Reciprocal Functions

These techniques involves sketching the graph of $y=\dfrac{1}{f(x)}$ from the graph of $y=f(x)$.

Technique 1

When $f(x)$ approaches towards $0$, $y=\dfrac{1}{f(x)}$ approaches towards $\infty$, the graph of $y=\dfrac{1}{f(x)}$ approaches the vertical asymptote(s).

Technique 2

The graph of $y=\dfrac{1}{f(x)}$ has vertical asymptotes at the $x$-intercepts of $y=f(x)$.

Technique 3

When $f(x)$ approaches towards $\infty$, $y=\dfrac{1}{f(x)}$ approaches towards $0$, the graph of $y=\dfrac{1}{f(x)}$ approaches the horizontal asymptote(s).

Technique 4

  • When $f(x)=\pm1$, $\dfrac{1}{f(x)}=\pm1$. The graphs are in the same quadrant.
  • When $f(x) \lt 0$, $\dfrac{1}{f(x)} \lt 0$.
  • When $f(x) \gt 0$, $\dfrac{1}{f(x)} \gt 0$.

Technique 5

  • The minimum turning point of $f(x)$ gives the maximum turning point of the reciprocal function.
  • The maximum turning point of $f(x)$ gives the minimum turning point of the reciprocal function.

Example 1

Draw the reciprocal graph of the following function.

Example 2

Draw the reciprocal graph of the following function.

Example 3

Draw the reciprocal graph of the following function.

Example 4

Draw the reciprocal graph of the following function.

Example 5

Draw the reciprocal graph of the following function.