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 .

 

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 19-Sep-2007