# How do you graph y=(x^3-8)/(6-x^2) using asymptotes, intercepts, end behavior?

Dec 21, 2016

Slant asymptote: $y = - x$. Vertical asymptotes $\uparrow x = \pm \sqrt{6} \downarrow$. x-intercept ( y = 0 ):2. y-intercept ( x = 0 ): $- \frac{4}{3}$

#### Explanation:

By actual division.

y = -x +(3-4/sqrt6)/(x-sqrt6)+(3+4?sqrt6)/(x+sqrt6), revealing

horizontal asymptote y = quotient = -x and vertical asymptotes

$x = \pm \sqrt{6}$, that makes $y \pm \infty a t r = \infty$.

The y-intercept is -4/3 and x-intercept is 2.

y' is 0 at $x = 0 , \pm 3 \sqrt{2}$.

Local maximum ( for y'' < 0 ) at x = 3sqrt2, local minimum ( for y''>0 )

at x = -3sqrt2 and point of inflexion ( for y'' = 0 and y''' not 0 ) at x = 0.

Look at the graph that illustrates all these aspects.

graph{y(6-x^2)-x^3+8 =0 [-38.28, 38.28, -19.14, 19.14]}