A polynomial function is given. Describe the end behavior of the polynomial function.

Question

Q(x) = −x²(x² − 4)

(a) Describe the end behavior of the polynomial function.

End Behavior: y---> ________ as x--->  ∞

y---> ________ as x --> - ∞

Answer 1

Fall back to what we know

Q(x) = −x²(x² − 4)

= -x⁴ + 4x²

Which is easier to work with

Now we can identify the degree of the polynomial as 4

We know that the degree of the polynomial - 1 is how many times the polynomial turns. For instance , the parabola x² falls turns and goes back up.

This is a good example to see how even polynomials turn back on themselves.

as an exponential function y increases faster as x increases

As an even polynomial they are both going to ∞ or -∞

To understand which x⁴ gets larger the fastest, but it's negative. Then that is where the function goes.

As x→ -∞ then at y→ -∞ and x→ ∞ then at y→ -∞

Answer 2

You can use this sketch to determine the end behavior:

The "governing" element of the polynomial is the highest degree. In this case, we need to multiply −x² with x² to determine what that is. −x² • x² = - x⁴ which fits the lower left sketch -x^{(even power)} so as x approaches -∞, Q(x) approaches -∞ and as x approaches ∞, Q(x) approaches -∞