To sketch \frac{1}{f(x)} from f(x):

At every x-value, take the reciprocal of the original y-value: y \to \frac{1}{y}.
Points where f(x)=0 become vertical asymptotes, since \frac{1}{0} is undefined.
If \left|f\left(x\right)\right| is large, then \frac{1}{f(x)} is small (close to the x-axis). Thus, vertical asymptotes of f(x) become horizontal asymptotes at y=0. Conversely, if \left|f(x)\right| is small, then \frac{1}{f(x)} is large.
Local maxima and minima "flip":
- a maximum at (a,b) on f(x) becomes a minimum at \left(a,\frac{1}{b}\right) on \frac{1}{f(x)};
- a minimum at (c,d) becomes a maximum at \left(c,\frac{1}{d}\right). This happens because reciprocal values invert magnitudes.

Example

Sketch the graph of y=\frac{1}{x^2+2x-3}.

The denominator is zero when x=1 or x=-3, so there will be vertical asymptotes there.
The quadratic x^2+2x-3 has a minimum at \left(-1,-4\right), so there will me a local maxima at \left(-1,-\frac14\right).
Since x^2+2x-3 gets larger and larger y as x gets larger, there will be a horizontal asymptote at x=0.

Transformations & asymptotes