Question 826426
The points on the x axis would be like (x,0), because y is zero on the x-axis.  The vertex point is directly in the middle of (0,2) and (x,0), being at (0,1).


You are looking for distances from (x,y) to (0,2) to be equal to distances from (x,y) to (x,0).  Use the Distance Formula for expressions of these two distances and equate them.


{{{sqrt((x-0)^2+(y-2)^2)}}}
{{{sqrt(x^2+y^2-4y+4)}}}
'
The other distance,
{{{sqrt((x-x)^2+(y-0)^2)}}}
{{{sqrt(0+y^2)}}}
{{{sqrt(y^2)}}}
'
{{{sqrt(x^2+y^2-4y+4)=sqrt(y^2)}}}
Square both sides,
{{{x^2+y^2-4y+4=y^2}}}
{{{x^2-4y+4=0}}}
{{{x^2+4=4y}}}
{{{4y=x^2+4}}}
{{{highlight(y=(1/4)x^2+1)}}}


Graph might help:
{{{graph(300,300,-10,13,-3,22,(1/4)x^2+1)}}}