Question

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

Answer 1

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]}