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.

Step-by-step explanation:

Here the given function,

$\frac{x^2+5x-6}{x^2-16}$

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 $x \rightarrow \infty$