Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

More Graphs

The principles illustrated for parabolas apply to other graphs as well. We illustrate them here for circles (which are a special case of ellipses) and hyperbolas.

Circles and Ellipses

Consider the circle of radius $ 1 $ around the origin. This particular circle is sometime called the unit circle. It is the set of all points $ (x,y) $ whose distance from the origin is $ $1. By the Pythagorean Theorem the unit circle is the graph of the equation

$\displaystyle x^2+y^2=1.\qquad(*) $

More generally, the circle of radius $ r $ around the origin is the graph of the equation

$\displaystyle x^2+y^2 = r^2. $

As discussed earlier for parabolas , the circle can be translated so that its center is the point $ (h,k) $ which gives the general equation of a circle of radius $ r $ with the center $ (h,k) $:

$\displaystyle (x-h)^2 + (y-k)^2 = r^2. $

Consider again the special case that the center of a circle is the origin and its radius is $ r $. Then its equation can be rewritten as

$\displaystyle \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1. $

If we now scale , possibly differently in the horizontal and vertical directions, we obtain the equation

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1. $

This describes an ellipse with its center at the origin, a horizontal axis of length $ 2a $, and a vertical axis of length $ 2b $. If $ a=b=r $ we obtain again the circle of radius $ r $ (around the origin). The longer of the two axes is called the major axis of the ellipse, and the shorter its minor axis. Textbooks often assume that $ a>b $ (and switch $ a $ and $ b $ if the vertical axis is the major axis).

Figure 1. Circles and Ellipses

Figure 1 shows the unit circle in red, the ellipse with $ a=1,~b=\frac{1}{2} $ in yellow, and the ellipse with $ a=\frac{1}{2},~b=1 $ in green.

An ellipse can be translated so that its center is at $ (h,k $) which gives rise to the equation

$\displaystyle \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2} = 1. $

which is sometimes called the standard equation of an ellipse.

Note how all the relevant information about the ellipse, i.e., its center and the lengths and orientations of its major and minor axis, can be read directly from this equation.

Hyperbolas

Figure 2. Two Hyperbolas and their Asymptotes

Consider again the equation $ (*) $ of the unit circle. If we replace addition by subtraction we obtain the equations

$\displaystyle x^2-y^2 = 1 \qquad(**) $

or

$\displaystyle y^2-x^2 = 1 \qquad(***) $

Each of these these equations describes a hyperbola. A hyperbola has two branches. In Figure 2, the yellow and blue curves form the graph of $ (**) $ the purple and turquoise curves form the graph of $ (***) $. Also shown are the lines $ y=x $ (red) and $ y=-x $ (green) which are the asymptotes of the hyperbolas. The graphs of the hyperbolas approach these lines arbitrarily closely without ever reaching them.

Like other graphs, hyperbolas can be scaled and translated.

Conic Sections

The discussion on this page is a very brief introduction to a much larger and richer subject: the study of conic sections. A conic section is the intersection of a plane and a cone. Such curves can be expressed as the graphs of the general quadratic equation

$\displaystyle Ax^2 +Bxy +Cy^2 + Dx + Ey + F = 0 $

where $ A\ldots F $are constants. Conic sections include ellipses, parabolas, hyperbolas, pairs of intersecting lines, and single points. Their detailed study is intricate, fruitful, and fascinating, but beyond the scope of this class.