Question

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

Answer 1

Vertical asymptote: `x=10`, horizontal asymptote: `y=-5`
x intercept: `x = -0.6` , y intercept: `x = 0.3`, end behavior: `y-> -5` as `x -> -oo and y-> -5` as `x -> oo`

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 (x->10^+) y - > -oo`. Vertical asymptote is `x=10`

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

`y= (5+(3/x))/(-1+(10/x)) , x -> +- oo , y -> -5`

Horizontal asymptote is at `y=-5`

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

`5 x +3= 0 or 5 x =-3 or x = -0.6 or (-0.6,0)` or

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

`y=3/10= 0.3` or (0,0.3)#

End behavior: `y-> -5` as `x -> -oo` and

`y-> -5` as `x -> oo`

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