Jigu Suanjing: Difference between revisions

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In [[analytic geometry]], a '''truncus''' is a [[curve]] in the [[Cartesian plane]] consisting of all points (''x'',''y'') satisfying an equation of the form


[[File:Unit Truncus.png|right|thumb|300px|A mathematical graph of the basic truncus forumula, marked in blue, with domain and range both restricted to [5, 5].]]


<math>
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f(x) = {a \over (x + b)^2} + c
</math>
 
where ''a'',&nbsp;''b'', and&nbsp;''c'' are given constants. The two [[asymptotes]] of a truncus are parallel to the coordinate axes. The basic truncus y = 1 / ''x''<sup>2</sup> has asymptotes at ''x''&nbsp;=&nbsp;0 and ''y''&nbsp;=&nbsp;0, and every other truncus can be obtained from this one through a combination of [[translation (mathematics)|translations]] and [[dilation (mathematics)|dilations]].
 
For the general truncus form above, the constant ''a'' dilates the graph by a factor of ''a'' from the ''x''-axis; that is, the graph is stretched vertically when ''a''&nbsp;>&nbsp;1 and compressed vertically when 0&nbsp;<&nbsp;''a''&nbsp;<&nbsp;1. When ''a'' < 0 the graph is reflected in the ''x''-axis as well as being stretched vertically. The constant ''b'' translates the graph horizontally left ''b'' units when ''b''&nbsp;>&nbsp;0, or right when ''b''&nbsp;<&nbsp;0. The constant ''c'' translates the graph vertically up ''c'' units when ''c''&nbsp;>&nbsp;0 or down when&nbsp;''c''&nbsp;<&nbsp;0.
The asymptotes of a truncus are found at ''x'' = ''b'' (for the vertical asymptote) and ''y'' = ''c'' (for the horizontal asymptote).
 
[[Category:Curves]]
 
{{Geometry-stub}}

Revision as of 04:52, 24 February 2014


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