# How do you graph f(x)=(-x-1)/(x+2) using holes, vertical and horizontal asymptotes, x and y intercepts?

Mar 16, 2017

see explanation.

#### Explanation:

$\textcolor{b l u e}{\text{Asymptotes}}$

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

$\text{solve "x+2=0rArrx=-2" is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ ( a constant)}$

divide terms on numerator/denominator by x

$f \left(x\right) = \frac{- \frac{x}{x} - \frac{1}{x}}{\frac{x}{x} + \frac{2}{x}} = \frac{- 1 - \frac{1}{x}}{1 + \frac{2}{x}}$

as $x \to \pm \infty , f \left(x\right) \to \frac{- 1 - 0}{1 + 0}$

$\Rightarrow y = - 1 \text{ is the asymptote}$

Holes occur if there is a duplicate factor on the numerator/denominator. This is not the case here hence there are no holes.

$\textcolor{b l u e}{\text{Intercepts}}$

$x = 0 \to y = - \frac{1}{2} \Rightarrow \left(0 , - \frac{1}{2}\right) \leftarrow \text{y-intercept}$

$y = 0 \to - x - 1 = 0 \to x = - 1 \Rightarrow \left(- 1 , 0\right)$
graph{(-x-1)/(x+2) [-10, 10, -5, 5]}