Question

What are the asymptotes of `y=2/(x+1)-4` and how do you graph the function?

Answer 1

This type of question is asking you think about how numbers behave when grouped together in an equation.

Explanation:

`color(blue)("Point 1")`

It is not allowed (undefined) when a denominator takes on the value of 0. So as `x=-1` turns the denominator into 0 then `x=-1` is an 'excluded value

`color(blue)("Point 2")`

It is always worth investigation when the denominators approach 0 as this is usually an asymptote.

Suppose `x` is tending to -1 but from the negative side. Thus `|-x|>1`. Then `2/(x+1)` is a very large negative value the -4 becomes insignificant. Thus limit as `x` tends to negative side of -1 then `x+1` is negatively minute so `y=-oo`

In the same way as x tends to the positive side of -1 then `x+1` is positively minute so `y=+oo`

`color(blue)("Point 3")`

As x tends to positive `oo` then `2/(x+1)` tends to 0 so `y=2/(x-1)-4` tends to - 4 on the positive side

You have the same as x tends to negative `oo` in that y tends to - 4 but on the negative side.
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`color(blue)("In conclusion")`

You have a horizontal asymptote at `y=-4`

You have a vertical asymptote at `x=-1`