# Re-scaling Functions under the Same Axes

• ecastro
In summary, the conversation discusses the graphing of two functions, ##f(x,y)## and ##g(px,qy)##, on the same graph. It is suggested to plot ##f(x,y)## and ##h(x,y)##, defined as ##h(x,y)\equiv g(px,qy)##, using dual scales on the x and y axes. It is clarified that the graph is a 3D graph. There is a discussion about the definition of the function ##g(px,qy)## and which map should be graphed.
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?

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.

I don't quite understand... Sorry.

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)##?

andrewkirk said:
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.

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)##.

## 1. What is the purpose of re-scaling functions under the same axes?

Re-scaling functions under the same axes allows for better comparison and analysis of different functions on the same graph. It helps to standardize the data and make it easier to identify patterns and relationships.

## 2. How do you re-scale a function under the same axes?

To re-scale a function under the same axes, you need to adjust the scaling of the axes to make the data points fit within the graph. This can be done manually by changing the scale of the axes or by using a software tool that allows for automatic re-scaling.

## 3. Can you re-scale functions under the same axes with different units?

Yes, you can re-scale functions under the same axes with different units. However, it is important to ensure that the units are compatible and can be converted to each other. Otherwise, the re-scaled graph may not accurately represent the data.

## 4. What are the benefits of re-scaling functions under the same axes?

Re-scaling functions under the same axes can help to visually compare and analyze multiple functions on the same graph. It also allows for easier identification of trends and patterns in the data. Additionally, re-scaling can help to make the data more visually appealing and easier to interpret.

## 5. Are there any limitations to re-scaling functions under the same axes?

One limitation of re-scaling functions under the same axes is that it can sometimes distort the data and make it difficult to accurately interpret. This is especially true when there is a wide range of values on the same graph. Additionally, re-scaling can also hide important details in the data if the scale is not chosen carefully.

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