Graph help!!! (1 Viewer)

[conman]

Can some one help me graph this equation x^3 + y^3 = 1. I've drawn the graph and my graph is exactly the same with the text book's solution. But text book also included oblique asympt y = -x. I can not figure out why they have that oblique asympt. Can some one explain for me????????

 

[airie]

In fact, this asymptote exists for any constant a where x³+y³=a³

This is because, after putting y as the subject,
y
= (a³-x³)^1/3
= [-x³(-a³/x³+1)]^1/3
= -x(-a³/x³+1)^1/3

And as x approaches +/- infinity, -a³/x³ approaches zero, thus -a³/x³+1 approaches 1. Therefore, the value of y approaches -x*1^1/3 = -x ie. there exists an oblique asymptote y=-x