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

Question

Sophie Adams · Accepted Answer

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# #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]

Christian Antunez · Answer

undefined To graph $ y = \frac{5x + 3}{-x + 10} $, you can:

1. Find the vertical asymptote by setting the denominator equal to zero and solving for $ x $.
2. Find the horizontal asymptote by observing the end behavior as $ x $ approaches positive and negative infinity.
3. Find the $ y $-intercept by setting $ x $ to zero and solving for $ y $.
4. Find the $ x $-intercept by setting $ y $ to zero and solving for $ x $.

Plot these points and use them to sketch the graph of the function.