# How do you graph y=(5x+3)/(-x+10) using asymptotes, intercepts, end behavior?

Aug 6, 2018

Vertical asymptote: $x = 10$, horizontal asymptote: $y = - 5$
x intercept: $x = - 0.6$ , y intercept: $x = 0.3$, end behavior: $y \to - 5$ as $x \to - \infty \mathmr{and} y \to - 5$ as $x \to \infty$

#### Explanation:

y= (5 x+3)/(-x+10 , Vertical asymptote occur when denominator

is zero. -x+10=0 :. x= 10; lim(x->10^(-) y -> oo

$\lim \left(x \to {10}^{+}\right) y - > - \infty$. Vertical asymptote is $x = 10$

Horizontal asymptote: lim (x->-oo) ; y =-5/1=-5

$y = \frac{5 + \left(\frac{3}{x}\right)}{- 1 + \left(\frac{10}{x}\right)} , x \to \pm \infty , y \to - 5$

Horizontal asymptote is at $y = - 5$

x intercept: Putting $y = 0$ in the equation we get,

$5 x + 3 = 0 \mathmr{and} 5 x = - 3 \mathmr{and} x = - 0.6 \mathmr{and} \left(- 0.6 , 0\right)$ or

y intercept: Putting $x = 0$ in the equation we get,

$y = \frac{3}{10} = 0.3$ or (0,0.3)#

End behavior: $y \to - 5$ as $x \to - \infty$ and

$y \to - 5$ as $x \to \infty$

graph{(5 x+3)/(-x+10) [-80, 80, -40, 40]}[Ans]