Plotting points and functions in one graphic

Question

I have this plot:

Plot[{0 x, -x, x, -Sqrt[2 - x^2], Sqrt[2 - x^2]}, {x, -3, 3}, 
 PlotStyle -> {Purple}]

Now I want to do the following. I want to plot the points: $(-1,0),(1,0),(0,0),(x,0),(1, \pm 1),(1,\pm \frac{1}{\sqrt 3}),(0, \pm \frac{2}{\sqrt 3}),(0, \pm \sqrt 2)$ in this graphic. I'm able to plot points using Graphics[Point[{x, y}]but I would like to plot those point in the same graphic. This is probably really easy, but I can't find it in the documentation.

Answer 1

Thanks to george2079 I was able to get what I needed:

Data = {{-1, 0}, {1, 0}, {0, 0}, {1, 1}, {1, -1}, {1, 
   1/Sqrt[3]}, {1, -1/Sqrt[3]}, {0, 2/Sqrt[3]}, {0, -2/Sqrt[3]}, {0, 
   Sqrt[2]}, {0, -Sqrt[2]}}
Show[Plot[{0 x, -x, x, -Sqrt[2 - x^2], Sqrt[2 - x^2]}, {x, -3, 3}, 
  PlotStyle -> {Purple}], Plot[0 x, {x, -3, 3}, PlotStyle -> {Blue}], 
 Graphics[{PointSize[Medium], Blue, Point[Data]}]]

Answer 2

For fun, let's describe the ellipse--which is actually a circle--in polar coordinates and specify the points more simply as complex numbers rather than pairs of reals. And then use David Park's Presentations add-on (http://home.comcast.net/~djmpark/DrawGraphicsPage.html) in order to draw the whole thing--without having to use PlotStyle, Graphics, or Show (at least, not overtly):

pts = {1, 1 + I, 1 + I/Sqrt[3], 2 I/Sqrt[3]};
pts = Join[{0}, pts, Conjugate[pts]];
Draw2D[{
        Thick, Purple, PolarDraw[Sqrt[2], {t, 0, 2 Pi}], 
        Thickness[0.0035], Draw[{-x, x}, {x, -3, 3}],
        Black, PointSize[Large], ComplexPoint /@ pts
       }, 
       Axes -> True, PlotRange -> 3]