Finding the shape of a function

[SqueeSpleen]

While I was doing exercises about this, I noticed that, for example $f(x,y)=e^{x^2+y^2}$ was very similar to $f(x,y)=x^2+y^2$
And something similar with:
$f(x,y)=e^{(1-x^2-y^2)^{(1/2)}}$ and $f(x,y)=(1-x^2-y^2)^{(1/2)}$
So I supposed that some transformations that preserves some properties kinda preserves also the shape of some functions.
So when I had to graph the function $e^{1-x^2-y^2}$ I explained that I draw something similar to a paraboloid, but the points that would be $z \in (-\infty,0)$ in a paraboloid, where in (0,1), the point of z = 0 was in 1, and the points $z \in (0, \infty)$ where in (1,∞), I think it's something like ploting the function in "logarithm scale".
I wouldn't put this explanation when it wasn't rigoruos at all, and I only was asked to sketch the function (I did this well :p), but this information was not only irrelevant but also wrong :P, so I lost points doing that.

I wanted to know how this is called and a formal explanation of this phenomenon because I didn't heard about this in class but I guess this was the sort of explanation I had to do about why I sketched that graph.

Offtopic questions I don't know where to ask or I guess they don'r deserve a new topic:
(*) What's the best online English-Spanish math dictionary?

(*) If you graph in a fixed area a hiperboloid, for example
x^2+y^2-1=z
with $x \in (-a,a)$ $y \in (-a,a)$ as "a" gets bigger, the graph of the hiperboloid will be closed to the graph of the cone x^2+y^2=z, same will happen with x^2+y^2+1=z
I guess same will happen with a lot of curves, not all, I think some curves with, for example trigonometric functions will never "converge".
There's some subject where this is studied? How it's called it?

 

[Simon Bridge]

I wanted to know how this is called and a formal explanation of this phenomenon because I didn't heard about this in class but I guess this was the sort of explanation I had to do about why I sketched that graph.

I would first make sure about what it is that was "not only irrelevant but wrong" - maybe you were not supposed to use a log scale?

It is true that sometimes when you have z=f(g(x,y)), f will preserve important properties of g or g will preserve the overall shape of f in some way.

But, in the examples, it looks like an accident - f and g had similar symmetry to start out with.

the hiperboloid will be closed to the graph of the cone ... I guess same will happen with a lot of curves, not all, I think some curves with, for example trigonometric functions will never "converge".
There's some subject where this is studied? How it's called it?

... "conic sections" - "solids of rotation"?
The hyperboloid is a conic section that has been rotated - in your case is is constrained to the cone traced out by rotating the assymptotes.
That does not have to be the case - it could be constrained to an elliptical "cone".

 

[SqueeSpleen]

It is true that sometimes when you have z=f(g(x,y)), f will preserve important properties of g or g will preserve the overall shape of f in some way.

What I would like to know is a way of knowing it without doing the graphs, when will g or f preserve it?
There's some easy way to know it?
Or the best I had to do was to make a table, explain that the level curves were circles because if we make a coordinates change x^2+y^2=r^2, then e^(1-r^2) then r is the only thing that determines the value of e^(1-r^2), and then sketch the graph?
I knew that g(x)=e^x preserved this properties with some functions so I used, but I guess I shouldn't.

Anyway I know the best figure out how exactly I was supposed to sketch it would be to ask this to teacher but I already finished the course, anyway I'll try to ask her, but I'm curious now :p

 

[Simon Bridge]

What I would like to know is a way of knowing it without doing the graphs, when will g or f preserve it?
There's some easy way to know it?

No - you just have to learn the general form of common transformations ... translation, rotation, scaling etc. ... that can be relied upon to preserve the shape of the graph. In function form those transformations are usually controlled by g rather than f. Look up "shape preserving transformations".

Or the best I had to do was to make a table, explain that the level curves were circles because if we make a coordinates change x^2+y^2=r^2, then e^(1-r^2) then r is the only thing that determines the value of e^(1-r^2), and then sketch the graph?
I knew that g(x)=e^x preserved this properties with some functions so I used, but I guess I shouldn't.

You seem to have over-generalized from a few examples.
What would have been best in your situation was whatever the person marking expected you to do. You should find out what that was.
The point of such exercises is not so much to complete the task in front of you (otherwise copying someone else would be fine) but to learn (or show you have learned) a particular set of lessons.

Anyway I know the best figure out how exactly I was supposed to sketch it would be to ask this to teacher but I already finished the course, anyway I'll try to ask her, but I'm curious now :p

That's right - ask the teacher what was intended. There should be clues in the course materials too.