User:Marty/UCSC Math 110 Fall 2008/Homework 5

For this homework, please turn in five pages. On each page, please follow these steps:
 * 1)  Draw a range topograph, beginning with the numbers $r,s,t$ given below.  Use a black pen or pencil!
 * 2)  Include at least 15 numbers in each range topograph.  You do not need to include the domain topograph information (i.e., which face is $\pm (2,3)$).
 * 3)  Find a quadratic form $Q(x,y)$ which takes the values $r,s,t$ at "home base", in other words:
 * 4) * $Q(1,0) = r$, $Q(0,1) = s$, and $Q(1,1) = t$.
 * 5)  Find the discriminant of the quadratic form.
 * 6)  Color the lakes and rivers with a blue pen or marker.
 * 7)  Label the edges with arrows or zeroes, based on "climbing".
 * 8)  How many times does $1$ appear on the range topograph?  (once, twice, infinitely many times, etc...?)
 * 9)  Write down an observation about the topograph, e.g., "There are only negative numbers in the topograph".

Please staple these five pages together. The five topographs must begin with the following five triples of integers at home base -- these are the "$(r,s,t)$" that you should use in the above directions.
 * 1)  $(1,2,3)$.
 * 2)  $(1,-1,5)$
 * 3)  $(1,-1,0)$
 * 4)  $(2,-2,4)$
 * 5)  $(1,1,0)$

Each topograph is worth two points. Please try a rough draft first, before drawing a final draft to turn in. Your final draft should be drawn in such a way that the most important features are easily seen.