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I'm wondering if there is a plotting tool out there that will work in symbolic math?

For example, if I plot $x+y=a$ and $y=x$ on a chart, I should be able to click on the point of intersection to get $\left(\large \frac{a}{2},\frac{a}{2}\right)$.

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How do you expect it to plot $x + y = a$ without knowing the value of $a$ (e.g. whether it's positive or negative, which affects where the line should be drawn)? – ShreevatsaR Jun 16 '11 at 20:52
BTW, not exactly what you want, but Geogebra is great. You can give it a specific value of $a$ like $a=5$ or something, ask it to plot the lines $x+y=a$ and $y=x$, ask it to compute their intersection, and it will will show you the coordinates and automatically update them as you modify $a$. – ShreevatsaR Jun 16 '11 at 20:54
I suppose Mathematica (or WolframAlpha) will do that, but that might be a bit heavier than what you need. – Jack Henahan Jun 16 '11 at 21:30
@ShreevatsaR: Do you know whether you can save your "Geogebra-ic" work an image file (.jpg, .png, .gif), etc., or is it only "save-able" as a Geogebra file that only Geogebra can open? Just curious. – amWhy Jun 16 '11 at 21:48
@amWhy: Geogebra can export files in .png, .bmp, and .eps, as far as I can remember. – Beni Bogosel Jun 17 '11 at 0:08

Another soft for geometric drawings, which is maybe better than Geometer's Sketchpad, and (more important) free is Geogebra. In Geogebra you can plot lines given by certain equations (unfortunately, I don't think symbolic equations work), and you have a left panel with all the objects you have drawn and also their equations. There is a tool for intersection of two objects, and a new point is created. Every point can be found in the left panel with its coordinates.

A program which does symbolic computation and symbolic plotting at a very advanced level is Maple. Unfortunately, this is not free.

To do symbolic plotting is impossible on computer, since all the data on the screen needs coordinates to look the way it is. The best compromise, which I think is available with Geogebra, is to introduce a parameter, which varies between some fixed values, and can be modified with a cursor. Maple has this function for sure; I know that I've used it a couple of times.

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Symbolic equations sort of work, i.e., they work if all the symbols have previously been given values. :-) (As you said in your last paragraph. E.g. you can give it a=0, b=0, r=1, and ask Geogebra to plot $(x-a)^2 + (y-b)^2 = r^2$, and changing the values of a, b, r will change the circle. This is about as much as you can expect a computer to do, as far as I can imagine.) – ShreevatsaR Jun 17 '11 at 7:58

Are you familiar with Geometer's Sketchpad?

There's a student version for $29.95, and I believe a free trial version. The resources are fabulous (tutorials, exchange forums, downloadable/accessible sketches created by others, lessons, etc.) and it is really quite user friendly. It may even be "usable" on-line (free). Not sure on that.

I believe it also allows you to create java applets on web pages, but it's usually what I turn to for plotting, geometric constructions, determining points of intersection, area, etc. Your "sketches" can be saved, reloaded to edit, saved as image files, etc.

If you have an iPhone (and iPod touch) or an iPad, there's a program ($30?) called Spacetime that you can purchase to download and use on your device (making it essentially a graphing calculator). You're better off with the Wolfram App mentioned below if you're interested in access to mobile graphing and much more.

I mention this mainly because if you go to the Spacetime website, there's also a free desktop version of the program (platform for Mac and Windows), which allows plotting,solving equations, etc., but isn't as powerful as Sketchpad. It probably doesn't do much more than Wolfram Alpha can do, but I believe you can save work with the desktop version and I don't know you can do that with Wolfram Alpha. For more details of the capabilities of Spacetime, see the Spacetime online manual.

By the way, for the iPhone (and iPod touch) and iPad, there's an app (almost free, $1.99 USD - one time payment) available for downloading mobile access to Wolfram Alpha to your i-Pad-Pod-phone.

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Can Geometer's Sketchpad plot $x+y=a$ without knowing what $a$ is (as the OP seems to want)? – ShreevatsaR Jun 17 '11 at 7:55
@ShreevatsaR: pretty much as you describe it for geogebra in your comment to another answer: it allows you to vary the value of a variable, which changes the corresponding function/graph, etc. – amWhy Jun 17 '11 at 19:22
@amWhy: Nice suggestions +1 – Amzoti May 7 '13 at 2:25
+111111111111111 – Babak S. Jul 9 '13 at 6:31

Another way is to use Maple 15, 16 or recently version 17. In this powerful software which is famous in symbolic computations, you can easily probe the point of intersection as well as other useful points in any certain plots. I could do that for you while $a=4$:

enter image description here enter image description here

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