Study Guide

# Algebra II

## Algebra II Introduction

• ### Introduction to Algebra II

We've made it through the trenches of Algebra. We've learned the ins and outs of variables, functions, and all of those rules. (So many rules.)

Now it's time to solve some of the great mysteries of life. What is the natural number, and why is it so friendly with logs? What is The Matrix? Are we inside it right now? To answer these questions, we need to learn more rules. (Yes, more rules.)

### Why Should I Care?

Algebra II is another stepping stone to holding all of the math power. Not only will we be able to solve for x, we'll be able to solve for x when it's in messy equations involving x2. Better yet, once we've mastered Algebra II, we'll have what it takes to handle all kinds of questions that the real world might throw at us. We'll be able to better understand concepts such as exponential growth. Ever wonder how fast bacteria really grow? It's exponential.

It'll also add more to our arsenal in order to work through any problem we'll encounter in Pre-calculus, Calculus, and beyond.

• ### What is Algebra II?

Algebra II is an extension of Algebra I. We'll still look at functions, graphs, and variables, but this time we'll step it up a notch.

One of the big boys in Algebra II is the quadratic equation. This guy seems pretty mean and tough at first, but it's really a teddy bear. It is, at least, once we use the magical quadratic formula on it. Poof. Done.

We'll also learn the ins and outs of the natural number and the natural log. By the end, dare we say, working through these types of problems will come naturally.

• ### Basic Elements

What makes Algebra II go 'round? One of the basic elements is the concept of the equation. That's right, you've seen them before, and you're about to see them again—and again. We'll get to know them really well, and we'll get to know new sides to them. We'll learn how to incorporate new types of terms, and we'll be able to eventually solve them without even flinching.

We'll also spend some more quality time with functions. These will show up in several ways, including in equations and graph forms. What's it mean to be one-to-one? This basic stuff will help get you through higher math, such as calculus, so take the time to learn it now.

Just like you can figure out the rules of a new game, Algebra II has its own set for you to learn. Remember adding and subtracting integers? Those were rules, and you mastered them. You'll learn a more complicated set of rules; just think of them as an expansion pack. Game on.

Two other basic elements that often go together, and are important to get a feel for, are sequences and series. These guys can be confusing, which is why it's important to practice them. You've come to the right place, because we have all kinds of practice problems ready to go.

All of these basic elements can be useful…in their own way. Enter the real-world applications.

• ### Applications in Science and Engineering

Algebra II has tons of real-world applications. Power functions are often used to model things in nature, like how populations grow. If we wanted to model the trends that make a newly uploaded video go viral, we could use what we learn in Algebra II to do just that. Future biologists, get your natural number on.

From supply and demand to compound interest, we need the basics of Algebra II to master the math we'll encounter in economics. Not only that, but Algebra II will help us understand the graphs we see on a day-to-day basis. These might include graphs of stock market trends or graphs of the cuteness of kittens. (The latter tends to be exponential.)

Matrices are also important in science. In physics, we can learn a little thing call the cross product that'll make electricity and magnetism (slightly) easier. Speaking of physics, it's possible to get through a lot of basic calculations once you've mastered this topic.

• ### What is Algebra II?

Algebra II is an extension of Algebra I. We'll still look at functions, graphs, and variables, but this time we'll step it up a notch.

One of the big boys in Algebra II is the quadratic equation. This guy seems pretty mean and tough at first, but it's really a teddy bear. It is, at least, once we use the magical quadratic formula on it. Poof. Done.

We'll also learn the ins and outs of the natural number and the natural log. By the end, dare we say, working through these types of problems will come naturally.

• ### Basic Elements

What makes Algebra II go 'round? One of the basic elements is the concept of the equation. That's right, you've seen them before, and you're about to see them again—and again. We'll get to know them really well, and we'll get to know new sides to them. We'll learn how to incorporate new types of terms, and we'll be able to eventually solve them without even flinching.

We'll also spend some more quality time with functions. These will show up in several ways, including in equations and graph forms. What's it mean to be one-to-one? This basic stuff will help get you through higher math, such as calculus, so take the time to learn it now.

Just like you can figure out the rules of a new game, Algebra II has its own set for you to learn. Remember adding and subtracting integers? Those were rules, and you mastered them. You'll learn a more complicated set of rules; just think of them as an expansion pack. Game on.

Two other basic elements that often go together, and are important to get a feel for, are sequences and series. These guys can be confusing, which is why it's important to practice them. You've come to the right place, because we have all kinds of practice problems ready to go.

All of these basic elements can be useful…in their own way. Enter the real-world applications.

• ### Applications in Science and Engineering

Algebra II has tons of real-world applications. Power functions are often used to model things in nature, like how populations grow. If we wanted to model the trends that make a newly uploaded video go viral, we could use what we learn in Algebra II to do just that. Future biologists, get your natural number on.

From supply and demand to compound interest, we need the basics of Algebra II to master the math we'll encounter in economics. Not only that, but Algebra II will help us understand the graphs we see on a day-to-day basis. These might include graphs of stock market trends or graphs of the cuteness of kittens. (The latter tends to be exponential.)

Matrices are also important in science. In physics, we can learn a little thing call the cross product that'll make electricity and magnetism (slightly) easier. Speaking of physics, it's possible to get through a lot of basic calculations once you've mastered this topic.

• ### Major Algebra II Themes

There are certain things that we see pop up time and time again in Algebra II. These themes include graphs, functions, equations, and new notation. It's also important to notice that there are certain types of graphs that we'll run into more than once.

We've met them all before, but we were just acquaintances back then. Sure, we'd wave "hello" to them in the hallway, but we never really got to know them. This time around, we'll approach these themes from all sorts of angles. Not only that, but we'll start to learn about how they're all related to each other. That's gonna be one bizarre family reunion.

• ### Graphs

When we start squaring things, graphs start to get interesting. Instead of just being straight lines, we start seeing curves. Functions with exponents often tend to be moody. They either see the glass half empty, or half full, depending on how they're feeling. There are lots of positives and negatives going around here. So many, that we'll start to get really good at drawing these: ±.

What happens when we throw a <, >, ≥, or ≤ into the mix? These graphs are more than pretty pictures. Or so we've been told.

• ### Functions

Just like in Algebra I, we'll be introduced to brand-spanking-new functions. Some of these will be familiar (linear functions) and some won't (crazy quadratic and exponential functions). As all new things in math, remember that these functions are new for everyone. Learn to love them, and they'll help you get through yet another level of math with flying colors.

• ### Equations Systems

Another major theme in Algebra II is groups of equations, all rolled into one. This might seem complicated at first, but there are ways to organize the chaos. One way is using matrices. Yes, the singular is matrix. And yes, we're all in The Matrix.

In addition to matrix notation, we'll also introduce the concepts of series and sequences. This section might go on…and on…and on…but it's another helpful way to keep track of lots of numbers and equations.

• ### New Notation

Some notation will be new in Algebra II. These include the dreaded matrix notation (disclaimer: it isn't that bad) and sigma notation. Others notations aren't so new, but we may have put them on the shelf hoping they'd never re-appear. Too bad—they're back. Exponents, roots, powers, and logarithms will all make an appearance.

• ### Graphs

When we start squaring things, graphs start to get interesting. Instead of just being straight lines, we start seeing curves. Functions with exponents often tend to be moody. They either see the glass half empty, or half full, depending on how they're feeling. There are lots of positives and negatives going around here. So many, that we'll start to get really good at drawing these: ±.

What happens when we throw a <, >, ≥, or ≤ into the mix? These graphs are more than pretty pictures. Or so we've been told.

• ### Functions

Just like in Algebra I, we'll be introduced to brand-spanking-new functions. Some of these will be familiar (linear functions) and some won't (crazy quadratic and exponential functions). As all new things in math, remember that these functions are new for everyone. Learn to love them, and they'll help you get through yet another level of math with flying colors.

• ### Equations Systems

Another major theme in Algebra II is groups of equations, all rolled into one. This might seem complicated at first, but there are ways to organize the chaos. One way is using matrices. Yes, the singular is matrix. And yes, we're all in The Matrix.

In addition to matrix notation, we'll also introduce the concepts of series and sequences. This section might go on…and on…and on…but it's another helpful way to keep track of lots of numbers and equations.

• ### New Notation

Some notation will be new in Algebra II. These include the dreaded matrix notation (disclaimer: it isn't that bad) and sigma notation. Others notations aren't so new, but we may have put them on the shelf hoping they'd never re-appear. Too bad—they're back. Exponents, roots, powers, and logarithms will all make an appearance.

• ### Key Skills

So here's the thing: this is a math class. That means one thing: there will be numbers. Okay, two things. There will be a lot of building up from the stuff we learned since…well…kindergarten.

There will be a lot of adding, subtracting, multiplying, and dividing. And yes, PEMDAS is still a thing.

We'll still be asked to solve for x. In addition to this, we will also be asked to solve for a bunch of other variables. There will even be several variables linked to several equations. We'll learn how to solve for them all.

We'll also use what we've learned about graphing and jump right in to more interesting problems. Join Shmoop, and get fired up.

• ### Algebra

It's probably not so surprising that all of the skills we learned in Algebra will be important to remember for Algebra II. The good news is that all those hours you spent studying won't be in vain. The bad news is that we might need to review a few things, or run into some topics that we weren't too thrilled about the first time around. Lucky for us, Shmoop's Algebra Guide will make it a breeze.

• ### Geometry

In geometry, we learned about triangles, circles, and all of the other shapes in which we can cut pizza and crust-less PB & J. In Algebra II, we'll be re-introduced to trigonometric functions ("trig" functions for short). You've probably spotted these before, and possibly blocked them from your mind: Sine (sin), Cosine (cos), Tangent (tan), and all of their inverses. Trig functions are useful for finding angles, or using known angles to find length. Knowing these will be especially handy once we get around to learning Pre-calculus.

• ### Algebra

It's probably not so surprising that all of the skills we learned in Algebra will be important to remember for Algebra II. The good news is that all those hours you spent studying won't be in vain. The bad news is that we might need to review a few things, or run into some topics that we weren't too thrilled about the first time around. Lucky for us, Shmoop's Algebra Guide will make it a breeze.

• ### Geometry

In geometry, we learned about triangles, circles, and all of the other shapes in which we can cut pizza and crust-less PB & J. In Algebra II, we'll be re-introduced to trigonometric functions ("trig" functions for short). You've probably spotted these before, and possibly blocked them from your mind: Sine (sin), Cosine (cos), Tangent (tan), and all of their inverses. Trig functions are useful for finding angles, or using known angles to find length. Knowing these will be especially handy once we get around to learning Pre-calculus.