Making Pre-Calculus fun tolerable
Chris is a Stanford-educated tutor with over 10 years experience tutoring Pre-Calculus 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.
Pre-Calculus
I. Exponents, Radicals & Factoring
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 i27, and rationalizing imaginary and complex denominators.
"Quadratic" means "squared", for some reason, so this chapter is about solving equations with x2'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.
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.
II. Functions, Graphing, Exponentials & Logs
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.
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).
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.
This is a very specific topic where you're told to find the "average rate of change of the function on the interval [a,b]." What that means in English is "plug and chug into your average value formula." And hey, why not find the secant line between those two points while you're at it?
III. Polynomials & Rational Functions
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 x3, x4 and x5. 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.
This chapter covers converting parametric equations to rectangular and back again, eliminating the parameter, parametric forms of circles and ellipses, and graphing them.
IV. 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 nCr and nPr 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.
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:
V. 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.
VI. 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 "tan2X-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.
VII. 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.
IX. 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!