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

It tends towards $- \infty$ on the left, and $\infty$ on the right. graph{6x^3+1 [-10, 10, -5, 5]}
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 $\infty$ as $x \to \infty$ (i.e., to the right), and towards $- \infty$ as $x \to - \infty$ (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, ${\left(- k\right)}^{3}$ will be negative, because a negative number cubed returns a negative.