Lesson The Parabola -- An Algebra Approach by Rapalje

There are several different ways to explain and graph parabolas.  Probably the most sophisticated method is presented in analytic geometry, where a parabola is defined as the set of all points at an equal distance from a given point (which is the focus of the parabola) and a given line.  The equation of such a parabola is in the form  (if a > 0, the graph opens upward, while if a < 0, the graph opens downward), or (if a > 0, the graph opens to the right side, while if a < 0, the graph opens to the left side).

 A second approach, the one you probably used in lower math classes, is to graph an equation in the form  or  by plotting enough points to get an idea what the graph looks like. 

 After completing Section 2.02 on “Graphing by Translation,” it seemed a good idea to explain graphing of parabolas by “translating” the well-known parabolas  or , by shifting these graphs up, down, right, left, or inverting the graphs.  In this section, equations in general form  are converted to the “translated” form  by the method of completing the square.  In this form, the vertex at (h, k) is easily identified.  However, the completing the square process can get downright ugly!

Then I realized that there is an easier way to find the vertex of .  The vertex will ALWAYS be at  .  In my opinion, this is the preferred method of graphing parabolas.  For a detailed explanation on this method, please click here to see the Math in Living Color section that pertains to this topic. 

Actually, I recommend that you use the  method, together with the methods of the graphing calculator, to solve the problems in this section instead of the method of completing the square.  Please see also Calculator Workshop Notes for the TI83/84.

ANSWERS 2.03

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