Question

What is the end behavior of the graph of `f(x)=-2x^4+7x^2+4x-4`?

Answer 1

`lim_(xtooo) f(x)=-oo, lim_(xto-oo) f(x)=-oo`

Explanation:

To determine the end behavior, let's take the limit as `xtooo` and `xto-oo`.

In our polynomial, `f(x)`, the first term is what will dominate the end behavior, because it has the highest degree. So we can find the limit of that:

`lim_(xtooo) color(red)(-2)color(blue)(x^4)=-oo`

As `x` gets very large, the blue term will always be positive, but the `-2` (red) will turn it negative. This is why our limit evaluates to `-oo`.

`lim_(xto-oo) color(red)(-2)color(blue)(x^4)=-oo`

As `x` gets very negative, the even exponent will make the term positive, but the red `-2` on the outside will make it negative. Thus, this limit will also evaluate to `-oo`.

In general, the function is downward opening because of the negative coefficient on the `x^4` term.

Hope this helps!