# Vertical asymptote

##### Intros

###### Lessons

##### Examples

###### Lessons

**Graphing Rational Functions**Sketch each rational function by determining:

i) vertical asymptote.

ii) horizontal asymptotes

**Identifying Characteristics of Rational Functions**Without sketching the graph, determine the following features for each rational function:

i) point of discontinuity

ii) vertical asymptote

iii) horizontal asymptote

iv) slant asymptote

- $a(x) = \frac{x - 9}{x + 9}$
- $b(x) = \frac{x^{2}-9}{x^{2}+9}$
- $c(x) = \frac{x^{2}+9}{x^{2}-9}$
- $d(x) = \frac{x+9}{x^{2}-9}$
- $e(x) = \frac{x+3}{x^{2}-9}$
- $f(x) = \frac{x^{2}+9}{x+9}$
- $g(x) = \frac{-x-9}{-x^{2}-9}$
- $h(x) = \frac{-x^{2}-9}{-x^{2}+9}$
- $i(x) = \frac{x^{2}-9}{x+3}$
- $j(x) = \frac{x^{3}-9x^{2}}{x^{2}-3x}$

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###### Topic Notes

For a rational function: $f(x) = \frac{numerator}{denominator}$

Provided that the numerator and denominator have no factors in common (if there are, we have "points of discontinuity" as discussed in the previous section), **vertical asymptotes** can be determined as follows:

$\bullet$equations of vertical asymptotes: x = zeros of the denominator

$i.e. f(x) = \frac{numerator}{x(x+5)(3x-7)}$; vertical asymptotes: $x = 0, x = -5, x = \frac{7}{5}$

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