# How do you graph #y=3/(x-3)+1# using asymptotes, intercepts, end behavior?

##### 1 Answer

Vertical asymptote :

Horizontal asymptote:

x-intercept ( y = 0 ) : 0. y-intercept ( x = 0 ): 0. See graph.

#### Explanation:

graph{(y-1)((y-1)(x-3)-3)=0 [-20, 20, -10, 10]}

The given equation has another form

This is an example to show that the indeterminate from

can take a finite limit, including 0.

As

the other factor

Likewise, as

the other factor

So, the asymptotes are given by x = 3 and y = 1.

If the limit of the product is 0, we directly get the pair of asymptotes

This is the logic behind the structure

#(y-ax-b)((y-a'x-b'x-c')=k

for the equation of a hyperbola that has the asymptotes given by

setting k = 0, in this form.