already do using what we learned in the lesson immediately prior. Variable expression n-th roots have a very basic simplifying process that we'll want to master before the upcoming lesson on multiplying these types of expressions.

  • Simplify monomial variable expressions involving n-th roots
  • Learn when to apply absolute value bars in your answers depending on the instructions in the problem

Radical and Rational Relationships - Working with Advanced Roots

Arithmetic with n-th Roots

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • Square Roots
  • Radical Expressions
  • n-th Roots
  • Simplifying

Priority: Normal

Now that we understand a bit about what n-th roots are and how to simplify them, we now turn to how to work with them when it comes to performing arithmetic. We'll multiply, add, and subtract various root expressions in this lesson.

  • Add and subtract root expressions if possible
  • When it is not possible to add or subtract, understand why
  • Learn to multiply n-th root expressions together and obtain the simplest final answer
Fractions and Rationalizing with n-th Roots

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • Radical Expressions
  • n-th Roots
  • Fractions
  • Rationalizing

Priority: Normal

This lesson focuses on applying n-th roots to fractions and rationalizing when the denominator is an n-th root, by using similar mechanics and ideas as the ones we saw in our study of applying square roots to fractions in Algebra One.

  • Learn how to take the n-th root of a fraction
  • Learn how to rationalize simple n-th root denominators
  • Get tips for special complex n-th root denominators
Fraction Exponents

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • Radical Expressions
  • n-th Roots
  • Fractions
  • Square Roots

Priority: VIP Knowledge

Now that we can take any root of an object and/or raise an object to any power, we introduce a very important exponent operation that lets you do both at once. This lesson introduces fraction exponents, which are a more consistent way to express a combination of roots and powers. For example, squaring a variable and then taking the third root can be done all at once, and the result is more easily obtained when we use fraction exponents. We will also see how to solve very simple equations that involve fraction exponents.

  • Understand why taking an n-th root is the same as raising to the $1/n$ power
  • Interpret fraction exponents of the form $a/b$ and know best practices for using them
  • Learn how to solve basic equations with a given variable raised to the $a/b$ power
Using and Graphing Radical Functions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • n-th Roots
  • Square Roots
  • Functions
  • Graphing

Priority: High

Having learned more about roots recently, we can study functions that have root expressions in them, and get to know and love the properties that such functions have.

  • Know the general properties of radical functions, including domain and range
  • Graph radical functions and know common graph properties
Power Functions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • Radical Expressions
  • n-th Roots
  • Functions
  • Graphing

Priority: Optional

Although we have thoroughly studied polynomial functions and their behavior, there is an "in-between" type of function that has terms that look similar to polynomial ones, but without integer exponents. These Power Functions have similar properties but have a few unique differences.

  • Learn what makes a general Power Function similar to and different from a polynomial function
  • Study the graphical properties that are unique to Power Functions
Solving Any Root Equation

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Working with Advanced Roots

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Tags

  • n-th Roots
  • Square Roots
  • Solving Equations

Priority: High

This lesson is a very comprehensive exploration into how to solve equations that have a root or radical expression in them. We'll start with square roots, and then use the rational expressions knowledge we have from the last lesson to solve radicals that are not square roots, both for single variable radicals as well as roots of entire expressions.

  • Review solving equations with just one radical from Algebra One
  • Practice the recently learned "exponent neutralizing" method of solving for variables raised to fraction exponent powers
  • Learn how to solve for $x$ when the equation has two square root expressions
  • Know that results often yield extraneous solutions and that all solutions of radical equations must be verified

Radical and Rational Relationships - Rational Expressions

Simplifying Polynomial Rational Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Factoring
  • Simplifying
  • Algebra Rules

Priority: VIP Knowledge

In this very important lesson, we'll see how to simplify rational expressions with objects like quadratics that can first be factored, which can create canceling linear factors. In short, we'll see how factoring allows us to reduce a fraction with variables that otherwise could not be simplified.

  • Learn methods for simplifying rational expressions that contain binomials or polynomials
  • Use factoring methods to either simplify rational expressions or classify them as irreducible
Multiplying and Dividing Rational Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Factoring
  • Simplifying
  • Multiplying
  • Dividing
  • Polynomials

Priority: High

When we multiply fractions that contain variable expressions, we might be wasting a LOT of time if we don't first factor them. When we factor first, the result might contain factors that cancel out and thus make the final answer much more simple. We also can handle the division of two rational expressions in a similar way, by first turning it into multiplication, and then proceeding with the same process.

  • Learn how to cancel common factors before multiplying rational polynomial expressions
  • Understand why failing to cancel factors before multiplying actually makes the problem impossible to simplify
  • Quickly turn the division of a rational polynomial expression into a multiplication problem
Common Denominators for Rational Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Factoring

Priority: Normal

To prepare ourselves to combine rational expressions, we will need to zone in on taking a set of rational expressions and identifying what their least common denominator is, as well as whichever extra factors each fraction will need to be turned into one with the least common denominator.

  • Revisit LCD and LCM for variable expressions
  • Identify the simplest common denominator of two fractions
  • Determine the missing factors that each original denominator needs to turn into the common denominator
  • Convert given rational expressions to have a common denominator
Adding and Subtracting Rational Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Algebra Rules
  • Adding
  • Subtracting

Priority: Normal

Rational expressions can be added and subtracted, just like numeric fractions. In fact, the same rule applies - the denominators must be the same before you can. In this lesson we'll see how to get rational expressions to have the same denominator, and then we'll add or subtract them.

  • Find the simplest common denominator in an addition or subtraction problem
  • Know why finding the "not simplest" common denominator will create a bad situation
  • Add or subtract expressions that have common denominators and simplify the answers completely
Complex Fractions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Algebra Manipulation
  • Simplifying

Priority: Normal

There are many ways to think about how fractions work, and many ways to make the math more manageable. From prior knowledge, we know that if we want to divide by a fraction, we can instead multiply by its reciprocal. This lesson shows you how to deal with similar sticky situations where fractions contain fractions.

  • Define and understand what a complex fraction is
  • Learn methods for simplifying complex fractions
  • Use the fraction bar and grouping symbols to correctly determine complex fraction numerators and denominators

Radical and Rational Relationships - Using Rational Expressions

Solving Rational Equations

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Using Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Solving Equations
  • Algebra Rules

Priority: High

With what we now know about rational expressions, we are ready to solve equations that contain them. This lesson looks at how to solve each of the three main flavors of these equations - purely numeric denominators, variable denominators with equations equal to zero, and variable denominators with equations equal to anything.

  • Get familiar with categorizing a given equation as rational or not
  • Learn the three major types of rational equations we solve in Algebra
  • Review least common denominator requirement and why other common denominators are not good enough
  • Understand why extraneous solutions often occur in rational equations, and consequently why we must always check solutions for validity
Rational Inequalities

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Using Rational Expressions

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Tags

  • Fractions
  • Rational Expressions

Priority: Optional

If a rational expression appears in an inequality with variables in the denominator, the complete solution set might not be obvious. Here we learn how to systematically solve the inequality thoroughly in such cases.

  • Learn and practice a strategy for solving inequalities that contain rational expressions
  • Recall and use the sign analysis technique with intervals on a number line
Using and Graphing Rational Functions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Using Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Functions
  • Graphing

Priority: VIP Knowledge

This lesson examines rational functions and their behavior, first in the case where the denominator is a linear factor, and then generally where each the numerator and denominator can be any polynomial. After this lesson, we will be able to identify several key properties for any given rational functions, including asymptotes, intercepts, and general graph shape.

  • Analyze one-variable rational functions that contain a rational expression with a linear denominator
  • Understand the three cases of this function form when the denominator is of the form $ax+b$
  • Examine the three cases of general rational functions where both the numerator and denominator are polynomials
  • Discern the domain and range of rational functions
  • Determine which types of rational functions do and do not have zeros
  • Find vertical, horizontal, and slant asymptotes, when they exist
  • Understand the common properties and shapes of the graphs of rational functions
  • Learn how to graph these functions by hand, as well as identify a function based on its graph
Proportions with Variable Expressions

Algebra Two  $\rightarrow$  Radical and Rational Relationships  $\rightarrow$  Using Rational Expressions

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Tags

  • Fractions
  • Rational Expressions
  • Review
  • Multiplying

Priority: Optional

This quick optional lesson shows us the finer points of how to handle cross multiplication when variable expressions are involved.

  • Recall what proportions are and how they work
  • Learn the Means Extremes Theorem and what its used for
  • Set up proportions with the variable in two places, and solve for the variable

Matrices - Matrices and Arithmetic

What Matrices Are

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Matrices and Arithmetic

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Tags

  • Matrices

Priority: Normal

Matrices are a somewhat oddball topic. Whether or not you have ever seen them before now, this lesson will help us understand exactly what matrices are, and see some important similarities and differences between matrices and numbers.

  • Better understand what matrices are as objects
  • Relate important similar concepts between numbers and matrices, such as the multiplicative identity and zero product property
  • Learn what scalars are and how they operate on matrices multiplicatively
Basic Matrix Arithmetic

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Matrices and Arithmetic

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Tags

  • Matrices
  • Adding
  • Subtracting

Priority: Normal

This lesson will outline how to add or subtract two matrices, as well as how to multiply a matrix by a scalar.

  • Learn which matrices can and cannot be operated on with addition and subtraction
  • Practice performing matrix addition and subtraction
  • Apply multiplicative scalars to matrices
Matrix Multiplication

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Matrices and Arithmetic

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Tags

  • Matrices
  • Multiplying
  • Algebra Rules

Priority: Normal

Multiplying matrices is a bit tedious, but its a staple item on the menu of things your teacher will serve up on a test, when you study matrices. It's also occasionally useful to know in the future, having a few important applications in probability and statistics. This lesson covers the basics and then some, for everything there is to know about multiplying two matrices together.

  • First, understand when two matrices can and cannot be multiplied together
  • Learn the pattern-based approach for systematically multiplying two matrices
  • Understand why the associative property of multiplication FAILS for matrices, and thus multiplication order matters
  • For square matrices, learn how the identity matrix and the zero matrix act just like the numbers 1 and 0 do for number multiplication, respectively
Matrix Arithmetic with Technology

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Matrices and Arithmetic

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Tags

  • Matrices
  • Adding
  • Subtracting
  • Multiplying
  • Calculator Skills

Priority: Optional

While you'll never ever get a test where your teacher intends for you to just plain add and multiply matrices with technology, sometimes we are allowed a calculator. This lesson will teach you how to get your TI (or some common equivalent calculators) to perform your matrix adding and multiplying for you.

  • Learn how to store matrices in the memory of a TI or equivalent graphing calculator, similar to how to store numbers in memory
  • Get your calculator to crunch your matrix addition and multiplication operations

Matrices - Solving Systems of Equations with Matrices

Solving Systems of Equations with Matrices

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Solving Systems of Equations with Matrices

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Tags

  • Matrices
  • Systems of Equations

Priority: VIP Knowledge

This valuable lesson will ultimately show you a commonly tested process - how to solve systems of equations with matrices. First we will turn a system into a special matrix. Then we will use row operations to manipulate the matrix until it is in a form that gives us the solution to the system.

  • Learn how to translate a two or three variable system of equations into an augmented matrix
  • Learn how to manipulate augmented matrices with row operations
  • Master the method of Gauss-Jordan Elimination to turn an augmented matrix into what we call "Row Echelon Form"
  • Turn Row Echelon matrices into identify form matrices so that the solution is plainly obtained
Matrix Determinants

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Solving Systems of Equations with Matrices

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Tags

  • Matrices
  • Number Properties

Priority: High

Matrices have a special number associated with them called a determinant, which is useful for a few upcoming concepts and shortcuts. For now, we will learn how to find the determinant of a matrix, as well as which kinds of matrices do and do not have a determinant.

  • Define what the determinant of a matrix is
  • Understand which types of matrices do and do not have determinants
  • Learn how to manually calculate the determinant of smaller matrices
  • Learn how to theoretically determine the determinant of larger matrices
Inverse Matrices

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Solving Systems of Equations with Matrices

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Tags

  • Matrices
  • Number Properties

Priority: Normal

Just like numbers have inverses (which we called reciprocals), so too do matrices. But just like multiplication of matrices is more complex than multiplication of numbers, so too are matrix inverses. This lesson defines what an inverse matrix is, and how we can use them in a practical way.

  • Understand what an inverse matrix is, and how they are like the matrix world equivalent to reciprocals in the world of numbers
  • Find the inverse matrix of a 2x2 matrix explicitly using a formula
  • Solve two variable systems of equations instantly without row operations by applying inverse matrices
Solving Systems with Cramer's Rule

Algebra Two  $\rightarrow$  Matrices  $\rightarrow$  Solving Systems of Equations with Matrices

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Tags

  • Matrices
  • Systems of Equations

Priority: Normal

If you are familiar and comfortable with finding matrix determinants, you can utilize Cramer's rule for solving two or three variable systems of equations, which is essentially a massive shortcut. In this lesson we will see how Cramer's rule works and practice using it.

  • See and understand how Cramer's Rule works
  • Practice solving systems of equations with matrices via Cramer's Rule for 2x2 or 3x3 (two or three variable, respectively) matrix systems.