# Thread: Asymmetry of X-axis and Y-axis, why?

1. ## Asymmetry of X-axis and Y-axis, why?

Throughout all of my education in math it was frequently implied (and IIRC occasionally stated, but NEVER explained) that it was ok for a plot to have multiple values of x for a given value of y, but not ok for there to be multiple values of y for a given value of x. [That, for example, if one were to graph X x Y = X that it should be massaged into Y=X/X and then into Y=1 and thus be a horizontal line ay Y=1 (possibly with a gap at X=0 if we stop the process at Y=X/X) rather than the cross shape that the original X*Y=X denotes]

I never understood the point of this, can someone explain it?  Reply With Quote

2. ## Re: Asymmetry of X-axis and Y-axis, why?

I suspect they are thinking of clarity of presentation - I have certainly seen graphs/charts wth multiple Y scales (usually only 2 - one deliniated to the left and one to the right).

Graphs with multiple X scales are much rare in my experience as you have nowhere to put the second scale.

Note for these graphs what is usually important is the shape of the curve - so if you are plotting how two different measures change over time you put time along the X axis and the measures on Y and then can compare the shape of the lines - direct comparison of the values being meaningless.

Actually that last point may be why they are saying don't do this - people tend to perform the direct comparison even when they should not.  Reply With Quote

3. ## Re: Asymmetry of X-axis and Y-axis, why?

This statement isn't necessarily true for arbitrary plots, but it is true for the type of plot that is most commonly considered in early math education: plots of functions. And that's the explanation that you are looking for:

A function is, by definition, a mapping of elements from two sets X and Y, where each element in X is mapped to exactly one element in Y. Thus, one value in Y can be the target of the mapping for multiple values in X but not the other way around.  Reply With Quote

4. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Khedrac I suspect they are thinking of clarity of presentation - I have certainly seen graphs/charts wth multiple Y scales (usually only 2 - one deliniated to the left and one to the right).

Graphs with multiple X scales are much rare in my experience as you have nowhere to put the second scale.

Note for these graphs what is usually important is the shape of the curve - so if you are plotting how two different measures change over time you put time along the X axis and the measures on Y and then can compare the shape of the lines - direct comparison of the values being meaningless.

Actually that last point may be why they are saying don't do this - people tend to perform the direct comparison even when they should not.
I don;t mean two diffetent scales I mean two different values of Y at a given value of x. So, for example, if we defined Y as the square root of X, the graph would contain the points (4,2) and (9,3) but for whatever reason it wouldn't contain (4,-2) and (9,-3)  Reply With Quote

5. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Bohandas I don;t mean two diffetent scales I mean two different values of Y at a given value of x. So, for example, if we defined Y as the square root of X, the graph would contain the points (4,2) and (9,3) but for whatever reason it wouldn't contain (4,-2) and (9,-3)
This isn't an X/Y situation so much as an input/output situation, where the convention happens to be using X as an input. Functions are defined such that each input has exactly 1 output, which is why for the function y=sqrt(x) square root is defined as the positive square root in particular. The same would apply for the function x=sqrt(y), where y is used as an input instead.

Functions being built this way is arbitrary at a deep level, but there are a number of useful properties which depend on it, especially when working with multiple functions at once. Other branches of mathematics don't necessarily do this.  Reply With Quote

6. ## Re: Asymmetry of X-axis and Y-axis, why?

Functions take Elements/Numbers from X and convert them to Y.
If you had programming, its pretty clear that you want your, say sqareroot function to always act the same. For one input value, you want one output value, that you can then use. If you get two different things out, its hard to catch them both and use them meaningfully.
Functions act much like black boxes or machines. To do more complicated stuff, you want to link them together in a chain. Like exp(sin(sqrt(x))).
This only works if each function gives only one output.  Reply With Quote

7. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Rydiro Functions take Elements/Numbers from X and convert them to Y.
They take elements/numbers from something and convert to something - X and Y are common conventions for functions of a single variable that produce a one dimensional output, but they aren't by any means required. For instance another common convention is for a function to output X,Y,Z coordinates using a t input, or a u and a v input in a more genericized sense.  Reply With Quote

8. ## Re: Asymmetry of X-axis and Y-axis, why?

{Scrubbed}. Mathematicians do deal with equations such as y²=x and x²+y²=1 and the graphs.  Reply With Quote

9. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Knaight They take elements/numbers from something and convert to something - X and Y are common conventions for functions of a single variable that produce a one dimensional output, but they aren't by any means required. For instance another common convention is for a function to output X,Y,Z coordinates using a t input, or a u and a v input in a more genericized sense.
You are right, I simplified a bit. I tried to avoid going into sets, set theory and category theory.  Reply With Quote

10. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Bohandas Throughout all of my education in math it was frequently implied (and IIRC occasionally stated, but NEVER explained) that it was ok for a plot to have multiple values of x for a given value of y, but not ok for there to be multiple values of y for a given value of x. [That, for example, if one were to graph X x Y = X that it should be massaged into Y=X/X and then into Y=1 and thus be a horizontal line ay Y=1 (possibly with a gap at X=0 if we stop the process at Y=X/X) rather than the cross shape that the original X*Y=X denotes]

I never understood the point of this, can someone explain it?
Other's have stated the basic answer: functions have a unique mapping, meaning a single input only leads to a certain output. So when you see a graph where x is the input (i.e. a graph of a function of x), each x value will map to a single unique value.

You do see other graphs, though. You brought up an example yourself: y = sqrt(x), or y^2 = x. If you were to plot all the points on the x/y plane that are a solution to that equality, you'd get a sideways parabola. It's a perfectly valid plot, you can do that, it just isn't a plot of a function.

In school you're usually dealing with functions, hence the insistence on the unique mapping. If you were to take more pure math courses, you'd probably see the distinction more often. In highschool and most of first year (as well as the "focused" classes you see sometimes, like calculus classes aimed at physics and engineering students, even in upper years) the distinction isn't as useful so it often doesn't get explained all that well.  Reply With Quote

11. ## Re: Asymmetry of X-axis and Y-axis, why?

Yeah, this is a "functions" thing. Basically, the idea of a function is that it's a thing where if you hand it some specific thing as an input, you get the same output each time you give it that input. The way it's usually easiest to explain it to my students is that it's like a vending machine: each button (input) better give you a predictable kind of can of soda (output). It's ok to have two buttons that both give you Coke, but not ok to have a button that sometimes gives you Coke and sometimes gives you Caffeine Free Diet Coke.

Since the convention when graphing is to graph the inputs (independent variables) as x and the outputs (dependent variables) as y, if it's a function it'll sometimes have more than one x value with the same y value (just like having two buttons on a vending machine that are both stocked with the same product), but you won't have the same x value have two y values (just like you wouldn't have multiple products stocked in the same slot in a vending machine).

Not all graphs work this way, of course. The most common non-function graph a lot of people would see relatively early on in their math education would be a scatter plot. A lot of those would have data points that share x values but not y values depending on the data. However, you would be in high school before you need to graph equations that aren't functions depending on your curriculum, and it's fairly easy to miss the non-function equations along the way depending on how far you go in math and the particular courses you take. (Where I teach now, I think the equation of a circle in geometry is the first non-function equation most students will end up doing much graphing of other than the trivial ones of x=some constant that they'll see when learning about equations for lines in middle school.)  Reply With Quote

12. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by Algeh Yeah, this is a "functions" thing. Basically, the idea of a function is that it's a thing where if you hand it some specific thing as an input, you get the same output each time you give it that input. The way it's usually easiest to explain it to my students is that it's like a vending machine: each button (input) better give you a predictable kind of can of soda (output). It's ok to have two buttons that both give you Coke, but not ok to have a button that sometimes gives you Coke and sometimes gives you Caffeine Free Diet Coke.

Since the convention when graphing is to graph the inputs (independent variables) as x and the outputs (dependent variables) as y, if it's a function it'll sometimes have more than one x value with the same y value (just like having two buttons on a vending machine that are both stocked with the same product), but you won't have the same x value have two y values (just like you wouldn't have multiple products stocked in the same slot in a vending machine).

Not all graphs work this way, of course. The most common non-function graph a lot of people would see relatively early on in their math education would be a scatter plot. A lot of those would have data points that share x values but not y values depending on the data. However, you would be in high school before you need to graph equations that aren't functions depending on your curriculum, and it's fairly easy to miss the non-function equations along the way depending on how far you go in math and the particular courses you take. (Where I teach now, I think the equation of a circle in geometry is the first non-function equation most students will end up doing much graphing of other than the trivial ones of x=some constant that they'll see when learning about equations for lines in middle school.)
I haven't opened my math book in too long   Reply With Quote

13. ## Re: Asymmetry of X-axis and Y-axis, why?

You do get graphs where both X and Y take multiple values, but they need to be controlled by a parameter that isn't X or Y, or the combination of the two has to hold constant.

E.g.:    Reply With Quote

14. ## Re: Asymmetry of X-axis and Y-axis, why? Originally Posted by RCgothic You do get graphs where both X and Y take multiple values, but they need to be controlled by a parameter that isn't X or Y, or the combination of the two has to hold constant.
Or it can just be the locus of all solutions to an equation, like your heart example. I've seen software that graphs those without parameterizing the path, by generating a point cloud of solutions. A cloud of pixels whose square the solution passes through, anyway.  Reply With Quote

15. ## Re: Asymmetry of X-axis and Y-axis, why?

Yes, because thats how functions are defined is the answer. But its worth looking into why functions are defined that way.

A lot of graphs are created to answer some question of the form, If I do this, what would happen?

The asymmetry of the graph (and the definition of a function) come from the inherent difference between cause (I do this) and effect (what would happen).  Reply With Quote

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