How to Superimpose Multiple Plots in Mathematica?

[abbeynewton]

Hi everyone, i have been recently trying to get familiar with the Mathematica software for a project am working on. In the course of that i encountered a problem on graphics. here's the question:
Superimpose the plots y=√(x^2+a), √a≤x≤√a+3 for the values a=1,a=4,a=9 in the same figure.

Now the challenge is to input this correctly into Mathematica. I tried doing it this way:

eqy1[x_]:=√(x^2+a)/; a -> 1
eqy2[x_]:=√(x^2+a)/; a -> 4
eqy3[x_]:=√(x^2+a)/; a -> 9
Plot[eqy1[x],eqy2[x],eqy3[x], {x, √a, √a+3}, PlotRange -> {-3,3}, PlotStyle->{red,green,blue}]

when i entered this ,it sent an error message saying that an option is needed beyond position 2 and all of that.

Please i need any clue, hint or help on how to input it correctly. I have only a month to learn everything i need to know in Mathematica for my final year project, and i need to get this problem before i go through. thanks

 

[CompuChip]

The function you want to plot is just the first argument. If you want to plot multiple functions in the same graph, you need to pass them as a list.
Try

Plot[{eqy1[x],eqy2[x],eqy3[x]}, {x, √a, √a+3}, PlotRange -> {-3,3}, PlotStyle->{red,green,blue}]

If you are using Mathematica 7 or higher, this will do just fine.

 

[CompuChip]

By the way, I think you meant to use /. instead of /; in your definitions.

expr /. a -> b

replaces a with b everywhere in the expression expr.

/; is more like a conditional operator, for example

abs[x_] := x /; x >= 0
abs[x_] := -x /; x < 0

You don't even really need it here, you could just define a function of two variables

f[x_, a_] := √(x^2+a)

or

Subscript[f, a_][x_] := √(x^2+a)

(which you can also write as f_{a_}[x_] using Control + -)
and then plot f[x, 1], f[x, 4] and f[x, 9] (or f₁[x], f₄[x] and f₉[x] respectively)

 

[abbeynewton]

thanks CompuChip...this was really helpful

 

[abbeynewton]

But i still have a problem with the limit...mathematica is telling me that sqrt[a] in the limit is not a machine size real number...what should i do?

 

[Bill Simpson]

If you can kill the kernel, restart it, evaluate the whole notebook, select all the cells with input, output and error message and paste that here then it should be possible to quickly identify exactly what the problem is.

 

[CompuChip]

The probelm is exactly what it says.
If you plot a graph then you can only plot it if you give Mathematica numerical boundaries.
For example it can plot f(x) between x = -3 and x = 3, or between x = 1200 and x = 1987, but it cannot plot between x = -a and x = a if it doesn't know what a is.

If this is what you want, you will probably end up doing something like

Show[{
  Plot[f[x, 1], {x, Sqrt[1], Sqrt[1+3]}, PlotRange -> {-3,3}, PlotStyle->red],
  Plot[f[x, 4], {x, Sqrt[4], Sqrt[4+3]}, PlotRange -> {-3,3}, PlotStyle->green],
  Plot[f[x, 9], {x, Sqrt[9], Sqrt[9+3]}, PlotRange -> {-3,3}, PlotStyle->blue]
}]

or you will have to plot them all on the smallest interval including them all, i.e.

Plot[{f[x, 1], f[x, 4], f[x, 9]}, {x, Sqrt[9], Sqrt[9+3]}, PlotRange -> {-3,3}, PlotStyle->{red, green, blue}]

This will plot all three of them between x = 3 and x = 2 sqrt(3), the PlotRange will take care of cutting off the functions on the vertical axis.

 

[abbeynewton]

thanks for the explanation...was really helpful...but does is mean that Mathematica does not recognize, squareroots of unknown variables, and you have to define them,even when using intervals??

 

[CompuChip]

How should it?

Can you plot f(x) = x² between x = -a and x = c?