The vertical asymptotes of \( y = \dfrac{1}{f(x)} \) occur at the zeros of the graph of \( y = f(x) \).

If \( x = a \) is a zero of \( y = f(x) \), then \( f(a) = 0 \) and \( \dfrac{1}{f(a)} \) will be undefined; therefore, \( x = a \) will be a vertical asymptote of \( y = \dfrac{1}{f(x)} \). The domain of \( y = \dfrac{1}{f(x)} \) is \( \{x~|~f(x) \neq 0, x \in \mathbb{R} \} \).

A polynomial function and its related reciprocal function have the same positive and negative intervals.

When the function \( y = f(x) \) is positive (above \( x \)-axis), then its reciprocal \( y = \dfrac{1}{f(x)} \) is also positive.

When the function \( y = f(x) \) is negative (below \( x \)-axis), then its reciprocal \( y = \dfrac{1}{f(x)} \) is also negative.

When the graph of the function \( y = f(x) \) is decreasing, the graph of \( y = \dfrac{1}{f(x)} \) is increasing and vice versa.

When the graph of the function \( y = f(x) \) is increasing, the graph of \( y = \dfrac{1}{f(x)} \) is decreasing and vice versa.

A local maximum point on \( y = f(x) \) is a local minimum point on \( y = \dfrac{1}{f(x)} \).

A local minimum point on \( y = f(x) \) is a local maximum point on \( y = \dfrac{1}{f(x)} \).

This follows from the fact that the intervals of increase of \( f(x) \) are the intervals of decrease on the reciprocal, and vice versa.

The end behaviour of all functions of the form \( y = \dfrac{1}{f(x)} \) (where \( f(x) \) is a polynomial) have \( y \rightarrow 0 \) in both directions, resulting in a horizontal asymptote of \( y = 0 \).

Since polynomials have end behaviour \( y \rightarrow \pm \infty \), \( \dfrac{1}{f(x)} \rightarrow 0 \).

The point \( (x, y) \) on the graph of \( y = f(x) \) maps to the point \( \left(x, \dfrac{1}{y} \right) \) on \( y = \dfrac{1}{f(x)} \).

\((-3, -3)\) on \(f(x)=x\) maps to \((-3, -1/3)\) on \(y=1/f(x)\).

Points on the polynomial function with a \( y \)-coordinate of \( 1 \) or \( -1 \) will remain stationary for the graph of the reciprocal.

\((1,1)\) and \((1,-1)\) remain stationary when \(f(x)=x\).

If \( y = f(x) \) is an even function, then its reciprocal is also even (see \( f(x) = -x^2 + 4 \)).

If \( y = f(x) \) is an odd function, then its reciprocal is also odd (see \( f(x) = x \)).

If \( y = f(x) \) is neither odd nor even, then its reciprocal is neither as well (see \( f(x) = -2x - 2\)).

Created with GeoGebra. Author: University of Waterloo. CC BY-NC-SA 4.0. https://ggbm.at/zgkdjeam