Describe the function‘s end behavior. f(x)=10−5x^3+7x^2

Question

as x→∞,f(x)→

as x→−∞,f(x)→

Answer 1

To determine end behavior, look at the highest exponent. In this problem it is 3. The parent function would be:

as x→∞,f(x)→∞

as x→−∞,f(x)→−∞

BUT, there is a negative sign in front of the coefficient. So the parent reflects on the x axis. This means:

as x→∞,f(x)→-∞

as x→−∞,f(x)→∞

Answer 2

With a polynomial, we want to identify the highest order term, the one with the largest exponent, and write it as

f_n(x) = axⁿ

The sign of a, represented as sign(a), tells us the answer to the first part of the question,

as x → ∞, f(x) → [sign(a)] ∞

Then, whether n is odd or even tell us whether the behavior of the function to the left matches the behavior to the right. One way to represent that is with (-1)ⁿ which is 1 if n is even and -1 if n is odd. So the answer to the second part is

as x → -∞, f(x) → [sign(a)] (-1)ⁿ ∞

This is the general behavior for any polynomial.