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178 results for "Yang Kuang, PhD"
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How to Graph a Rational Function with Numerator and Denominator of Equal Degrees
After you calculate all the asymptotes and the x- and y-intercepts for a rational function, you have all the information you need to start graphing the function. Rational functions with equal degrees in [more…]
Found in: Functions -
How to Invert a Function to Find Its Inverse
If you’re given a function and must find its inverse, first remind yourself that domain and range swap places in the functions. Literally, you exchange [more…]
Found in: Functions -
How to Solve a Quadratic Equation When It Won’t Factor
When asked to solve a quadratic equation that you just can’t seem to factor (or that just doesn’t factor), you have to employ other ways of solving the equation, such as by using the quadratic formula. [more…]
Found in: Pre-Calculus -
How to Find Imaginary Roots Using the Fundamental Theorem of Algebra
The fundamental theorem of algebra can help you find imaginary roots. Imaginary roots appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign [more…]
Found in: Pre-Calculus -
How to Graph Polynomials When the Roots Are Imaginary Numbers — An Overview
In pre-calculus and in calculus, certain polynomial functions have non-real roots in addition to real roots (and some of the more complicated functions have [more…]
Found in: Pre-Calculus -
How to Simplify an Expression Using Co-function Identities
If you take the graph of y = sin x and shift it to the left by pi/2 units, it looks exactly like the graph of y= cos x. The same is true for tangent and cotangent, as well as secant and cosecant. That's [more…]
Found in: Trigonometry Basics -
How to Apply the Sine Sum and Difference Formulas to Trig Proofs
When dealing with sine sum and difference formulas in a trig proof, you need to make one side of the given equation look like the other. You can work on both sides to get a little further if need be, but [more…]
Found in: Trigonometry Basics -
How to Find Trigonometric Functions of an Angle by Using Pythagorean Identities
You can use Pythagorean identities to find the trig function of an angle if you know one trig function of the angle and are looking for another. For example, if you know the sine of an angle, you can use [more…]
Found in: Trigonometry Basics -
How to Solve a Triangle When You Know Two Consecutive Side Lengths (SSA)
In some trig problems, you may be given two sides of a triangle and an angle that isn't between them, which is the classic case of SSA, or Side-Side-Angle [more…]
Found in: Trigonometry Basics -
How to Identify the Four Conic Sections in Graph Form
Each conic section has its own standard form of an equation with x- and y-variables that you can graph on the coordinate plane. You can write the equation of a conic section if you are given key points [more…]
Found in: Graphing Equations -
How to Find the Limit of a Function Algebraically
When your pre-calculus teacher asks you to find the limit of a function algebraically, you have four techniques to choose from: plugging in the x value, factoring, rationalizing the numerator, and finding [more…]
Found in: Limits -
How to Identify Even and Odd Functions and their Graphs
Knowing whether a function is even or odd helps you to graph it because that information tells you which half of the points you have to graph. These types of functions are symmetrical, so whatever is on [more…]
Found in: Functions -
How to Graph and Transform an Exponential Function
Graphing an exponential function is helpful when you want to visually analyze the function. Doing so allows you to really see the growth or decay of what you’re dealing with. The basic parent function [more…]
Found in: Functions -
How to Calculate Values for the Six Trigonometric Functions
In pre-calculus, you need to evaluate the six trig functions — sine, cosine, tangent, cosecant, secant, and cotangent — for a single angle on the unit circle. For each angle on the unit circle, three other [more…]
Found in: Trigonometry Basics -
How to Simplify an Expression Using Periodicity Identities
Periodicity identities illustrate how shifting the graph of a trig function by one period to the left or right results in the same function. The functions of sine, cosine, secant, and cosecant repeat every [more…]
Found in: Trigonometry Basics -
How to Graph a Tangent Function
The tangent function has a parent graph just like any other function. Using the graph of this function, you can make the same type of transformation that applies to the parent graph of any function. The [more…]
Found in: Trigonometry Basics -
How to Graph a Hyperbola
Think of a hyperbola as a mix of two parabolas — each one a perfect mirror image of the other, each opening away from one another. The vertices of these parabolas are a given distance apart, and they open [more…]
Found in: Graphing Equations -
Right Triangles and Trig Functions for Pre-Calculus
If you’re studying pre-calculus, you’re going to encounter triangles, and certainly the Pythagorean theorem. The theorem and how it applies to special right triangles are set out here: [more…]
Found in: Calculus -
How to Format Interval Notation in Pre-Calculus
In pre-calculus you deal with inequalities and you use interval notation to express the solution set to an inequality. The following formulas show how to format solution sets in interval notation. [more…]
Found in: Calculus -
Comparing Radicals and Exponents
Radicals and exponents (also known as roots and powers) are two common — and oftentimes frustrating — elements of basic algebra. And of course they follow you wherever you go in math, just like a cloud [more…]
Found in: Pre-Algebra & Algebra Basics -
How to Graph Polynomials
Although it may seem daunting, graphing polynomials is a pretty straightforward process. Once you have found the zeros for a polynomial, you can follow a few simple steps to graph it. [more…]
Found in: Pre-Calculus -
How to Factor a Perfect Square
FOIL stands for multiply the first, outside, inside, and last terms together. When you FOIL a binomial times itself, the product is called a perfect square. [more…]
Found in: Pre-Calculus -
How to Find the Tangent of a Doubled Angle
The double-angle formula for tangent is used less often than the double-angle formulas for sine or cosine; however, you shouldn't overlook it just because it isn't as popular as its cooler counterparts [more…]
Found in: Trigonometry Basics -
How to Express Products of Trigonometric Functions as Sums or Differences
If you can break up a product of trig functions into the sum of two different terms, each with its own trig function, doing the math becomes much easier. In pre-calculus, problems of this type usually [more…]
Found in: Trigonometry Basics -
How to Prove an Equality by Using Periodicity Identities
Using the periodicity identities comes in handy when you need to prove an equality that includes the expression (x + 2pi) or the addition (or subtraction) of the period. For example, to prove [more…]
Found in: Trigonometry Basics
