## Simple Algebra II Graph Symmetries Discussion and Examples

[caption id="attachment_17854" align="alignright" width="480"] Typical functions in two variables.[/caption] College preparatory classes in high school often include Algebra and Algebra II. Perhaps the most memorable aspect of Algebra II is the two-dimensional (2D) graphing of mathematical functions in two variables. This is typically introduced beginning with the Cartesian coordinate system. The generic function is written y = f (x). This reads y equals a function of x. See the illustration for some examples of functions. Cartesian Coordinate System In the Cartesian system, two variables, often x and y, are assigned their own line, one horizontal (x), one vertical (y). The intersection between the two axes is called the origin, and is assigned the value (0, 0). The value of x is the value written on the left in the brackets;…

## Understanding the Spherical Polar Coordinate System

[caption id="attachment_9496" align="alignright" width="480"] A cone in the spherical polar coordinate system. CC-SA 3.0 Unported by Lantonov[/caption] Do you have a basic knowledge of the spherical polar coordinate system? A coordinate system provides a way to describe and plot math functions using two or three variables. If there are two variables the graph is 2D. If there are three variables, the graph is 3D. The Cartesian Coordinate System The most familiar coordinate system is the Cartesian coordinate system. Typical variable names are x and y in 2D (although variables can have any name), and x, y, and z in 3D. Every point of every 2D function has a unique value in (x, y). Every 3D function similarly has a unique value in (x, y, z). The Polar Coordinate System This…

## Not Infinitesimal Point Coordinates

[caption id="attachment_7388" align="alignright" width="480"] A coordinate system.[/caption] Infinitesimal point coordinates or not? Do you occasionally enjoy speculating? Even if one has technical training, if he speculates outside his field of expertise, he is opening himself up for possible difficulty. What he speculates, if he voices it, could result in his being labeled someone who doesn’t know what he is talking about. But I never was bright enough to avoid speculation, though I always acknowledge it for what it is. So humor me here… So if you know something of all this and what I'm writing either doesn't jive, or you can 'supercharge' it, please let me know. Mathematicians’ Points – Reality? Generally, a point in space is seen as a dot in space, having infinitesimal point coordinates, that is, no…

## Discrete Mathematics: What is a Point and What a Line?

[caption id="attachment_7234" align="alignright" width="440"] Discrete points? An abstract.[/caption] Most technically minded people will probably take me to task over what I am going to say in this article. That is OK, though. I’m used to it. Not only are my writings quirky—I am quirky. Hey! This is QuirkyScience. I want to talk about points and lines in the real world—in other words, discrete mathematics. What is Reality, What Fantasy? To a mathematician, the point may be a dimensionless object in 3D space, a mathematical object with an x, y, z coordinate in Euclidean space. A line would be a collection of such dimensionless points lined up all in a row. But that is in the world of the mathematician. In the real world, there can be no such thing. Rather,…

## Introduction to Polar Coordinates

[caption id="attachment_25281" align="alignright" width="403"] Polar rose: r = 2 sin (4*θ)[/caption]Frequently used in analytical geometry is the standard 2-dimensional x, y coordinate system called the Cartesian coordinate system (named after famous mathematician, René Descartes). It's time to branch out to a different system, the polar coordinates system. In fact, there are any of a number of ways of locating points in 2-D space. Conversion from Cartesian Coordinates The polar coordinates system utilizes an angle and a radius. It is relatively simple to change from the x-y system to an r-θ system. Drawing a circle centered at the origin on an x-y plane and then drawing a right triangle with the radius of the circle equaling r, then by definition, the side adjacent to the angle divided by the hypotenuse (longest…

## Determining the Equation for a Line from Two Points

Lines can be drawn in three dimensions, but most analytical geometry courses stick to lines in two dimensions, generally using the Cartesian or XY coordinate system. The generic equation for a line may follow the form: y = mx + b where m is the slope (measure of tilt or steep-ness) of the line, while b is its intercept or intersection with the y-axis. Equation for a Line from Two Points A line can be determined and an equation derived from two points. In the Cartesian system, for instance, take two points, (2 , 3) and (– 1 , 5). The first number in each pair represents the x-value of a point and the second number in each pair represents the y-value. Writing these points into the general equation y…