How do you find the end behavior of #f (x) = 3x^4 - 5x + 1#?

Question

James Allen · Accepted Answer

See the explanation below.

First, graph the function. graph{3x^4-5x+1 [-10, 10, -5, 5]} Based on the graph, you can see that as #x -> oo#, #f(x) -> oo#. As #x->-oo#, #f(x) -> oo#. You can also write this using limits: #lim_(x->oo) f(x) = oo# #lim_(x->-oo) f(x) = oo#

Anna Allen · Answer

undefined To find the end behavior of a polynomial function like $f(x) = 3x^4 - 5x + 1$, you look at the term with the highest degree, which is the $3x^4$ term in this case.

1. When the leading term has an even exponent (like $x^4$), the end behavior as $x$ approaches positive or negative infinity is the same sign as the leading coefficient. So, as $x$ approaches positive infinity, $f(x)$ also approaches positive infinity, and as $x$ approaches negative infinity, $f(x)$ also approaches positive infinity.

2. When the leading term has an odd exponent (like $x^3$), the end behavior as $x$ approaches positive infinity is the same sign as the leading coefficient, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

Therefore, for the function $f(x) = 3x^4 - 5x + 1$, as $x$ approaches positive infinity, $f(x)$ approaches positive infinity, and as $x$ approaches negative infinity, $f(x)$ approaches positive infinity.