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Elleyne Kase
Articles & Books From Elleyne Kase
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 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
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
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.
Article / Updated 03-26-2016
Cosecant is almost exactly the same as secant because it's the reciprocal of sine (as opposed to cosine). Anywhere sine has a value of 0, you see an asymptote in the cosecant graph. Because the sine graph crosses the x-axis three times on the interval
you have three asymptotes and two sub-intervals to graph.
The reciprocal of 0 is undefined, and the reciprocal of an undefined value is 0.
Article / Updated 03-26-2016
Knowing how to graph trig functions allows you to measure the movement of objects that move back and forth or up and down in a regular interval, such as pendulums. Sine functions are perfect ways of expressing this type of movement, because their graphs are repetitive and they oscillate (like a wave).
The waves crest and fall over and over again forever, because you can keep plugging in values for
for the rest of your life.