# Making College Algebra fun *tolerable*

# Chris is a Stanford-educated tutor with over 10 years experience tutoring College Algebra to students of all abilities, from students struggling to get from a C to a B, to go-getters trying to move an A- up to an A, to struggling students just hoping to pass. In that time he got a lot of experience learning how to explain this stuff in a way it actually makes sense to non-math people. Through his videos he has helped countless students, and he can do the same for you.

# College Algebra

##### Colleges vary in how they break down Algebra and Math Analysis topics, so I've put everything before Pre-Calc in one place. Whatever your school calls each class, the stuff you learn is pretty much standard, so you can count on the chapters below pretty much having you covered.

# I. Numbers & Calculations:

#### In this chapter we'll get into the basics of prime numbers and "prime factoroization", which we'll use in fractions and roots throughout Algebra and Algebra 2. I also get into divisibility rules, which show up -- among other places -- on the SAT.

#### Everything you need to know about fractions: reducing, least common denominators, addition, subtraction, multiplication, division, converting improper fractions to mixed numbers, converting to decimals. Also the tricky problem of converting repeating decimals to fractions.

#### In this chapter we'll introduce the concept of exponents, as well as rules for combining and canceling exponents. Also, we'll talk about the special cases of exponents of 0 and 1, and what happens with negative exponents.

#### We'll start by comping percentages using basic ratios, then we'll move on to using a decimal-based approach since that's what you'll need for more advanced problems. I also point out the most common types of trick questions about percents, including common types of SAT questions, and computing interest. And if you're super-advanced, "compound interest", where you have to calculate how interest builds over n weeks, months, years, etc.

#### In this chapter we'll learn how to reduce radicals using a couple different methods. If you have a tough teacher and you have to know how to rationalize denominators and deal with variables under the root, then you'll need our Algebra 2 Roots & Radicals page.

#### In this chapter we'll introduce Order of Operations, often referred to simply by its ultra-useful acronym PEMDAS. We'll also get into how to do PEMDAS on your calculator, and a few common calculator mistakes to avoid.

# II. Geometry:

#### This chapter covers the basic formula and how to use it on right triangles, as well as the most common mistake students make with it. Also included is a video about simplifying radicals, since that comes up a lot with Pythagorean.

#### The videos in this topic introduce you to the X-Y plane (aka "rectangular coordinate system"), explain what the "origin" is, explain how to graph points, explain "the four quadrants", and tell you how to plot points.

#### If you're new to X-Y graphs, this is the place for you. We'll start with segments and lines without equations (for equations of lines from algebra, click here). Then we'll look at graphs of circles and rectangles, and x- and y-intercepts.

#### This chapter teaches you the formulas for finding the distance and midpoint between two points on the X-Y plane, as well as word problems using these formulas.

# III. Variables & Equations:

#### In this chapter I explain what "x" is, what expressions are, how to plug in values to "evaluate" expressions, and how to turn word problems into equations so you can solve them. We'll also go over how to check whether a number is a solution of an equation or not.

#### These videos explain the process of solving algebraic equations by addition/subtraction, multiplication/division, combining like terms, and any combination of the three (multi-step problems like 3x-5=7).

#### Combining "like terms", adding and subtracting polynomials and monomials. Also covered: distribution (a.k.a. "multiplying stuff out"), where you have a number or variable in front of a set of parentheses that has to get multiplied by each term, such as 2(x-5).

#### Cross "multiplication" refers to what you do when you have two fractions across an equal sign from each other. You cross "cancel" when you are multiplying fractions. Confused yet? This chapter will sort you out.

#### Ratios are mostly word problems that lead to cross multiplication equations, and most students find them pretty basic. In this chapter we'll work some typical problems about marbles and pizzas, and I'll emphasize the one trick they always pull on the SAT.

#### Problems where you have to simplify a giant fraction which has more fractions inside the numerator and denominator.

# IV. "Linear" Equations, Inequalities & Variation:

#### Rate, time and distance show up in these word problems (Rate x Time = Distance). Example problems include: How far did someone drive in 3 hours? If two trains leave stations and different times, how long until they pass each other? What units should my answer be, and how do I convert?

#### Slope-Intercept Form, Point-Slope Form, Standard Form, Vertical Lines, Horizontal Lines, Perpendicular Lines: in this chapter, we experience the splendor of all the different types of linear functions, and master the equations and graphing of each.

#### When a word problem talks about two things being "proportional", this is what they're talking about. Or maybe they're "inversely proportional". "Joint" variation just refers to when you have three things that are proportional or inversely proportional instead of just the two, which is basically just ratios in disguise.

#### Inequalities are just equations with an "<" or ">" instead of "=". In this chapter, we'll look at how to solve a few different types of "linear" inequalities: ones with just X, where you present the answer in Interval Notation, and ones with X & Y, where you shade one side of the line or another. We'll also cover systems of inequalities, "and" vs "or", etc. For other types of inequalities, try this page.

### Solving Absolute Value Equations & Inequalities

*Absolute Value Equations*

Absolute Value Inequalities

#### Absolute value signs (i.e. |x+3|) wreak havoc on equations and inequalities, often resulting in multiple answers and interval notation, but I'll give you simple steps to memorize for dealing with them. If you need to graph absolute value functions, check out the library functions page.

#### Whenever you're given two or three equations at the same time, they're "simultaneous equations. This chapter covers "elimination" and "substitution" techniques to solve for X & Y, and explains finding the intersection of lines (or not as in the case of parallel & coincident lines). I also demonstrate solving three equations, three unknowns.

# V. Exponents, Factoring & Canceling:

#### Combining exponents, canceling terms, multiplying rational expressions and equations, multiplying and dividing variables with various exponents, negative exponents: if it's got an exponent, this chapter covers it.

#### In these videos we'll cover all forms of factoring polynomials, from "factoring stuff out" to quadratics to sum and difference of cubes. We'll also learn factoring by u-substitution & grouping.

#### This chapter covers everything you'll ever be asked to do to or with a root or a "rational" (fraction) exponent. Topics covered: simplifying roots & radicals, reducing roots, dividing roots, adding-subtracting-multiplying-and-dividing radicals, and rationalizing denominators.

#### All is not as it seems in this exciting and short chapter. We're talking square roots of negative numbers, finding high exponents of "i" like i

^{27}, and rationalizing imaginary and complex denominators.#### Problems where you have to simplify a giant fraction which has more fractions inside the numerator and denominator.

# VI. Solving Equations, Systems & Inequalities:

#### These videos explain the process of solving algebraic equations by addition/subtraction, multiplication/division, combining like terms, and any combination of the three (multi-step problems like 3x-5=7).

#### Whenever you're given two or three equations at the same time, they're "simultaneous equations. This chapter covers "elimination" and "substitution" techniques to solve for X & Y, and explains finding the intersection of lines (or not as in the case of parallel & coincident lines). I also demonstrate solving three equations, three unknowns.

#### Inequalities are just equations with an "<" or ">" instead of "=". In this chapter, we'll look at how to solve a few different types of "linear" inequalities: ones with just X, where you present the answer in Interval Notation, and ones with X & Y, where you shade one side of the line or another. We'll also cover systems of inequalities, "and" vs "or", etc. For other types of inequalities, try this page.

### Solving Absolute Value Equations & Inequalities

*Absolute Value Equations*

Absolute Value Inequalities

#### Absolute value signs (i.e. |x+3|) wreak havoc on equations and inequalities, often resulting in multiple answers and interval notation, but I'll give you simple steps to memorize for dealing with them. If you need to graph absolute value functions, check out the library functions page.

#### "Quadratic" means "squared", for some reason, so this chapter is about solving equations with x

^{2}'s in them. You'll have to learn several tricky techniques -- square rooting, factoring, completing the square, and/or the Quadratic Formula -- but I give you tips to make them easier, and to decide which to use in different situations.#### In this chapter we take a look at how to solve equations where the variable is under a square root or a radical. Often we'll be able to simply square both sides of the equation, but we'll always have to be careful to check for extraneous solutions.

#### Rational Equations are the problems where you have a bunch of x's and x

^{2}'s in the denominator of a giant fraction, and you have to find the least common denominator and simplify in order to solve for X.#### In this chapter we'll return to the Big Three of inequalities -- number lines, test points, and interval notation -- for perhaps the final time (nostalgic yet?).

# VII. Word Problems & Applications:

#### Rate, time and distance show up in these word problems (Rate x Time = Distance). Example problems include: How far did someone drive in 3 hours? If two trains leave stations and different times, how long until they pass each other? What units should my answer be, and how do I convert?

#### Rational Equations are the problems where you have a bunch of x's and x

^{2}'s in the denominator of a giant fraction, and you have to find the least common denominator and simplify in order to solve for X. And I teach a great method using tables to solve nasty word problems involving stuff like boats rowing upriver and faucets filling tubs.# VIII. Functions

#### Slope-Intercept Form, Point-Slope Form, Standard Form, Vertical Lines, Horizontal Lines, Perpendicular Lines: in this chapter, we experience the splendor of all the different types of linear functions, and master the equations and graphing of each.

#### In this chapter we'll introduce functions, the vertical line test, function notation (i.e. plugging numbers into functions), graphing functions the easy way (by plugging in). Also introduced are domain, range, finding inverse functions, x-intercepts, y-intercepts, and graphing functions.

#### Usually in math, the names don't make any sense. But this is an exception: "even" and "odd" refer to whether the exponents on the x's are even or odd!

#### Time to master graphing all kinds of standard functions (a.k.a. library functions or parent functions) using transformations. Vertical stretch, horizontal stretch, translating/moving graphs up down left right. We'll also cover those Frankenstein-esque combo functions: piecewise functions.

#### By popular demand, this short video explains the process of finding x-intercepts and y-intercepts for any function. These are also known as "zeroes" of a function, and you'll see why by the end of this.

#### These videos tell you what translations, rotations, and dilations are, as well as showing you the steps to do reflect a shape or function across a line using graphical techniques.

# IX. Exponentials & Logarithms:

#### As soon as the variable in an equations moves up to the exponent, you've got yourself an exponential and you may need logs (logarithms or logarithmic equations). In chapter we'll analyze and graph them, and look at some common types of problems such as compound interest.

#### In this chapter you'll get all the basics on logarithms (logs) and log equations, as well as how to graph them and use them to solve tough exponential equations. I also devote a video to the difference between graphing logs vs graphing exponentials.

# X. Polynomials & Rational Functions:

#### In this chapter we'll focus on the anatomy of parabolas: vertex, axis of symmetry, vertex form, x-intercepts, roots, and the discriminant. We'll also cover word problems where you are asked to maximize/minimize the area or volume of a shape (minima/maxima).

#### In this chapter we'll learn a somewhat tedious process of dividing polynomials by each other, a skill that's kind of fun once you get the hang of it and which will serve you well in Pre-Calc & Analysis.

#### In this chapter we'll put our synthetic division skills to the test by using "p/q" to fully factor higher-power polynomials containing x

^{3}, x^{4}and x^{5}. Plus calculator graphing tips!#### Rational functions bring with them a crazy list of math terms: asymptotes (horizontal, vertical, oblique & slant), domain, range, and intercepts. You're welcome.

#### In this chapter we'll return to the Big Three of inequalities -- number lines, test points, and interval notation -- for perhaps the final time (nostalgic yet?).

#### In this chapter we'll learn a somewhat tedious process of splitting up a perfectly good rational expression into a couple fractions with A's and B's in the numerator.

# XI. Conic Sections

#### In this chapter we'll emphasize the similarities and differences between the equations of these four shapes, and we'll discuss why conic sections are called that.

# XII. Probability, Sequences & Permutations:

#### Common confusion: a "series" is just a sequence with plus signs between the terms instead of commas. All other questions, check out the chapter page, which includes a free printable pdf of all the formulas for arithmetic and geometric sequences.

#### Make

_{n}C_{r}and_{n}P_{r}pay for what they've done by mastering them and using them to execute on your upcoming test. Also in this chapter: brush up for this common SAT question.#### This chapter "probably" (lol) covers mutually exclusive events, dependent probability, and, or, colored rocks, coin flips, regular dice, weighted dice, and even the Binomial probability formula.

# XIII. Matrices:

#### This chapter covers the basics - matrix addition, subtraction, multiplication, and determinants - along with advanced moves like solving systems with row operations and Cramer's Rule.

# And of course there's Trig:

# XIV. SohCahToa & The Unit Circle

#### With this chapter we'll start trig off on the right foot -- triangles -- which has worked great for my tutoring students over the years. Also covered: what SohCahToa is (other than a weird abbreviation for the sinusoidal functions); what opposite, adjacent, and hypotenuse mean; how to find sine, cosine and tangent; and how to work a bunch of "solving triangles" problems.

#### In this chapter we'll get into the 30-60-90 and 45-45-90 triangles, with special emphasis on how to find their sides, do SohCahToa on them, and not get confused between which is which. Also, some super-tricky examples for students with tough teachers.

#### These new trig functions are just the reciprocal (flip) of sine, cosine and tangent, but they can be confusing, so we'll emphasize always writing them in the correct order each time, and we'll do lots of examples. I'll also show you how to do these on your calculator, which doesn't have buttons for these.

#### The longest chapter in trig. We'll start off slow, developing understanding by using SohCahToa to derive only the first quadrant of the unit circle at first. Then we'll work through reference angles, sign tricks, negative angles, co-terminal angles and undefined functions until you can calculate the six trig functions for any angle. We'll finish up with some tricks for memorizing the Unit Circle Chart.

#### Now that you've learned the Unit Circle in degrees, we're ready for Radians. This chapter the radian version of everything: reference angles in radians, negative angles in radians, etc. I also demonstrate common test problems like converting radians to degrees and degrees to radians, finding the six trig functions of angles with radians, and tricks for memorizing the radian unit circle.

# XV. Formulas, Equations & Identities

#### Also called Arcsin, Arccos, Arctan, etc., these are problems like sin

^{-1}(1) where they give you the sin/cos/tan of an angle and you're supposed to give the angle*in the correct quadrant*. Key for solving trig equations, I explain how to do these with the unit circle or a calculator.#### Trig equations are problems where you're solving for X or Theta but they're hidden behind a trig function, like "2sinX-1=0" or "tan

^{2}X-1=0". Lots of vocab in this one -- specific solutions, general solutions, 2npi, 360n -- and lots of factoring to do too. Easy to get confused between these and inverse trig functions!#### Proofs strike fear in many a heart because most students were traumatized by proofs in Geometry. But have no fear. In this chapter I'll show you how to easily memorize the key identities you'll need, as well as take you through the Big Three techniques that will help you spot and solve most proofs.

#### Double-angle, half-angle, sum, difference, even-odd properties... I cover all the odds and ends here in one place. Most of these you'll never see again, so I focus on getting you through it quick so you can move on.

#### Two trig identities which reduce a sin^2 or cos^2 to first-order expressions, a key skill for calculus.

# XVI. Graphing Trig Functions

#### These problems are frustrating for students because they pretty much all look the same, yet you can lose big points if you get one little shift or label wrong. Plus there's the annoying vocab: Amplitude, "b", phase shift, vertical shift. No worries, the videos in this chapter will sort it all out using explanations and techniques my students have found helpful, and I'll point out common errors to avoid.

#### I put these four in a separate chapter from Sine & Cosine for two reasons. First, many non-honors students don't even have to do these, so why scare you. Second, you should really get good at sine & cosine graphs first, since these four badboys are way easier if you base them on sine and cosine graphs, which is the approach I find helps students the most.

# XVII. Applications of Trig

### Laws of Sines & Cosines

*Basic Law of Sines problems*

The Law of Sines "Ambiguous Case"

Law of Cosines

#### Earlier in Trig, we've already had a few videos about solving right triangles for missing sides, so how is this chapter different? It no longer has to be a right triangle! These problems are way more complicated than SohCahToa, yet in this chapter students often seem relieved to actually be "doing something" again rather than learning "a bunch of stuff you'll never see again".

#### Basics of vector addition, subtraction, multiplication, dot product, scalar product, magnitude, unit vectors, cross multiplication, and components.

#### This chapter covers converting parametric equations to rectangular and back again, eliminating the parameter, parametric forms of circles and ellipses, and graphing them.

#### This chapter covers kinematics projectile motion problems as you would see in Pre-Calculus or Algebra 2 math classes. This topic is covered in more depth on the physics page. One-dimensional and two-dimensional gravity problems, range, vector components of velocity, etc.

#### This chapter covers the basics of arc length and sector area, as well as difficult word problems about bike gears, vehicle speed, planets, and velocity of rotating objects.

#### This chapter covers everything from graphing polar coordinates and functions to converting equations between polar and Cartesian x-y coordinates. Also, how to do these on your calculator!

##### (There's lots more geometry on the Geometry page.)