Rectangular Hyperbolas
When the function is in the form 1/xn, the graph is a rectangular hyperbola. There are two types:
When n = 2k + 1, ie. n is an odd integer
y = 1/x
When n = 2k, ie. n is an even integer
y = 1/x2
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.
Curve Sketching | Quadratic Equation | Rectangular Hyperbolas | Conic sections | Modulus Functions
