The Locus of Vertices for a Family of Parabolas

By

Kenneth E. Montgomery

Equation 1 is the standard form for the equation of a parabola.

Equation 1:



Holding a and c constant (a = c = 1) we let b = n, for  in this GCF file which animates the parabola as n varies between these values. This animation can also be viewed in movie format, by opening this AVI file. The graphs of Equation 1 are overlaid in Figure 1, for { | -3 < b < 3}.
 
 


 

Figure 1: , with a = c = 1 and { | -3 < b < 3}



We hypothesize from the animation and from the graphs in Figure 1 that the locus created by the vertices of each parabola in the family of Equation 1, for  is itself a parabola. If we overlay the plot of Equation 2 (in black) onto Figure 1, we see that the locus indeed appears to be this parabola (Figure 2). A general proof that the locus is the parabola of Equation 2 is readily provided, using Calculus.
 
 


 

Figure 2: and  , with a = c = 1 and 



Proof:

For  let  where then  For the slope of this equation, we have  Setting this first derivative equal to zero

and solving for x, we have

 

This equation describes the x coordinates of the locus of points. We wish to show that this locus forms a parabola. Substituting back into the equation,
 
 

with  and



we have
 
 



and squaring the first term yields
 
 



Simplifying the first term and multiplying the second term by  , gives
 
 



Combining like terms, we have
 
 



Multiplication of the first term by yields
 
 



Factoring the a, we have
 
 



Squaring the negative root yields
 
 



for which we then substitute x, resulting in the desired parabolic equation
 
 


 

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