Pre-Calculus—Semester A
It's the circle of math, and it moves us all.
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Type: Basic
- Category:
- Math
- High School
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Shmoop's Pre-Calculus 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.
This course has also been certified by Quality Matters, a trusted quality assurance organization that provides course review services to certify the quality of online and blended courses.
Ever wanted to see the future? To know when and where the robot apocalypse will start, or at least whether you'll win the next hand of Seven-Card Stud?
No, we're not talking about fortune-telling. It's Pre-Calculus, y'all!
Whether it's anticipating the end behavior of a gnarly-looking polynomial or predicting the odds that we'll draw the Queen of Hearts from a standard deck, our first semester of Pre-Calc is all about telling the future. Nostradamus has got nothing on us.
With readings, quizzes, and activities up the wazoo, our semester-long, Common Core-friendly course will get you up-close and personal with
- graphing polynomial functions and predicting their end behavior, especially ones fancier and more hoity-toity than your typical quadratics.
- stacking polynomials on top of each other Jenga-style to form rational functions (and anticipating the shapes of their graphs, too).
- a more in-depth look at the unit circle and its best buds, the trig functions.
- graphing those trig functions on a coordinate plane (including their evil twins, inverse trig functions).
- using probability distributions and expected values to make Sherlock Holmes-like predictions (and get the upper hand in games of chance).
So why live in the busted old present when you can predict how your math homework will look before you've even done anything? The future is now!
P.S. Pre-Calculus is a two-semester course. You're looking at Semester A, but you can check out Semester B here.
Unit Breakdown
1 Pre-Calculus—Semester A - Polynomial Functions
After a general introduction to polynomial functions and a quick rundown of some of their parts, we'll learn about inverses: how they work, and how to find them. We'll also see a few methods for factoring polynomials and finding roots, like synthetic division, the rational zeros theorem, Descartes' rule of signs, and bounds on zeros. We'll put these techniques to use by seeing how roots relate to graphs, touching on complex zeros and multiplicity. You won't exactly get rich when you rock a polynomial function, but you will gain a wealth of math-knowledge that'll save you a bunch of time in the long run. And time is money, right? Right?
2 Pre-Calculus—Semester A - Rational Functions
If you're looking for functions with a bit more height, with an entire second level worth of function on top, you might be interested in rational functions: the double-decker buses of functions. After getting to know the domains and ranges of rational functions, as well as complex fractions and partial fraction decomposition, we'll take a look at asymptotes and discontinuities. These are the gaps that often show up in rational functions, but don't worry. Unlike the "gaps" that might appear in a double-decker bus, no one's going to sprain their ankle falling through an asymptote.
3 Pre-Calculus—Semester A - The Unit Circle
Before Netflix, before HBO, before red carpet, before writing was invented (much less People magazine), the stars were the original entertainment celebrities. (Hence the name, we presume.) The dance of the heavenly spheres was so important to ancient people that they invented trigonometry just to be able to keep track of it better. Now it's our turn to follow in their footsteps and dance the trigonometry dance. And at the heart of it all sits the unit circle. It's not quite a heavenly sphere, but it'll do in a pinch.
4 Pre-Calculus—Semester A - Trigonometry
Now that we've gotten nice and cozy with the unit circle, we can use all the fancy doodads and whatsits we've gathered from its geometry and apply them all to algebra. Drop that compass and grab the nearest graphing calculator (tell your neighbor it'll only be a second), because we've moved on from trig ratios to trig functions. This unit's chock-full of identities, formulas, inverses, and even (gasp!) proofs, so make sure you've got the trig and algebra chops to swim, not sink. And if you do find yourself sinking, the unit circle doubles as a great flotation device.
5 Pre-Calculus—Semester A - Probability & Statistics
Statistics is so hard to get rid of because it's so applicable. We use it to make decisions, set trends, and gamble away our life's savings. (Fun!) The question is: should we be followers, or leaders, in this next round of probability and statistics, a unit jam-packed with dice, cards, coin flips, and all the usual probabilistic suspects? Maybe we'll check in with our new friend, Expected Value, and see what he has to say.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson 2.04: Rewriting Complex Fractions
Now that you know you can make expressions where the numerator and denominator are both polynomials, you might ask yourself: what about an expression where the numerator and denominator are both rational functions? Is it possible to construct ever more complex expressions made up of other expressions, with the structural elegance of a Russian nesting doll? Or is it more like just shoving bags full of bags into other bags till you have a whole bunch of bags (in a whole bunch of bags)?
As we'll see, the truth is somewhere in between. These hypothetical expressions made out of rational expressions do exist: they're called complex fractions. They're not the most elegant things in the world, though, and we'll spend a lot of energy trying to simplify them. It's harder than opening up Russian dolls, but not as hard as untangling a big pile of bags.
Sample Lesson - Reading
Reading 2.2.04: We Have a Simplifying Complex
To start, let's figure out what a complex fraction is. A complex fraction is a rational expression, where the numerator and denominator are themselves rational expressions. They're not the prettiest things in the world; 
is pretty standard as complex fractions go. These things are monsters.
As you might imagine, these fractions can be pretty tough to deal with, so we generally try to simplify complex fractions so that we can work with them more easily. Try to get rid of common factors, change the quotient into a division problem, or do whatever it takes to get a handle on these monstrous complex fractions.
To see this process in action, let's consider an example:
Oh boy. First things first: let's try to simplify this complex fraction by making it into a division problem rather than this uber-fraction:
At this point, it's worth seeing if we can do some factoring and canceling. It takes a little work, but we can do it:
We now see some opportunities for cancellation: (x – 3), (x + 2), and 7 can all be canceled out, yielding a simpler fraction:
Boom. Doesn't that look nicer?
Recap
A complex fraction is a rational expression with a nasty case of inception—the numerator and denominator of a complex fraction are themselves rational expressions.
We generally try to simplify complex fractions so that we can work with them more easily. We can turn the fraction into a division problem, turn that into a multiplication problem, look for things to factor and cancel—whatever it takes.
Sample Lesson - Activity
- Credit Recovery Enabled
- Course Length: 18 weeks
- Course Type: Basic
- Category:
- Math
- High School
Schools and Districts: We offer customized programs that won't break the bank. Get a quote.