Question 610967
First, when you are asked to determine the "end behavior" of some expression, you are being asked: "What happens to the value of the expression when the variable becomes very large, positive or negative?" Note: You can get one answer to this question for large positive x's and another for large negative x's.<br>
Second, {{{-x^4+x^2}}} is a polynomial. When determining the end behavior of a polynomial, ignore all terms except the one with the highest exponent. (This is done because for large values, the highest exponent term will "overwhelm" whatever values come from the other terms.)<br>
So we are going to examine what happens to {{{-x^4}}} for large positive and large negative numbers. For large positive x's the {{{x^4}}} part of {{{-x^4}}} will become an extremely large positive number. And with the "-" in front, {{{-x^4}}}, becomes an extremely large negative number.<br>
For large negative x's the {{{x^4}}} part of {{{-x^4}}} will become an extremely large positive number (remember how even exponents work on negative numbers!). And with the "-" in front, {{{-x^4}}}, becomes an extremely large negative number.<br>
So for both large positive and large negative x's, {{{-x^4}}} (and therefore {{{-x^4+x^2}}}), will take on extremely large negative values. Graphically this means that the graph of {{{y = -x^4+x^2}}} will go down (to large negative y values) on the right (where x is large and positive) and on the left (where x is large and negative).