Question

How do you find the end behavior of `f(x)= -x^4+x^2`?

Answer 1

`f(x)->-\infty` as `x->\pm \infty`

Explanation:

The function `f(x)=-x^4+x^2` is a polynomial with a degree of 4 (the largest exponent), which is even. Also, the coefficient of the highest powered term is negative.

These facts are enough to conclude that `f(x)->-\infty` as `x->\pm \infty`. This means that the graph of `f` goes down forever and ever without bound as `|x|` gets larger and larger without bound. To be a bit more precise, the graph of `f` will go below and stay below any given horizontal line by choosing `x` to be sufficiently far from zero.