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

Question

asked Mar 25, 2019 212k views

1 Answer

Shinese · Answer 1 · 2019-03-30T13:21:57+0000

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
$x \rightarrow \infty$