What is the end behavior of the graph #f(x)=x^5-2x^2+3#?

To find end behavior, we could always graph and function and see what is happening to the function on either end. But sometimes, we can also predict what will happens.

#f(x)=x^5-2x^2+3# is a 5th degree polynomial- We know that even degree polynomials somewhat mirror eachother in general tendency on either side. So if you have a positive leading coefficient, both sides will go "up" and if you have a negative leading coefficient, both sides will go "down". So they behave like quadratics. With odd degree polynomials, like the one we have, it's different- one side will typically go up while the other will go down- behaving like cubic functions.

The general rule for odd degree polynomials is:
Positive polynomials: They start "down" on the left end side of the graph, and then start going "up" on the right end side of the graph.
Negative polynomials.They start "up" on the left end side of the graph, and then start going "down" on the right end side of the graph.

#f(x)=x^5-2x^2+3# is a postive odd degree polynomial. Therefore,we predict that it will decrease down on the left side of its graph, and then increase up on the right side.

Let's graph it to check:

graph{x^5-2x^2+3 [-20, 20, -10, 10]}

As you can see, the graph does indeed become infinitely negative of as x gets smaller on the left side, and then becomes infinitely positive as x gets larger on the right side.