Question

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

Answer 1

Vertical asymptote : `uarr x = 3 darr`.
Horizontal asymptote: `larr y = 1 rarr`.
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

`(y-1)(x-3)=3`

This is an example to show that the indeterminate from `0 X oo`

can take a finite limit, including 0.

As `x to +-oo`,

the other factor `(y-1)` ought to `to 0`, giving `y to 1`.

Likewise, as `y to +-oo`,

the other factor `(x-3) to 0`, giving `x to 3`.

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

`(y-1)(x-3)=0`.

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.