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

Mar 31, 2018

Vertical asymptote is 6
End behaviour (horizontal asymptote) is 5
Y intercept is $- \frac{7}{2}$
X intercept is $\frac{27}{5}$

#### Explanation:

We know that the normal rational function looks like $\frac{1}{x}$

What we have to know about this form is that it has a horizontal asymptote (as x approaches $\pm \infty$) 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

$\frac{1}{x - C} + D$

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

$0 = 5 + \frac{3}{x - 6}$

$- 5 = \frac{3}{x - 6}$

$- 5 \left(x - 6\right) = 3$

$- 5 x + 30 = 3$

$x = - \frac{27}{-} 5$

So you have the co-ordiantes $\left(\frac{27}{5} , 0\right)$

To find the y intercept set x to 0

$y = 5 + \frac{3}{0 - 6}$

$y = 5 + \frac{1}{-} 2$

$y = \frac{7}{2}$

So we get the co-ordiantes $\left(0 , \frac{7}{2}\right)$

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