Question

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

Answer 1

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:

`-5/x-7=0`

Adding `7` to both sides gives us

`-5/x=7`

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

`-5=7x`

Dividing both sides by `7` gives us

`x=-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_(xto-oo)(-5/x-7)`

Since we will be dividing by a more and more negative number, `-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]}