December 9th, 2011, 08:15 AM  #1 
Newbie Joined: Apr 2010 Posts: 12 Thanks: 0  Sketching graphs in R3
Sketch the graph . Let z = 0: Now when drawing the graph on the y and x axis, I understand that there are hyperbolas which intersect yaxis (as letting you get ). What I don't get is in my book their hyperbolas drawn, (on the same graph), which intersect the xaxis. This shouldn't be possible since letting you get , which can't be solved. 
December 9th, 2011, 12:47 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,343 Thanks: 534  Re: Sketching graphs in R3
Your question is confusing. Where is z? Also k=0 is possible  that's how you get the lines crossing the origin.

December 10th, 2011, 12:10 AM  #3 
Newbie Joined: Apr 2010 Posts: 12 Thanks: 0  Re: Sketching graphs in R3
What I meant to say is when you let (to get the view from above in R3) you get: This would be graph of the hyperbolas intersecting the yaxis, but in the book they are hyperbolas which also intersect the xaxis. Don't understand why there are hyperbolas that intersect the xaxis (as letting , you get which can't be solved). 
December 10th, 2011, 12:29 AM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Sketching graphs in R3
The hyperbolas intersecting the xaxis represent negative values of k. When 0 < k they intersect the yaxis, when k < 0 they intersect the xaxis, and when k = 0, you get degenerate cases, the lines y = ±x. This is the family of curves satisfying: 
December 16th, 2011, 01:47 AM  #5 
Newbie Joined: Apr 2010 Posts: 12 Thanks: 0  Re: Sketching graphs in R3
For k < 0, how can you draw , when the righthandside has a negative 1? (The standard equation of a hyperbola has a positive 1 on the righthandside).

December 16th, 2011, 05:03 AM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Sketching graphs in R3
The hyperbolas of the form: have vertices at (a,0) and (a,0), while the hyperbolas of the form: have vertices at (0,a) and (0,a). In our case, we have , so we have: or When k < 0, we may choose the form: so that now we have a positive value on the right and vertices on the xaxis. When 0 < k, we choose the form: so that we have a positive value on the right and vertices on the yaxis. In both cases, the asymptotes are the lines . When k = 0, we have: 

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