Explanation of what a parameter is

[Hannah7h]

Please could someone explain, pretty simply, what a parameter and also its relation to graphs? And if possible give a couple of examples. I have trawled through the internet and can't find anything which I understand.

Thank you very much.

 

[Mark44]

Please could someone explain, pretty simply, what a parameter and also its relation to graphs?

A parameter is usually a constant of some kind, one that may or may not be known. One of the simplest equations with two variables is y = mx + b. The graph of this equation is a straight line whose slope is m and whose y-intercept is b. In this equation m and b are parameters, and x and y are the variables.

 

[fresh_42]

Please could someone explain, pretty simply, what a parameter and also its relation to graphs? And if possible give a couple of examples. I have trawled through the internet and can't find anything which I understand.

Thank you very much.

What @Mark44 wrote is one usage of the term parameter as the numerical set-up of a situation. Here parameters mean the numeric frame, as room temperature in an experiment, or a certain range of tension which a screw has to tolerate if used normally. You can even consider $(m,b)$ in his example as parameters of lines, which means all pairs $(m,b)$ describe a set of lines and a certain pair fixes one line.

This leads to the other meaning (graphs). A parameter is a variable which accompanies a way along this graph. We say a curve is parameterized, which means a curve is a function $f\, : \,[0,1] \longrightarrow \mathbb{R}$ and $t$ is a parameter of the points $(t,f(t))$, i.e. a walk of $t$ through $[0,1]$ corresponds to a walk through $\{(t,f(t))\}$. It is not by chance, that those parameters are often noted as $t$, because this reflects its association with time: like a walk in real life, where the parameter time on your watch corresponds to a certain location of your walk.

Other parameters can be temperature, pressure, distance, height or anyone dimensional quantity which serves as a measure. The term parameter is used, when the object of consideration is "walk" even if this walk is a metaphor like in the case of a drawn curve.

So the least common meaning is, that a parameter is a kind of measure, a numeric framework that describes a certain element in a set that contains all elements of a kind. That is a certain line $(m_0,b_0)$ in a set of lines $\{(x,y)\,\vert \,y=mx+b\; ; \;m,b \in \mathbb{R}\,\}$ as in Mark's example, or a certain point $t_0$ in the set of points on a line $\{(x,y)\,\vert \,y(t)=mt+b\, ; \,t\in [0,1]\,\}$ in my example. Therefore the confusion you've seen online might result from varying perspectives, i.e. sets of elements which are parameterized and as in our examples whether the lines are considered or the points on a line.

 

[Hannah7h]

A parameter is usually a constant of some kind, one that may or may not be known. One of the simplest equations with two variables is y = mx + b. The graph of this equation is a straight line whose slope is m and whose y-intercept is b. In this equation m and b are parameters, and x and y are the variables.

Thank you for replying, so are m and b parameters here because they have a fixed value (are constants) for this particular line equation?

 

[Mark44]

Thank you for replying, so are m and b parameters here because they have a fixed value (are constants) for this particular line equation?

Yes, for a given line, m and b would have fixed values. As @fresh_42 mentioned, there are other uses of the term "parameter," but as you didn't give any context for your question, my explanation is for only one of the uses of this term.

 

[Hannah7h]

Yes, for a given line, m and b would have fixed values. As @fresh_42 mentioned, there are other uses of the term "parameter," but as you didn't give any context for your question, my explanation is for only one of the uses of this term.

I see now, thank you very much for your help.

 

[Hannah7h]

What @Mark44 wrote is one usage of the term parameter as the numerical set-up of a situation. Here parameters mean the numeric frame, as room temperature in an experiment, or a certain range of tension which a screw has to tolerate if used normally. You can even consider $(m,b)$ in his example as parameters of lines, which means all pairs $(m,b)$ describe a set of lines and a certain pair fixes one line.

This leads to the other meaning (graphs). A parameter is a variable which accompanies a way along this graph. We say a curve is parameterized, which means a curve is a function $f\, : \,[0,1] \longrightarrow \mathbb{R}$ and $t$ is a parameter of the points $(t,f(t))$, i.e. a walk of $t$ through $[0,1]$ corresponds to a walk through $\{(t,f(t))\}$. It is not by chance, that those parameters are often noted as $t$, because this reflects its association with time: like a walk in real life, where the parameter time on your watch corresponds to a certain location of your walk.

Other parameters can be temperature, pressure, distance, height or anyone dimensional quantity which serves as a measure. The term parameter is used, when the object of consideration is "walk" even if this walk is a metaphor like in the case of a drawn curve.

So the least common meaning is, that a parameter is a kind of measure, a numeric framework that describes a certain element in a set that contains all elements of a kind. That is a certain line $(m_0,b_0)$ in a set of lines $\{(x,y)\,\vert \,y=mx+b\; ; \;m,b \in \mathbb{R}\,\}$ as in Mark's example, or a certain point $t_0$ in the set of points on a line $\{(x,y)\,\vert \,y(t)=mt+b\, ; \,t\in [0,1]\,\}$ in my example. Therefore the confusion you've seen online might result from varying perspectives, i.e. sets of elements which are parameterized and as in our examples whether the lines are considered or the points on a line.

Thank you very much, I understand it better now

 

[Kumar8434]

I have a great explanation of parameters. I read it on a website:

For every degree above 70, our convenience store sells $x$ bottles of sunscreen and $x^2$ pints of ice cream.

We could write the algebra relationship like this:

And it’s correct… but misleading!

The equation implies sunscreen directly changes the demand for ice cream, when it’s the hidden variable (temperature) that changed them both!

It’s much better to write two separate equations

that directly point out the causality. The ideas “temperature impacts ice cream” and “temperature impacts sunscreen” clarify the situation, and we lose information by trying to factor away the common “temperature” portion. Parametric equations get us closer to the real-world relationship. Here, 'temperature' is the parameter.

Here is the full article: https://betterexplained.com/articles/a-quick-intuition-for-parametric-equations/