Cheat Sheet

Pre-Calculus Workbook For Dummies

From Pre-Calculus Workbook For Dummies, 2nd Edition by Yang Kuang, PhD, Michelle Rose Gilman

Pre-calculus uses the information you know from Algebra I and II and ratchets up the difficulty level to prepare you for calculus. This cheat sheet is designed to help you review key formulas and functions on the fly as you study. It includes brief definition reviews of logarithmic functions, even-odd identities, conic section equations, and decoding hyperbolas, as well as adding and subtracting matrices.

Logarithm Basics

Logarithms are simply another way to write exponents. Exponential and logarithmic functions are inverses of each other. For solving and graphing logarithmic functions (logs), remember this inverse relationship and you'll be solving logs in no time! Here's the relationship in equation form (the double arrow means "if and only if"):

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Observe that x = by > 0.

Just as with exponential functions, the base can be any positive number except 1, including e. In fact, a base of e is so common in science and calculus that loge has its own special name: ln. Thus, logex = lnx.

Similarly, log10 is so commonly used that it's often just written as log (without the written base).

Even-Odd Identities in Trigonometric Functions

All functions, including trig functions, can be described as being even, odd, or neither. Knowing whether a trig function is even or odd can help you simplify an expression. These even-odd identities are helpful when you have an expression where the variable inside the trig function is negative (such as –x). The even-odd identities are as follows:

sin(–x) = –sinx csc(–x) = –cscx
cos(–x) = cosx sec(–x) = secx
tan(–x) = –tanx cot(–x) = –cotx

Completing the Square for Conic Sections

When the equation of a conic section isn't written in its standard form, completing the square is the only way to convert the equation to its standard form. The steps of the process are as follows:

  1. Add/subtract any constant to the opposite side of the given equation, away from all the variables.

  2. Factor the leading coefficient out of all terms in front of the set of parentheses.

  3. Divide the remaining linear coefficient by two, but only in your head.

  4. Square the answer from Step 3 and add that inside the parentheses.

    Don't forget that if you have a coefficient from Step 2, you must multiply the coefficient by the number you get in this step and add that to both sides.

  5. Factor the quadratic polynomial as a perfect square trinomial.

Finding the Key Parts of All Hyperbolas

A hyperbola is the set of all points in the plane such that the difference of the distances from two fixed points (the foci) is a positive constant. Hyperbolas always come in two parts, and each one is a perfect mirror reflection of the other. There are horizontal and vertical hyperbolas, but regardless of how the hyperbola opens, you always find the following parts:

  • The center is at the point (h, v).

  • The graph on both sides gets closer and closer to two diagonal lines known as asymptotes. The equation of the hyperbola, regardless of whether it's horizontal or vertical, gives you two values: a and b. These help you draw a box, and when you draw the diagonals of this box, you find the asymptotes.

  • There are two axes of symmetry:

    • The one passing through the vertices is called the transverse axis. The distance from the center along the transverse axis to the vertex is represented by a.

    • The one perpendicular to the transverse axis through the center is called the conjugate axis. The distance along the conjugate axis from the center to the edge of the box that determines the asymptotes is represented by b.

    • a and b have no relationship; a can be less than, greater than, or equal to b.

  • You can find the foci by using the equation f 2 = a2 + b2.

Rules for Adding and Subtracting Matrices

To add or subtract matrices, you have to operate on their corresponding elements. In other words, you add or subtract the first row/first column in one matrix to or from the exact same element in another matrix. The two matrices must have the same dimensions; otherwise, an element in one matrix won't have a corresponding element in the other.

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