Question

What is the end behavior of `f(x)=6x^3+1`?

Answer 1

It tends towards `-oo` on the left, and `oo` on the right. graph{6x^3+1 [-10, 10, -5, 5]}

Explanation:

This is simply an `x^3` function that has experienced a veradical stretch by a factor of 6 (our `x^3` coefficient), and a vertical shift of 1 unit upwards (from the +1). Neither of these alterations change the end behavior; if our 6 were instead -6, that would have an effect, but the coefficient is instead positive.

The elementary `x^3` function tends towards `oo` as `x->oo` (i.e., to the right), and towards `-oo` as `x->-oo` (to the left). If you wish to test this, choose an arbitrarily large constant `k>0`. `k^3` will be positive, because k was positive, and there is no negative coefficient. On the other hand, `(-k)^3` will be negative, because a negative number cubed returns a negative.