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Yang Kuang
Articles & Books From Yang Kuang
Article / Updated 02-20-2019
Every good thing must come to an end, and for pre-calculus, the end is actually the beginning — the beginning of calculus. Calculus includes the study of change and rates of change (not to mention a big change for you!). Before calculus, everything was usually static (stationary or motionless), but calculus shows you that things can be different over time.
Article / Updated 02-20-2019
Functions can be categorized in many different ways. Here, you see functions in terms of the operations being performed. Here, though, you see classifications that work for all the many types of functions. If you know that a function is even or odd or one-to-one, then you know how the function can be applied and whether it can be used as a model in a particular situation.
Article / Updated 08-14-2023
As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you find ten habits that should be a part of your daily math arsenal. Perhaps you’ve been told to perform some of these tasks since elementary school — such as showing all your work — but other tricks may be new to you.
Step by Step / Updated 02-20-2019
Here you find some pretty amazing curves that are formed from some pretty simple function equations. The trick to drawing these polar curves is to use radian measures for the input variables and put the results into a polar graph. A polar graph uses angles in standard positions and radii of circles; it’s not your usual rectangular coordinate system.
Cheat Sheet / Updated 07-24-2021
When you study pre-calculus, you are crossing the bridge from algebra II to Calculus. Pre-calculus involves graphing, dealing with angles and geometric shapes such as circles and triangles, and finding absolute values. You discover new ways to record solutions with interval notation, and you plug trig identities into your equations.
Article / Updated 03-26-2016
You can graph a secant function f(x) = sec x by using steps similar to those for tangent and cotangent. As with tangent and cotangent, the graph of secant has asymptotes. This is because secant is defined as
The cosine graph crosses the x-axis on the interval
at two places, so the secant graph has two asymptotes, which divide the period interval into three smaller sections.
Article / Updated 03-26-2016
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"):
Observe that x = by > 0.
Article / Updated 04-19-2017
Logarithmic equations take different forms. As a result, before solving equations that contain logs, you need to be familiar with the following four types of log equations:
Type 1. In this type, the variable you need to solve for is inside the log, with one log on one side of the equation and a constant on the other.
Article / Updated 07-08-2021
In Pre-Calculus, you're going to come across triangles with right angles that vary in degree. This article covers two of the most common right triangles you'll find.
45-45-90 degree triangles
All 45-45-90-degree triangles (also known as 45ers) have sides that are in a unique ratio. The two legs are the exact same length, and the hypotenuse is that length times the square root of 2.
Article / Updated 03-26-2016
The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality (like fraternal twins). Cosine graphs follow the same basic pattern and have the same basic shape as sine graphs; the difference lies in the location of the maximums and minimums. These extremes occur at different domains, or x values, 1/4 of a period away from each other.