Re-scaling Functions under the Same Axes

[ecastro]

Consider two functions $f\left(x, y\right)$ and $g\left(px, qy\right)$, where $p$ and $q$ are known. How can I plot the two functions on the same graph (i.e. the same axes)? The function $f\left(x, y\right)$ will have axes with values $x$ and $y$, while the other will have axes with values $px$ and $qy$. Should the function $g\left(px, qy\right)$ have its $x$ and $y$ values as $\frac{x}{p}$ and $\frac{y}{q}$, respectively?

 

[andrewkirk]

This is a 3D graph you are drawing, right?

Why not just plot the functions $f(x,y)$ and $h(x,y)$ where $h$ is defined by $h(x,y)\equiv g(px,qy)$?

That's effectively what you'll be plotting anyway if you use dual scales on the x and y axes where the x-axis tick mark for $x$ for the $f$ graph is the same as for $\frac{x}{p}$ for the $g$ graph (and for y-axis $y\to \frac{y}{q}$). Having two dual scales on two axes of a 3D graph is getting just a bit too busy.

 

[ecastro]

I don't quite understand... Sorry.

 

[andrewkirk]

Is it a 3D graph you are drawing? You describe your functions as $f(x,y)$, which implies three dimensions, so that $f(x,y)$ is graphed as a curved surface whose height above the x-y plane is given by $z=f(x,y)$. Is that what you are doing?

If not, then what are you trying to say when you write that you want to graph $f(x,y)$?

 

[ecastro]

Is it a 3D graph you are drawing? You describe your functions as $f(x,y)$, which implies three dimensions, so that $f(x,y)$ is graphed as a curved surface whose height above the x-y plane is given by $z=f(y)$. Is that what you are doing?

If not, then what are you trying to say when you write that you want to graph $f(x,y)$?

Yes, it's a 3D graph.

 

[andrewkirk]

What do you mean by 'the function $g(px,qy)$'? A function is a map from one set (the domain) to another (the range). In your case it looks like the domain is $\mathbb{R}\times\mathbb{R}$ (ie the set of all ordered pairs of real numbers) and the range is $\mathbb{R}$. That much is clear.

But which of the following is the map that you want to graph?

1. The map that, for an element of the domain identified by the ordered pair of real numbers $(u,v)$, maps to an element in the range that is the real number $g(u,v)$;
OR
2. The map that, for an element of the domain identified by the ordered pair of real numbers $(u,v)$, maps to an element in the range that is the real number $g(pu,qv)$.