The Locus of a Class of Parabolic Functions
by

 Gayle Gilbert & Greg Schmidt


 

 

Problem: Consider the locus of the vertices of the set of parabolas graphed from

 

.

 

 

Show that the locus is the parabola

 

.

 


 

Consider , where , and the corresponding graphs:

 

 

Software: Microsoft Office

 

 

Now using calculus to find the minimum of each of the graphs we have for the minimum x-values:

,

since

.

 

Now, , and so we have that for each of our graphs

 

The minimum y-values of:

 

.

 

So for the vertex of each of our graphs is:

 

.

 

 

Now consider the function .  By overlaying it on our previously graphed functions we notice that it appears as though  intersects every graph at its vertex.

 

 

 

Software: Microsoft Office

 

 

Indeed, this is the case.  Consider again our general , and set it equal to . 

 

So we have

 

 

 

.

 

Hence,  intersects each  at its respective x-coordinate minimum.

Similarly, we check what we already suppose to be true, namely that

 

,

 

which is the y-minimum of each of our respective .

 

Whereby, we have shown that the locus of  is given by the function

 

.

           


Return