## Analytic Geometry: The Parabola

What is a parabola, and of what use is it? If a sheet of paper is likened to an infinite plane in space, the x- and y-axes, drawn at right angles to each other, provide a means of describing each point on the plane of the paper in terms of an x and a y value. Thus the point (1, 3) tells us that beginning at the origin or center where the x-axis crosses the y-axis, if we travel one unit to the right and then three units up, we will have reached the point we seek. Introducing the Parabola Now we will pass on to describing a parabola.1 The parabola is an important mathematical "curve," inasmuch as it describes, mathematically, the behavior of a number of important actions, such…

## Analytic Geometry: The Hyperbola

[caption id="attachment_6399" align="alignright" width="380"] Simple hyperbola with asymptotes. Image by author.[/caption] The parabola, ellipse, circle, and hyperbola are all termed conic sections. This means that a plane that cuts into a cone in just the right way will generate one of these figures. We will consider the basic equation of a hyperbola and graph one. Equation of an Hyperbola It may be recalled the equation for an ellipse centered at the origin is, x2/a2 + y2/b2 = 1 where 2a is the length of the ellipse and 2b is its height. The equation for an hyperbola centered at the origin is very similar, x2/a2 - y2/b2 = 1 The graph of this function is completely different from that of an ellipse. Let's look at a very basic one, for which…

## Analytic Geometry Coordinate Axes and Drawing a Line

In analytical geometry (usually taught in high school), two lines are drawn on a paper that are perpendicular to each other. The vertical line represents the "y-axis," and the horizontal line represents the "x-axis." Using these two axes, every point on the paper can be given a value that defines where the point is. If the place where the two lines cross is the zero point or origin, its coordinates (x, y) are simply, (0, 0). Along the horizontal x-axis, starting to the right of the (0, 0) point, write little numbers like a ruler has, 1, 2, 3, and so forth. To the left of that point, write, -1, -2, -3, and so on. For the y-axis, the 1, 2, 3, and such go upward, whereas the -1, -2,…

## Math Equations for Parallel and Perpendicular Lines

It's fun and very instructive to figure out the math equations for parallel and perpendicular lines. The basic mathematical equation for a line is, ax + by = c Here are three examples of line equations: 2x + 3y = 6 4x – 2y = –5 –x/3 + 2.47y = √3 Slope-Intercept Form One of the most useful formats for the equation of a line is the slope-intercept form. That form is written, y = mx + b The variables here are x and y. The letters m and b are constants that represent the rise or tilt of the line (slope, m) and the point at which the line crosses the y-axis (intercept, b). So the first of the three equations for a line listed above is written in…

## 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…