# How do you graph y=(8-5x)/(x-5) using asymptotes, intercepts, end behavior?

Dec 27, 2016

see explanation.

#### Explanation:

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

The denominator of y cannot be zero as this would make y 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.

solve : $x - 5 = 0 \Rightarrow x = 5 \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$y = \frac{\frac{8}{x} - \frac{5 x}{x}}{\frac{x}{x} - \frac{5}{x}} = \frac{\frac{8}{x} - 5}{1 - \frac{5}{x}}$

as $x \to \pm \infty , y \to \frac{0 - 5}{1 - 0}$

$\Rightarrow y = - 5 \text{ is the asymptote}$

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1 ) Hence there are no oblique asymptotes.

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

$x = 0 \to y = \frac{8}{- 5} \Rightarrow \left(0 , - \frac{8}{5}\right)$

$y = 0 \Rightarrow 8 - 5 x = 0 \Rightarrow x = \frac{8}{5} \Rightarrow \left(\frac{8}{5} , 0\right)$
graph{(8-5x)/(x-5) [-20, 20, -10, 10]}