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

Dec 29, 2016

Vertical asymptote : $\uparrow x = 3 \downarrow$.
Horizontal asymptote: $\leftarrow y = 1 \rightarrow$.
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

$\left(y - 1\right) \left(x - 3\right) = 3$

This is an example to show that the indeterminate from $0 X \infty$

can take a finite limit, including 0.

As $x \to \pm \infty$,

the other factor $\left(y - 1\right)$ ought to $\to 0$, giving $y \to 1$.

Likewise, as $y \to \pm \infty$,

the other factor $\left(x - 3\right) \to 0$, 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

$\left(y - 1\right) \left(x - 3\right) = 0$.

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.