Are you in Algebra 1? Are you in Algebra 2? This is the playlist for you! Here all of your algebra-related needs will be addressed!

You may have this scary looking upper case sigma before. But don't worry, it just means to add some things up! Once we understand how series work, we can do all kinds of other things. Also, have you ever seen numbers with exclamation points after them? No it, doesn't mean that you shout the number. Those are factorials! We will learn about those in here as well. Lots of good stuff in here!

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OVERSEAS

Sequences, Factorials, and Summation Notation

Now that we have slope-intercept form under our belt, let's learn the standard form for the equation of a line. This one makes it easy to find the intercepts, and also graph the line very quickly, so it has its advantages! We will also learn how to convert between the two forms.

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Graphing Lines in Standard Form (ax + by = c)

What are the odds when you play the lottery? When you bet on a horse race? When you flip a coin? That you'll have a good day today? Some of these are simple questions. Some, more difficult. But most events involve probabilities that can be assessed and quantified. Let's look at a few, starting with easy ones, and then trickier ones, while defining permutations and combinations along the way!

Special thanks to Heather Ferris of Vegas Aces! Visit her here:
http://www.vegas-aces.com/
   / thevegasaces  

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Theoretical Probability, Permutations and Combinations

We just learned what polynomials are, but that's not going to be enough! We have to learn how to add, subtract, multiply, and divide them. Let's start with the easy operations, addition and subtraction!

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Adding and Subtracting Polynomials

Before we dive into solving all kinds of equations, we want to just quickly go over some number properties and how they will apply to algebraic expressions just like they did for arithmetic ones. Most of these will be familiar, like the commutative and associative properties, as well as the distributive property, but they will become even more important in algebra. We also go over a few other basic ones that are completely intuitive, but still worth mentioning. Take a look!

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Basic Number Properties for Algebra

We're almost done with this first round of graphing now! We just have to learn about the relationships between parallel and perpendicular lines in terms of slope and other characteristics, so that we can more easily graph them, and write the equations for such lines using some nifty tricks. Opposite reciprocals, yo!

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Graphing Parallel and Perpendicular Lines

Now that we know how to use logarithms, we are ready to solve a whole new class of equations that we couldn't before! Whether these are logarithmic equations or exponential equations, using logs will allow us to get the answers we need.

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Solving Exponential and Logarithmic Equations

What are logs? No not like the ones from trees, silly! We need to know about logarithms. They represent the inverse operation of exponentiation. How do they work? How do we evaluate them? How do we graph them? Let's go through the basics now.

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Logarithms Part 1: Evaluation of Logs and Graphing Logarithmic Functions

We know how to solve algebraic equations, but what about systems that contain two equations? Is there a solution that satisfies both equations? There might be! Let's go over a few different strategies to find this solution.

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Solving Systems of Two Equations and Two Unknowns: Graphing, Substitution, and Elimination

Alright, enough messing around! You need to know how to solve algebraic equations, and I'm here to explain it to you. Let's start with the very basics. What are we doing when we solve equations? We are systematically undoing all the operations that a variable is participating in. What does that mean? Take a look!

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Algebraic Equations and Their Solutions

We've already learned a little bit about set theory when we first started using interval notation. Let's expand on this a little bit to include sets that merely list some elements, so that we can understand how to find the union and intersection of two sets. This is good notation to know right here!

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Set Theory: Types of Sets, Unions and Intersections

What are these funny little symbols? They're roots! Taking the square root of something give you the number that, when squared, gives the original number. Cube roots are the same, but they are the opposite of cubing. I'm not explaining it very well right now, but that's what this clip is for, so watch it!

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Square Roots, Cube Roots, and Other Roots

We know about functions, so what are inverse functions? Let's find out!

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Inverse Functions

We know a lot about functions now, so let's look at some special cases where functions get weird and jump around.

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Continuous, Discontinuous, and Piecewise Functions

By now, graphing lines seems trivial, and even graphing quadratics is a piece of cake. But what about higher-degree polynomials like cubics, quartics, and quintics? And even higher than that? It might seem quite daunting, but there are some handy tricks that will help you graph even these very complex functions. These involve using the leading coefficient test to discern the end behavior of the function, as well as plotting zeros and assessing their multiplicities in order to get a rough sketch of the function. It's not as bad as it sounds, check it out!

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Graphing Higher-Degree Polynomials: The Leading Coefficient Test and Finding Zeros

We already know how to graph lines, but it's going to get much trickier than that moving forward! We are going to graph higher-degree functions, and we will need to discuss things like domain and range, maxima and minima, and other good stuff. Let's quickly run through a few details regarding the way we will approach graphing from now on.

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Graphing Algebraic Functions: Domain and Range, Maxima and Minima

Now that we can simplify expressions with roots and exponents, we can pretty easily solve equations that contain them, too. We still are just undoing operations that the variable is participating in, we just have to take square roots instead of adding and subtracting. Give it a try, you'll do great!

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Solving Algebraic Equations With Roots and Exponents

We've already introduced the concept of the inequality, but let's take it to the next level. We need to be able to solve and graph inequalities that involve rational functions, as well as functions with two variables. Check it out!

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Solving and Graphing Polynomial and Rational Inequalities

Alright, now that we know what polynomials are, it's time to learn how to solve them! But how can we do that? There's terms of different degrees, it's a mess! Well as it turns out, there are a few techniques we can learn, and let's start with factoring.

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Solving Quadratics by Factoring

We've graphed tons of lines, but now it's time to graph more complex figures. Conic sections, which are literally cross-sections of a cone, will be important to be able to graph. Let's learn what these conic sections are, and learn how to graph them one by one, starting with circles.

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Graphing Conic Sections Part 1: Circles

Are you tired of algebra yet? Come on, it's fun! Before we dive into graphing, let's just do a little bookkeeping with rational expressions. We have to be able to easily simplify, add, subtract, multiply, and divide these, otherwise the more complex algebra that comes later will be impossible. Don't worry, it's not too bad!

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Manipulating Rational Expressions: Simplification and Operations

That's enough with shapes, let's get back to equations and variables! I know you've missed it so. Jumping back into algebra, the first thing we want to do is understand what a function is, as this will be the main algebraic construct we concern ourselves with from now on. Let's learn what a function is, and what we can do with them.

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Back to Algebra: What are Functions?

Almost done with logarithms! It's a hefty topic so we have to round out the trilogy. We will definitely need to know how to manipulate logarithmic expressions, whether expanding or condensing them, and to do that, we have to know about the properties of logarithms. So let's learn those now!

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Logarithms Part 3: Properties of Logs, Expanding Logarithmic Expressions

Now that we know the basics regarding graphing algebraic functions, it's time to learn some tricks that will come in handy as we graph different kinds of functions. Specifically, there are terms inside functions that do very predictable things to graphs, and if we learn what these are, we can look at certain functions and quickly graph them without making any tables, even functions that look super difficult. Who doesn't like a shortcut? Check this out!

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Transforming Algebraic Functions: Shifting, Stretching, and Reflecting

Now we have all these pesky terms with exponents and roots, and they come in all these complicated looking expressions. Square this, cube root that, whatever will we do? Don't worry, it's usually pretty easy to simplify these until they get into a more palatable form, and it's quite satisfying to get them there! Let's learn all the tricks.

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Simplifying Expressions With Roots and Exponents

Now we know how to factor polynomials, but sometimes that just won't work. We need a different method that might seem a little trickier, but it works every time! It's called completing the square, and it might sound like geometry, but it has to do with manipulating an expression to generate a polynomial that is a perfect square. What am I doing explaining it to you here, just watch the darn thing!

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Solving Quadratics by Completing the Square

We've mentioned in passing some different ways to classify numbers, like rational, irrational, real, imaginary, integers, fractions, and more. If this is confusing, then take a look at this handy-dandy guide to the taxonomy of numbers! It turns out we can use a hierarchical scheme just like we do for biological organisms or fundamental particles, and it's really useful for understanding the properties of different types of numbers.

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What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational

We just learned how to add and subtract polynomials, so now we have to learn how to multiply them. This is a little tricky at first, but once we master the FOIL method, it's easy as pie!

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Multiplying Binomials by the FOIL Method

The equations for lines come in different forms, and slope-intercept form is the classic! It's got everything you need right there, a slope, a y-intercept, what more could you ask for? Let's learn how to find the equation of a line in this form given two points on the line.

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Graphing Lines in Slope-Intercept Form (y = mx + b)

You remember this one, don't you? Opposite of b, plus or minus... what was it? Well instead of just telling you, why don't we derive the formula? As it turns out, it's like completing the square, but with variables. Will you just watch this already? I can't explain it without the pictures.

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Solving Quadratics by Using the Quadratic Formula

By now we are experts at solving quadratics by a number of different strategies. But what about cubics? And quartics? And quintics? Seems pretty daunting, but believe it or not there is a reliable method to solve these higher degree polynomials as well. It's a little more time-consuming, but it can be done! Check it out.

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Solving Higher-Degree Polynomials by Synthetic Division and the Rational Roots Test

When we learn math in school, for a while it's just called math, but then one day around 8th grade, it's suddenly called algebra, and every math class after that has a special name, too. But don't worry, it's not so scary! It's just a new set of symbols and algorithms to become familiar with. But don't forget that at one point you had learn the plus and minus symbols and what those meant. Soon, algebra will be just as easy as arithmetic! I promise.

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Introduction to Algebra: Using Variables

What are these funny little brackets? They represent absolute values. What's that? Well are you here to read this, or watch the clip? Let's go?

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Absolute Values: Defining, Calculating, and Graphing

In getting through algebra, we never talked about complex numbers, but they are important so let's discuss them now! These are numbers with a real component and an imaginary component, so we first have to understand what i means, in a mathematical context. It's the square root of negative one. Then we need to learn how to do arithmetic with these expressions, and finally we can understand how we get answers in this form when using the quadratic formula but with a negative discriminant. Much to discuss, check this out!

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Complex Numbers: Operations, Complex Conjugates, and the Linear Factorization Theorem

Now that we know the basics about solving algebraic equations, we will start looking at more and more difficult examples. But don't worry, just one step at a time! First, a few that are just a little more complicated than the easiest ones.

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Algebraic Equations With Variables on Both Sides

You might think we are all set with graphs, but you're wrong! We will learn about many other types of functions as well as how to graph them. First up is rational functions, which are quotients of polynomials. How can we graph these? We have to find their asymptotes. What's that? Sounds sexy. Well, it is, kind of.

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Graphing Rational Functions and Their Asymptotes

The last conic section is the weirdest looking one. It's the hyperbola! Sometimes it kind of looks like a double parabola, but the equation is totally different. It's actually much more like an ellipse in terms of how the equation works. Let's learn about these things!

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Graphing Conic Sections Part 4: Hyperbolas

The parabola is back! I know we've learned a bit about these, but they are one of the conic sections, so let's go a little deeper and learn some new ways to define a parabola. Also, we know how to graph these in vertex form, but we haven't learned how to graph them in standard form, so let's take care of that too, while we're at it!

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Graphing Conic Sections Part 3: Parabolas in Standard Form

The next conic section is like a circle, but it's like someone stretched a circle out in a particular direction. It's the ellipse! These are very important in astronomy as most celestial objects follow elliptical paths. Let's learn all about ellipses and how to graph them.

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Graphing Conic Sections Part 2: Ellipses

Algebra brings us a new and interesting way to see how math can solve real-world problems! We just have to learn how to construct equations that say something meaningful about a given situation, and then solve them using the techniques we already know. Learn all about how to do this here!

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Algebraic Word Problems

Now that we are familiar with the coordinate plane, it's time to learn more about lines, as these are the simplest things to graph. Lines have slopes, and they usually have intercepts, so lets learn what these words mean!

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Graphing Lines in Algebra: Understanding Slopes and Y-Intercepts

We have seen a nice variety of algebraic equations by now, so let's hone in on a specific kind that we will be working with a lot. These are called polynomials. Let's quickly define what they are and some of their properties, so that we can learn all about how to solve them in the next few clips!

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Introduction to Polynomials

Alright, we've avoided this long enough! To do algebra, we can't just be all about solving equations, we eventually have to graph some stuff as well. Let's start with the easiest things to graph, lines. Here is a little introduction to the coordinate plane and how to graph lines.

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Graphing in Algebra: Ordered Pairs and the Coordinate Plane

Now that we know what functions are, what can we do with them? Can we do operations with them? Yes! Can we plug functions into other functions? Yes! Oh, what won't these functions do? Let's play with a few of them now.

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Manipulating Functions Algebraically and Evaluating Composite Functions

We know about logs, so what are natural logs? Again, no, they're not made of wood. They are logs of base e! What is e? It's this crazy irrational number. Just watch this.

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Logarithms Part 2: Base Ten Logs, Natural Logs, and the Change-Of-Base Property

We know about functions where exponents are raised to different variables, to produce quadratics, cubics, quartics, and so forth. But what about when the variable itself is an exponent? This is called an exponential function, and they are important for modeling things like population growth and other real-world scenarios. Let's learn about them!

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Evaluating and Graphing Exponential Functions

We've seen lots of equations, which are constructs that equate two things, but there are also inequalities, which state that two things are not equal, but rather that one is greater than or less than the other. We can solve this just like equations, but there are one or two little differences that we have to go over. Check it out!

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Solving Algebraic Inequalities

One thing we may want to do in algebra is compute the distance between two points on the coordinate plane. Luckily, there is a handy dandy equation for doing just that! Let's derive it first, and then practice using it.

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Theoretical Probability, Permutations and Combinations

シリーズ

Algebra 1 & 2 #48

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