Question

How do you describe the end behavior of `y= x^4-4x^2`?

Answer 1

Well, the end behavior can be found by the degree of the polynomial and the sign of the first term.

The degree for this polynomial is 4 because `x^4` has the highest exponent

The sign of this term is positive.

To understand end behaviors, there are 4 possibilities

Odd degree, positive
Odd degree, negative
Even degree, positive
Even degree, negative

This equation is clearly even, positive.

Now, the best way to think of their end behaviors is to think about parent functions

odd, positive = `y = x`, which has an end behavior from quadrant 3 to quadrant 1

odd, negative = `y = -x`, which has an end behavior from quadrant 2 to quadrant 4

even, positive = `y = x^2`, which has an end behavior from quadrant 2 to quadrant 1

Even, negative = `y = -x^2`, which has an end behavior from quadrant 3 to quadrant 4.

Therefore, this equation has arrows pointing in quadrants 2 and 1.

Remember: Just picture the graph for the parent functions above and you will understand end behaviors much more clearly.