# Asymptotes

pronounced ass-im-tote (the p is silent)

An asymptote is a line which the graph approaches but never touches.

## Drawing Graphs with Asymptotes

Care should be taken when drawing a graph approaching an asymptote.

- The line should approach but not cross or touch the asymptote.
- The line should never curl away from the asymptote.

## Finding the asymptotes for graphs of rational functions

A **vertical asymptote** will be formed when the denominator of the fraction becomes zero

- (because at that point the fraction is undefined}

#### Example

… … The graph of $y=\dfrac{1}{x-6}$

… … will have a vertical asymptote where $x - 6 = 0$

… … so it will have a **vertical asymptote** when $x = 6$

… … notice the hyperbola also has a **horizontal asymptote** at $y = 0$, see below

## Rational Graphs

Graphs in the form $y = \dfrac{a}{f(x)} + b$ where f(x) is a polynomial

Will have a **horizontal asymptote** at $y = b$

- this is because, as the denominator of the fraction approaches infinity
- the fraction as a whole will approach zero (but never get there)

… … as $f(x) \rightarrow \pm \infty , \quad \dfrac {a}{f(x)} \rightarrow 0$

this includes both

- the hyperbola and
- the truncus

… … So the above graph $y=\dfrac{1}{x-6}$

… … has a horizontal asymptote at $y = 0$

## Oblique or Curving Aysmptotes

More complicated graphs may have an oblique (sloping) or curved asymptote.

If an equation is in the form $y = \dfrac{f(x)}{g(x)} + h(x)$

- where f(x) has lower degree than g(x)

The asymptote will be what is left after the fraction is replaced with zero

- (which removes the entire fraction from the equation)

- ie: asymptote is $y = h(x)$

#### Example 1

For the equation: $y=\dfrac{2}{x-3}+2x-3$

- state the asymptotes
- sketch the graph

**Solution:**

- Vertical asymptote, set the denominator of the fraction to zero

… … $x - 3 = 0$

… … $x = 3$

- Horizontal asymptote, replace the fraction with zero

… … $y = 0 + 2x – 3$

… … $y = 2x - 3$

#### Example 2

For the equation $y = \dfrac{-1}{ \big( x + 4 \big)^2} + \dfrac{x^2}{4}$

- state the equations of the asymptotes
- sketch the graph

**Solution**

- Vertical Asymptote: set the denominator of the fraction to zero

… … $\big( x + 4 \big)^2 = 0$

… … x + 4 = 0

… … x = -4

- Horizontal Asymptote: replace the fraction with zero

… … $y = 0 + \dfrac{x^2}{4}$

… … $y = \dfrac{x^2}{4}$

.