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

1 Answer

Answer:

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.