# How do you graph y=4/(x-6)+19 using asymptotes, intercepts, end behavior?

Nov 13, 2016

#### Explanation:

Multiply both sides by $x - 6$:

$x y - 6 y = 4 + 19 x - 114$

$x y - 6 y = 19 x - 110$

This is a rotated Hyperbola.

Arrange in accordance with the quadratic equation in the reference:

$x y - 6 y - 19 x - 110 = 0$

Here are the values of their coefficients:

${A}_{x y} = \frac{1}{2} , {B}_{x} = - \frac{19}{2} , {B}_{y} = - 3 , C = - 110 , \mathmr{and} {A}_{\text{xx}} = {A}_{y y} = 0$

Find the center:

D = | (A_("xx"), A_(xy)), (A_(xy),A_(yy)) | = | (0,1/2), (1/2,0) | = -1/4

x_c = -1/D | (B_x, A_(xy)), (B_y,A_(yy)) | = 4| (-19/2, 1/2), (-3,0) | = 4(3/2) = 6

y_c = -1/D | (A_("xx"), B_x), (A_(xy),B_y) | = 4| (0, -19/2), (1/2,-3) | = 4(19/4) = 19

The center is $\left(6 , 19\right)$

The calculation for the vertices is very long so I will just give them to you $\left(4 , 17\right)$ and $\left(8 , 21\right)$

Here is a graph: