Algebra II—Semester B

We put the "function" in "dysfunctional."

  • Credit Recovery Enabled
  • Course Length: 18 weeks
  • Course Type: Basic
  • Category:
    • Math
    • High School

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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.

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.


Yeah, that's right. Algebra's back for round two—and this time, it means business.

It's beefed up its stock of algebraic weapons—from sequences and series to trig functions—so it's going to take a whole lot more than the quadratic formula to get this guy to cry uncle. Good thing you've got Shmoop on your side. (And it doesn't hurt that you've been pumping a bit of iron yourself.)

Semester B is all about building, analyzing, and interpreting functions of all kinds. Once we've learned about exponential functions, logarithms, and trigonometry, we'll delve deep into functions and their inverses (a.k.a. evil twins). Then we'll get to know the end behavior of all kinds of functions and describe it using numbers, words, and interpretive dance. Finally, we'll end the semester with statistics and probability. Well, probably.

We've got activities, quizzes, projects, and more that cover:

  • graphing, analyzing, and interpreting different types of functions;
  • building and modeling with functions;
  • exponential, logarithms, and trigonometric functions;
  • the normal distribution and real-world applications of statistics;
  • combinations, permutations, and probability.

P.S. Algebra II is a two-semester course. You're looking at Semester B, but you can check out Semester A here.


Here's a sneak peek at a video from the course. BYOP (bring your own popcorn).


Unit Breakdown

7 Algebra II—Semester B - Exponential and Logarithmic Functions

We won't lie to you: we've packed a lot into this unit. We'll start off with logarithms and exponents and their rules. Soon after, we'll throw these bad boys into equations and solve them in every way under the sun: algebraically, graphically, and with technology.

8 Algebra II—Semester B - Trigonometric Functions

Here we'll give you the keys to the trig kingdom. After introducing the unit circle and a brief review of the basic trig ratios, we'll take things to the next level. We're talkin' graphing trig functions and showing you how to shift and flip them all over the coordinate plane. And how could we forget about applications? We'll wrap things up with some activities on what you can actually do with your new found trig knowledge.

9 Algebra II—Semester B - Building Functions

This unit is for the practically minded everyman in all of us. We're actually going to get our hands dirty (well...not literally) by building functions. Taking all the functions we've learned about so far, we'll throw 'em together in every which way we can think of to make new ones.

10 Algebra II—Semester B - Statistics

In this unit, we'll run the statistical gamut—from summary stats and the normal distribution to populations, samples, and simulations. We'll lay down the foundation for properly collecting real-world data and even get our hands dirty summarizing it. By the end of it all, your head will be fit to burst with all kinds of statistical knowledge.

11 Algebra II—Semester B - Probability

This unit is all about taking what we know about a situation and making the best possible guesses and extrapolations about it that we can. Some will be wrong, but we're also betting that a good number of them will be right. And given that our universe is a giant floating bubble of randomness and chaos, we think being able to predict events through math is pretty neat.


Recommended prerequisites:

  • Algebra I—Semester A
  • Algebra I—Semester B
  • Geometry—Semester A
  • Geometry—Semester B

  • Sample Lesson - Introduction

    Lesson 11.04: Theoretical Probability: AND versus OR

    Really? You don't have the popped kind? What are we, squirrels?

    (Source)

    If we're going to totally rely on the power of our calculations when we do math, we had better be confident in our ability to model every conceivable situation. For instance, say we're at the movies with two friends and they ask us to purchase the snacks. Our first friend is pretty easygoing. She'd be fine with Jujubes or Raisinets or Milk Duds, but our other friend is kind of greedy. He wants Jujubes and Raisinets and Milk Duds. It seems like the probabilities that either friend will be happy with what we bring back are pretty different, but why?

    What's causing this whole problem lies with two little words: "and" and "or." They may be short, tiny words, but their impact is huge, especially when it comes to probabilities. When we say "and," we're limiting our options. We're making the desirable specific event more specific. When we say "or," we're opening up the specific event to more possibilities. There's a greater number of outcomes that would make us happy.

    This lesson's reading will detail how our formulas must change to accommodate these different scenarios. Keep in mind that when we use "and," our probability will go down, and when we use "or," it will go up. You're more likely to be happy if you're open-minded about the things that can make you happy. There's a lesson in there somewhere. Lower your standards?


    Sample Lesson - Reading

    Reading 11.11.04: Either/Or (Or And)

    We're familiar with the word "and" from our experiences as speakers of the English language. It's often used to connect two things that are both true: if some guy says, "Today I'm going biking and swimming," then he'd better end up on a bicycle and in a large body of water before midnight, or he'll be a bit of a liar.

    Talking About "And"

    In probability, we use some fancy symbols to express pretty much the same idea. For two events A and B, we write the probability of A and B as P(AB). That hump is called "the intersection" of A and B, but we'll sometimes just put it in English: P(A and B). It's all the same.

    Whatever we call it, this is the probability that both A and B occur. So if we're trying to calculate P(AB), we count all of the outcomes that satisfy both A and B, and divide this number by the total number of outcomes. Sound simple? It is.

    Say we have a deck of cards, and we're curious what the probability of randomly drawing a red king is. We've got two events, A and B; the first, A, stands for "the card is a king," while B is "the card is red." We're trying to figure out P(AB). So, we count how many outcomes satisfy both A and B.

    In a standard 52-card deck, there are only four kings, and only two of those kings are red: the king of hearts and the king of diamonds. With 52 possible cards to pick from, .

    This process gets even easier when A and B are mutually exclusive; that is, if A and B can't both occur at the same time. If that's the case, then there's no way for them to both occur, so no matter what A and B are, P(AB) = 0.

    We can see this if we look back to our deck of cards. Being a king and being red aren't mutually exclusive, but being a king and being a queen are. If we try to calculate P(AB), where A means "the card we draw is a king" and B is "the card drawn is a queen," then we can be confident that P(AB) = 0. (We almost want to say that you can bet on it, but Shmoop doesn't gamble, even when the probabilities are indisputable.)

    How About "Or"

    "Or" is a different ballgame, but one that also borrows some of its rules from plain old English. It's often used to connect two things when only one of them needs to be true. If that same guy from earlier says, "Today I'm going biking or swimming," then all he needs to do to be speaking the truth is spend some time biking, or some time swimming. He could even do both, and we wouldn't hold it against him.

    The same general idea applies when we're talking in terms of probabilities. For two events, A and B, the probability of A or B is called the union of A or B, P(AB). It's the probability that A will occur, or B will occur, or both will occur.

    "Or" is kind of like the bus of probability, because it gives everything a ride. We just add up all of our events and throw them up top in our probability fraction, over the total number of outcomes. Let's grab our handy deck of 52 cards and take "or" out for a spin.

    What's the probability that we grab a queen or a red card? There are four queens. There are 26 red cards (half of the deck). So we just add those up and head to Fractionland, right?

    Not so fast.

    We accidentally counted the red queens twice: once when we counted queens, and once when we counted the red cards. We need to subtract the overlap (the two red queens) from our total before we fraction it up. We end up with 28 cards that are red or queens. Our probability then is .

    In terms of a formula (formulas are always good), P(AB) = P(A) + P(B) – P(AB). This is pretty much just a mathematical translation of the ideas we've been teasing out: add the probability of one event to the probability of the other, make sure to subtract any doubled outcomes, and you're done. It's a bit more elegant as a formula, though.

    Recap

    The intersection of two events, A and B, is the probability of both of them occurring together. In symbols, that's P(AB).

    To calculate P(AB), we count all of the outcomes that satisfy both A and B, and divide this number by the total number of outcomes.

    When A and B are mutually exclusive, P(A and B) = 0.

    The union of A and B refers to the probability that A occurs, B occurs, or they both occur. You'll recognize it on the street by its notation, P(AB). If you want a fancy formula for calculating the union (and you know you do), use this one: P(AB) = P(A) + P(B) – P(AB).


    Sample Lesson - Activity

    1. How do we find the probability of one event and another event happening together?

    2. How do we find the probability of one event or another event happening?

    3. What is the probability of pulling a card that is red and a number card (2-10) from a standard deck of cards?

    4. What is the probability of pulling a card that is red or a number card (2-10) from a standard deck of cards?

    5. When reaching into a bag of clothing, how would we find the probability of randomly pulling out a pair of red pants based on the contents of the bag?

    6. When reaching into a bag of clothing, how would we find the probability of randomly pulling out a pair of pants or something that is red based on the contents of the bag?

    7. When randomly selecting a number between 1 and 20, what is the probability of picking a number that is prime or odd?

    8. When rolling a standard fair, 6-sided die, what is the probability that you roll a number greater than 5 and an odd number?

    9. When picking a number between 1 and 20, what is the probability that it is bigger than 15, less than 5, or the number 20?
    10. When picking a card out of a standard deck of cards, what is the probability that the card chosen is a face card or a diamond?
    11. When picking a card out of a standard deck of cards, what is the probability that the card chosen is a face card and a diamond?
    12. What does it mean for events to be mutually exclusive? If A and B are mutually exclusive events, what is P(AB)?