Question

How do you graph `(6x)/(x^2-36)` using asymptotes, intercepts, end behavior?

Answer 1

`color(brown)("Very detailed explanation given.")`
Vertical asymptotes at `x=+-6`
As `x->-oo: y->0^-`
As `x->+oo: y->0^+`

Explanation:

Let a minute amount of `x` be designated by the symbol `deltax`
I have deliberately used this symbol as in introduction to calculus without actually using calculus

Set to `" "y=(6x)/(x^2-36)`

Notice that the denominator is the difference of two squares. So we have:

`y=(6x)/((x-6)(x+6))`

Note that the use of `->` in this explanation is used as meaning 'tends to'.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the vertical asymptotes")`

This becomes 'undefined if the denominator is zero thus we have vertical asymptotes at those points.

Vertical asymptotes at `x=+-6`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the behaviour either side of the asymptotes")`

`color(brown)("Set "x=+6)`

Let `deltax >0` but minute

Set `y=(6(x+deltax))/((color(white)(.)[x+deltax]-6)color(white)(.)(color(white)(.)[x+deltax]+6))`

Now `deltax` is so small that it is almost not there so
`(color(white)(.)[6+delta6]-6 )` is positive but so small then it almost zero. Consequently the whole denominator is only just bigger than 0 but on the positive side. Dividing this into `6(6+delta6)` yields a number that is tending to `+oo`

...................................................................................................
Let `deltax < 0` but minute

Following the same methodology of thinking we find that in this case the denominator is only just smaller than 0 so when divide into the numerator the whole is tending to `-oo`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Set "x=-6)`

As in the processes above:

When `deltax >0` we have `(6(6+delta6))/(0^+)->+oo`
When `deltax <0` we have `(6(6+delta6))/(0^+)->-oo`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Determine the behaviour at the extremities "x->+-oo)`

As `x` becomes increasingly larger and larger the -36 has less and less effect. So the equation behaves as if it where `y=6x/x^2 = 6/x`

The net consequence is that `y->+-oo` in keeping the sign of `x`

For `x<0: y->-oo`
For `x>0: y->+oo`