Looking for in depth reasoning behind why f(x)=+sqrt(x-2) does not have a vertices, is it because half parabolas can never have a vertex? Does a vertex always need to be a curve and/or a vertical parabola? Thanks!
I am not sure there is a general definition of a vertex. There is a vertex of a parabola of course. It is a point. A parabola does not have vertices: it has a vertex, which is a point with specific properties. What properties are you assigning to a vertex without reference to some curve?
Yes, it is reasonable to say that the equation \(\displaystyle y = \sqrt{x - 2}\)
is in the shape of a half parabola because \(\displaystyle x \ge 2\ and\ y \ge 0 \implies x = y^2 + 2.\)
So if what you mean by vertex is an extremum (either a minimum or maximum), you could say that (2, 0) is a minimum for both x and y and therefore a vertex.
If, however, what we mean by vertex is where an unbounded curve changes from increasing to decreasing or from decreasing to increasing, there is no vertex at all for your function.
What I think your quite reasonable question tells us is that you have to be very careful about definitions. With respect to a parabola, a vertex is the point where there is an extremum and the curve changes from decreasing to increasing or from increasing to decreasing. There is no point where the curve of your function is anything but increasing. Change definitions, and you change answers.
