# How do you graph y=(7x)/(-x-15) using asymptotes, intercepts, end behavior?

Dec 12, 2016

see explanation.

#### Explanation:

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 : $- x - 15 = 0 \Rightarrow x = - 15 \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$y = \frac{\frac{7 x}{x}}{- \frac{x}{x} - \frac{15}{x}} = \frac{7}{- 1 - \frac{15}{x}}$

as $x \to \pm \infty , y \to \frac{7}{- 1 - 0}$

$\Rightarrow y = - 7 \text{ is the asymptote}$

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

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

$x = 0 \to y = \frac{0}{- 15} = 0 \Rightarrow \left(0 , 0\right)$

$y = 0 \to 7 x = 0 \Rightarrow \left(0 , 0\right)$
graph{(7x)/(-x-15) [-40, 40, -20, 20]}