# How do you graph y=-5/x-7 using asymptotes, intercepts, end behavior?

Jul 4, 2018

See below:

#### Explanation:

We immediately notice that if our denominator is equal to zero, we'll be undefined and have a vertical asymptote at $x = 0$.

Since we have a vertical asymptote, this means the graph never intercepts the $y$-axis since the function goes unbounded.

What about $x$-intercepts? We can easily find these by setting $y$ equal to zero:

$- \frac{5}{x} - 7 = 0$

Adding $7$ to both sides gives us

$- \frac{5}{x} = 7$

Let's multiply both sides by $7$ to get

$- 5 = 7 x$

Dividing both sides by $7$ gives us

$x = - \frac{5}{7}$

This is where our graph intercepts the $x$-axis. So already, we have a sense of how our graph looks, but end behavior can tell us more:

Since we know our function has a vertical asymptote, we know it goes unbounded towards infinity. What about negative infinity?

Let's evaluate the following limit:

${\lim}_{x \to - \infty} \left(- \frac{5}{x} - 7\right)$

Since we will be dividing by a more and more negative number, $- \frac{5}{x}$ will just go to zero, and we're left with $- 7$. We can think of this as our horizontal asymptote.

Putting together all we know about our function, we can graph!

graph{-5/x-7 [-9.71, 10.29, -10.42, 0]}