Conics
There are four
conic sections which are often discussed in algebra classes, these include: the parabola, the circle, the ellipse, and the hyperbola.
Given an equation, you should be able to determine which of the four conic
sections the equation represents. It is fairly easy to determine the
type of conic section from the equation. The parabola is an equation
in two variables (usually x and y) where
either the x
or the y variable is a squared term.
For example, 
and 
are equations whose graphs would be parabolas. For the
circle, the ellipse, and the hyperbola
both the x and
the y variables are squared terms. These conics can best be
distinguished by observing the coefficients of the equation in the form
. If a = b, then the graph of the equation would be a
circle. For example, 
is an equation whose graph would be a circle. If a
≠ b, but both a and b have the same sign, then the graph of the equation
would be an ellipse. For example,
and 
are equations whose graphs would be ellipses. Finally,
if a and b have different signs, then the graph of the equation would be a
hyperbola. For example, 
and 
are equations whose graphs would be hyperbolas. Once
again, I am limited to using FrontPage to create this review; therefore, I
cannot show examples of the graphs. However, I can give you a few
pointers on the graphs.
The circle
needs to be in a form where the center and radius may be determined. The
general form for a circle is 
where (h, k) represents the center and r represents the
radius. Thus, 
would be a circle with a center at (2, 3) and a radius
of 4. 
would be a circle with a center at (–3, 0) and a radius
of 5. To sketch the circle, locate the center point and then plot points
above, below, to the right and to the left of the center point, the
appropriate distance as determined by the given radius.
The graphing form for the ellipse is
. To transform the ellipse
from the standard form to the graphing form, divide all
terms by the constant number …
and 
.
Once this is accomplished, the a will designate the
x-intercepts and the b will designate the y-intercepts of the ellipse
“centered” at (0,0). Thus, the x-intercepts are at ±2 and the
y-intercepts are at 
.
The graphing form for the hyperbola is
. To transform the hyperbola
from the standard form to the graphing form, divide all
terms by the constant number …
and 
.