How do you graph #y=5+3/(x-6)# using asymptotes, intercepts, end behavior?

1 Answer
Mar 31, 2018

Answer:

Vertical asymptote is 6
End behaviour (horizontal asymptote) is 5
Y intercept is #-7/2#
X intercept is #27/5#

Explanation:

We know that the normal rational function looks like #1/x#

What we have to know about this form is that it has a horizontal asymptote (as x approaches #+-oo#) at 0 and that the vertical asymptote (when the denominator equals 0) is at 0 as well.

Next we have to know what the translation form looks like

#1/(x-C)+D#

C~Horizontal translation, the vertical asympote is moved over by C
D~Vertical translation, the horizontal asympote is moved over by D

So in this case the vertical asymptote is 6 and the horizontal is 5

To find the x intercept set y to 0

#0=5+3/(x-6)#

#-5=3/(x-6)#

#-5(x-6)=3#

#-5x+30=3#

#x=-27/-5#

So you have the co-ordiantes #(27/5,0)#

To find the y intercept set x to 0

#y=5+3/(0-6)#

#y=5+1/-2#

#y=7/2#

So we get the co-ordiantes #(0,7/2)#

So sketch all of that to get
graph{5+3/(x-6) [-13.54, 26.46, -5.04, 14.96]}