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Contents
Rectangular Hyperbolas
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Rectangular
Hyperbolas
When the function is in the form 1/xn, the graph is a rectangular hyperbola. There are two types:
Properties
of the rectangular hyperbola:
1.
There is a vertical asymptote at x = 0.
2.
There is a horizontal asymptote at y = 0.
3.
There are no stationary points.
4.
There are no intercepts (since the axes are asymptotes. This is NOT TRUE
when the axes are not asymptotes.)
5.
For n = odd integer, the gradient is always decreasing throughout the
defined portion of x.
6.
For n = even integer, the gradient when x < 0 is always increasing,
whilst the gradient when x > 0 is always decreasing.
 Example:
Sketch the graph of 
Solution:1.
Intercepts: when x = 0, y = -1
when y = 0, x = 1
2.
Turning point:
There are no turning points for this kind of graph.
3.
Asymptotes:
As x tends to infinity, y tends to 1.
As y tends to infinity, x tends to -1.
x = -1 is a vertical asymptote and y = 1 is a horizontal asymptote.
4.
Gradient:
dy/dx is always positive.
The gradient is always increasing.
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When
n = 2k + 1, ie. n is an odd integer
