Algebra II—Semester A (2014-2015)
Nothing complex here...except complex numbers.
Shmoop's Algebra II course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.
You had so much fun in Algebra that you had to come back for more? Yeah, we don't blame you.
Algebra II has all the expressions and equations you've seen before…and then some. You're sure to see some old familiar friends along the way (we're looking at you, polynomials), along with a few unfamiliar faces (we're looking—or at least trying to look at you, imaginary numbers).
Semester A starts off with expressions, polynomials, and a beautiful thing we've all seen before: factoring. After being able to rearrange polynomials in more ways than a contortionist, we'll venture into the land of the imaginary. (Feel free to extend an invite to your imaginary childhood friend, Maurice.) We'll finish up the semester by working with more equations and inequalities than you can shake a dotted line at.
With loads of readings, problem sets, and activities, we'll cover:
- interpreting, factoring, and manipulating expressions;
- multiplying and dividing polynomials and rational expressions;
- working with imaginary and complex numbers;
- creating linear, quadratic, exponential, and absolute value equations;
- graphing and solving nonlinear equations and inequalities.
P.S. Algebra II is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Unit 1. Seeing Structure in Expressions
We'll start the course off by diving headfirst into expressions! With the help of factoring, radicals, and even sequences and series, we'll be able to pluck out different parts of an expression and figure out if two expressions are two peas in a pod or two peas in…separate pods. Because sometimes, one pod is just too cramped.
Unit 2. Arithmetic with Polynomials and Rational Expressions
You may have dealt with polynomials and rational expressions in the past, but you ain't never seen 'em like this before. Not only will we gain some new tips and tricks to help us deal with these pesky expressions, we'll learn and even prove a few theorems along the way. Yeah, we mean business.
Unit 3. The Complex Number System
You might think imaginary numbers are about as helpful as Maurice, your imaginary friend from kindergarten. We'll fill you in on a little secret: while Maurice can't deal with negatives under the radical or tell you about the Fundamental Theorem of Algebra, imaginary numbers can—and will. Then again, Maurice can make a mean imaginary apple cobbler. We all have our strengths.
Unit 4. Creating Equations
This unit is all about using the magic of the equal sign to create and solve problems. (We'd use the magic of Houdini, but we're still working on that rope escape trick.) By creating and graphing every type of equation you can possibly think of, we'll learn how to tell them apart and understand which type of equation is applicable to which situation.
Unit 5. Reasoning with Equations and Inequalities
For this unit, we're going how to put all the equations we created to use. We're going to graph our creations—both separately and together—and see how to use them to solve problems. More often than not, they can do a significant portion of the work for us, and who doesn't love outsourcing their work? So put your feet up, grab a piña colada, and let equations and inequalities do the work for you. (Not really.)
Sample Lesson - Introduction
Lesson 5: Plotting Complex Numbers on the Coordinate Plane
Optical illusions can break our brains. At first, you think you see one thing, but then you realize (or are told) that things are not what they appear to be.
Our mind is about to shift again. The optical illusion: an ol' fashioned coordinate plane.
We might have always looked at an axis one way, thinking they are all the same. The x-axis represents the x-values and the y-axis represents the y-values, right? When it comes to complex numbers, it isn't that simple.
The usual Cartesian plane is only useful when we're working with real numbers. We're upgrading our numbers, though, going from real to complex, so we'll have to upgrade our plane, too.
When we throw complex numbers into the mix, we need to find a place to plot the i's. Imaginary numbers prefer hiding in dark places like under the bed or in the attic. Silly imaginary numbers.
We're going to let the imaginary numbers hang out along the imaginary-axis. That means that the real numbers will be along the x-axis. When real numbers and imaginary numbers mix it up, we'll be ready to plot them on the complex plane.
Sample Lesson - Reading
Reading 3.5: Plotting Complex Numbers on the Coordinate Plane
Graphing complex numbers is a little different than graphing real numbers. First of all, the graphs are different. If there is an imaginary component to a complex number it can't be graphed on a regular old real number axis. That wouldn't be complex enough.
We're joking, though. Really, graphing on the complex number plane isn't that bad. Like the real number plane, the complex number plane also has an x- and y-axis. However, each axis represents something different in the complex plane.
Remember that complex numbers can be split into a real and imaginary component: a + bi. The x-axis is going to be pretty much the same as ever. The y-axis, though, is going to represent the imaginary part. In other words, we can label it i, 2i, 3i, etc., going up from the origin, and -i, -2i, -3i, etc. going down.
Adding i's to the y-axis has made it possible to take any complex number in a + bi form and to locate it on the plane. The x-coordinate is the "a" and the y-coordinate covers the "bi" part. Please, hold their applause.
So, to plot 3 + 7i, we start at the origin. We go over 3 units on the x-axis, and up 7i units on the y-axis. Our point would be in the Quadrant I, right here.
If our point was -4 – 11i, we would go 4 to the left and 11i down, and plot the point in the Quadrant III. See—even though we've made things more complex, they aren't really any more complex. Y'know, in a manner of speaking.
Since complex numbers have both a real and imaginary component, we graph each of these components separately. The real component is plotted along the x-axis while the imaginary coordinate is plotted along the y-axis.
Sample Lesson - Activity
Activity 3.5c: Problem Set
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 10, 11, 12
- Course Type: Basic
Algebra I—Semester A
Algebra I—Semester B
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following standards are covered in this course:A-REI.4b