Question

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

Answer 1

Slant asymptote: `y = - x`. Vertical asymptotes `uarr x = +-sqrt6 darr`. x-intercept ( y = 0 ):2. y-intercept ( x = 0 ): `-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 = +-sqrt6`, that makes `y +-oo at r = oo`.

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

y' is 0 at `x = 0, +-3sqrt2`.

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]}