Pre-Calculus—Semester B
We'll make a conehead out of you.
- 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.
Let's face it: most of us humans are visual creatures. Ever since our ancestors starting drawing sweet, sweet buffalo all over their cave walls, Homo sapiens have done best when we can see an idea. That's why we invented maps, and visual art, and video games. And why should our math be any different?
For round two of Pre-Calculus, we'll be turning all sorts of equations and concepts into something we can actually see. Visually representing weird ideas like exponents, logarithms, and vectors will be super helpful if we're going to deal with these guys. We can even throw an entire system of equations into a nice, neat matrix to help us conquer it.
In our second semester of eye-poppin', Common Core-aligned math goodness, we'll cover
- building exponential and logarithmic equations and graphs, as well as sketching out their backstories a bit more.
- slicing and dicing a cone into the various conic section graphs—parabolas, hyperbolas, ellipses, and our old friend the circle.
- using matrices to solve a whole system of equations in one fell swoop.
- vectors, and all the things they can represent, like velocity, movement in a certain direction, and a centaur's bow and arrows.
- the properties of limits, which help us see when quantities zoom off to infinity or shrink down to negative infinity.
Stick with us and you'll be a Pre-Calc Pro in no time flat!
P.S. Pre-Calculus is a two-semester course. You're looking at Semester B, but you can check out Semester A here.
Unit Breakdown
6 Pre-Calculus—Semester B - Exponential & Logarithmic Functions
Tired of wondering why some functions don't have a constant slope or a steady speed? Sick to death of those dull, bland linear graphs? Good news! If you want to expand something hugely or shrink it way, way down, we've got one super-powered math gadget that'll do the trick: the exponential function. But wait—there's more! Act now, and we'll also throw in a brand-new tool: the infamous logarithm. If an exponent makes a mess, the logarithm is there to clean it up. This offer ends soon, so don't delay. We'll even throw in free shipping.
7 Pre-Calculus—Semester B - Conic Sections
If you've mostly been daydreaming through Pre-Calculus, we'd understand, but now it's time to wake up. The road ahead is filled with obstacles—mostly in the shapes of circles, parabolas, ellipses and hyperbolas. All four types of conic sections are blocking the road to our Post-Pre-Calculus destination. And you know what that means: keep calm and stay foci—er, focused. Because we don't mean to get hyperbolic about new equations, but the first time someone asked us to find the equation of a hyperbola, we basically passed out.
8 Pre-Calculus—Semester B - Systems of Equations and Matrices
We've dealt with systems of two linear equations already in Algebra 1 and 2, but now we'll dive into systems with more than two variables. And with these guys, let's just say we're going to want a few different options. That'll lead us to matrix addition, multiplication, and even inverse matrices. By the end, we'll get matrices to do some of the heavy lifting for us. Because, well, we don't want to throw our backs out.
9 Pre-Calculus—Semester B - Vectors in Two and Three Dimensions
Think of Shmoop as your surly yet lovable archery instructor. We're going to walk you through the ins and outs of wielding the mighty vector, both standing and on horseback. Vectors are powerful mathematical tools, so we'll need to iron out all your fundamentals first: proper stance, holding the bow, aiming straight. Before you know it, you'll be launching vectors at their targets like some kind of eagle-eyed Elf.
10 Pre-Calculus—Semester B - Limits
Limits are like the leading politicians for all of the calculus-based material to follow. Not just because they make all the policy decisions and definitions for the calculus that comes after them, but also because when they're campaigning on the Cartesian plane, they tend to get closer and closer to the point they're trying to make—but never actually seem to get there. It's actually impressive how long they can stall without saying anything new.
Recommended prerequisites:
Sample Lesson - Introduction
Lesson 7.04: The Equation of an Ellipse
Freeze-dried ice cream, awesome puffy suits, no gravity….If you were ever a little kid (and we're betting you were), you probably once dreamt of growing up to become an astronaut. More recently, you might have dreamed of being a character in Ender's Game. Don't even try to say we're the only ones; we know.
Sadly, the closest we ever got to achieving our galactic dreams was using the equation of an ellipse—which is, after all, the shape of the path that the planets in our solar systems take as they travel around the sun. It's also the shape of the path we make after spinning around ten times fast and then trying to pin the tail on the donkey. Not really our shining moment.
But there's still time for you. All you need is a cone, major and minor axes, two foci, and a butter knife. Why the butter knife? Well to slice the cone with, obviously.
Sample Lesson - Reading
Reading 7.7.04: Running On the Ellipsical Machine
Ellipses are just circles that look like they got sat on by an elephant. Or by us, after eating four gallons of ice cream. They're a little squat, and instead of one uniform radius, like a circle, they have a major axis and a minor axis. Get the low-down on all the basics in this reading.
Ellipses have a lot of moving parts. Actually, all the parts are stationary, but we're going to need all of them in order to construct the whole shebang, meaning the conic form equation for an ellipse. Take a look at it, and a whole lot more info on ellipses, in this reading. Just stop when you get to the first Sample Problem, okay?
One last word on the foci of an ellipse. They share a special relationship with the points on the curve of the ellipse. Drum roll, please: the sum of the distances from the foci to any point on the ellipse is constant.
Okay, now that's all we need. Once we know the center (h, k) and the length of the semi-major and semi-minor axes, a and b, we've got all we need for the equation of an ellipse.
Everything, that is, except for a big stack of flash cards, which we'll need in order to remember all this new vocabulary.
Recap
An ellipse is the set of all points where the sum of the distances from each focus is constant. It has a major axis, a minor axis, vertices, and two foci that we can use to derive the equation of the ellipse.
It has the equation 
, where the center is at (h, k), and a and b will represent the lengths of the semi-minor and major axes.
We can derive the equation given just the foci, using the idea that the sum of the distances from the foci to any point on the ellipse is constant. (And a bunch of algebra, and the Pythagorean Theorem.) We are going to cover that in more detail in a few lessons, though. Try to contain your excitement until then.
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.