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Parabolas: From Definition to Equation

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides practice problems related to developing formulas for parabolas.

We have talked about the geometric definition of a parabola: “all the points in a plane that are the same distance from a given point (the focus) that they are from a given line (the directrix).” And we have talked about the general equations for a parabola:

Vertical parabola: y = a ( x h ) 2 + k y=a(xh ) 2 +k

Horizontal parabola: x = a ( y k ) 2 + h x=a(yk ) 2 +h

What we haven’t done is connect these two things—the definition of a parabola, and the equation for a parabola. We’re going to do it the exact same way we did it for a circle—start with the geometric definition, and turn it into an equation.

A parabola whose focus is the origin (0,0) and directrix is the line y=-4.

In the drawing above, I show a parabola whose focus is the origin (0,0) and directrix is the line y = -4 y=-4. On the parabola is a point ( x x, y y) which represents any point on the parabola.

Exercise 1

d 1 d1 is the distance from the point ( x x, y y) to the focus (0,0). What is d 1 d1?

Exercise 2

d 2 d2 is the distance from the point ( xx,yy) to the directrix (y = -4y=-4). What is d 2 d2?

Exercise 3

What defines the parabola as such—what makes (xx,yy) part of the parabola—is that these two distances are the same. Write the equation d 1 = d 2 d1=d2 and you have the parabola.

Exercise 4

Simplify your answer to #3; that is, rewrite the equation in the standard form.

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