Question

How do you graph `y=2/(x+9)-7` using asymptotes, intercepts, end behavior?

Answer 1

answers in explanation,

Explanation:

`y=2/(x+9)-7`
So from examination, you can determine that the function is a hyperbola, with asymptotes moved 9 units left and 7 units down and with a scale factor of 2.

But even with this I'll explain the process of how to actually find this out.
Firstly the vertical asymptote . This is any value that x cannot equal.

so since we are dividing by x in this function one possible point of where the function doesn't exists is when it is dividing by 0.
so,

`x+9 != 0`

`x!=-9` (that's our vertical asymptote )

Next Intercepts
Sub in y=0 for x-intercepts and x=0 for y-intercepts

It might be easier to use the expanded form for this step which is in this case.
`y=(-7x-61)/(x+9)`
I got this by subtracting the 7 into the fraction.

(y=0)

`0=-7x-61`

`x=61/-7` (this is our x-axis intercept)

(x=0)

`y=-61/9` (this is our y-axis intercept)

End Behaviour
Using your simplified form we know that
`y=-7+2/(x+9)`
this means that y will never equal -7 as there will always be a small amount added to it. so `y=-7` is our horizontal asymptote or end behaviour.

to find this out from a more complex rational function use algebraic long division to determine the whole number or function that divides evenly.

so if you need more points to make your graph more accurate just sub in values of x and plot those points.

graph{-7+2/(x+9) [-22, 6.47, -11.02, 3.22]}

Hope this helped