# 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.