How to Sketch the Surface Represented by \( z = x^2 - y^2 \)?

[Madou]

z = x²-y²
How do I draw it?

 

[e(ho0n3]

How would you draw any graph? Answer: By making a table of values and plotting them.

 

[Madou]

yes I say that z = const and look further

 

[Mark44]

yes I say that z = const and look further

?

But z is not a constant, according to the equation in your first post.

A more sophisticated approach than that of the other responder is to look at a variety of cross-sections. The graph of z = x^2 - y^2 is a three-dimensional surface. The intersection of the surface with the x-z plane (where y = 0) is a parabola, and so is the intersection of the surface with the y-z plane (where x = 0).

Look at the intersections with the planes y = +/-1, +/-2 and so on, as well as the planes x = +/-1, +/-2. These might help you visualize what the surface looks like.

 

[icantadd]

yes I say that z = const and look further

I wouldn't. I would set y = 0. Draw that like you would in high school z = x^2, looks like a parabola in the x going up /wrt z. Do the same for y = 0, except you get a parabola going down in the yz plane. I would say you can stop there. Draw one of the parabolas in xyz system. Suppose you choose z = -y^2. Then you get a parabola that is going down around the z axis. Now attach one of your z=x^2 parabolas to some points that lie on the z = -y^2 parabola. You can generally get a frame for a 3d graph like this. If you are really careful, the graph might look like a horses saddle.

 

[Madou]

yeah, that's what i was thinking - it might be a horse saddle !
But I wasn't sure about how to draw it carefully. Thank you all.