Question

Which of the following is the graph of the rational function y= x^2+5x-6/x^2-16

Answer 1

Answer: The below graph shows the given function.

Explanation:

Here the given function,

`(x^2+5x-6)/(x^2-16)`

Which is the rational function.

For vertical asymptote,

Denominator = 0

⇒
`x^2 - 16 = 0`

⇒
`x^2 = 16`

⇒
`x = \pm 4`

Thus, the vertical asymptotes of the given function are (4, 0) and (-4,0).

Also, x-intercept of the given function are ( 1,0) and (-6,0)

y-intercept of the given function are (0,0.375)

End behavior :

Since, The function has three intervals,

`(-\infty, -4)`

`(-4,4)`

`(-\infty, -4)`

In interval
`(-\infty, -4)` ,
`f(x)\rightarrow 1` as
`x \rightarrow -\infty` and
`f(x)\rightarrow -\infty` as
`x \rightarrow -4`

In interval
`(-4,4)`,
`f(x)\rightarrow \infty` as
`x \rightarrow -4` and
`f(x)\rightarrow -\infty` as
`x \rightarrow 4`

And, In interval
`(-\infty, -4)`,
`f(x)\rightarrow \infty` as
`x \rightarrow 4` and
`f(x)\rightarrow 1` as
`x \rightarrow \infty`