# How do you graph y=-3/(x-4)-1 using asymptotes, intercepts, end behavior?

Apr 12, 2018

vertical asymptote at $x = 4$
horizontal asymptote at $y = - 1$
$\left(0 , - \frac{1}{4}\right)$ and $\left(1 , 0\right)$ are intercepts

#### Explanation:

Vertical Asymptote at $x = 4$

We can see that there is a vertical asymptote at $x = 4$.

When $x$ is just slightly less than 4 (eg. 3.9999),

$x - 4$ is a very small negative number which makes

$\frac{1}{x - 4}$ a very large negative number which makes

$- \frac{1}{x - 4}$ a very large positive number which makes

$y$ a very large positive number.

When $x$ is just slightly more than 4 (eg. 4.0001),

$x - 4$ is a very small positive number which makes

$\frac{1}{x - 4}$ a very large positive number which makes

$- \frac{1}{x - 4}$ a very large negative number which makes

$y$ a very large negative number.

End Behavior

When $x$ is largely negative (eg. $- {10}^{10}$), $- \frac{3}{x - 4}$ becomes very close to 0, so $y$ is close to $- 1$.

When $x$ is largely positive (eg. ${10}^{10}$), $- \frac{3}{x - 4}$ becomes very close to 0, so $y$ is close to $- 1$.

So the plot approaches a horizontal asymptote at $y = - 1$ as its end behavior.

$y$-intercept

When $x$=0, $y = - \frac{3}{0 - 4} - 1 = - \frac{1}{4}$.

There is a $y$-intercept at $\left(0 , - \frac{1}{4}\right)$

$x$-intercept

When $y$=0,

$0 = - \frac{3}{x - 4} - 1$.

Solve for $x$ to get the $x$-intercept.

$x - 4 = - 3$

$x = 1$

There is an $x$-intercept at $\left(1 , 0\right)$

graph{-3/(x-4)-1 [-5, 15, -20, 20]}