Explain how to sketch a graph of the function f (x) = x^3+2x^2-8x. Include end behavior, zeros, and intervals where the function is positive and negative

Step-by-step explanation:

f (x) = x³ + 2x² - 8x

To find the end behavior, take the limit as x approaches ±∞. Since the leading coefficient is positive, and the order is odd:

lim (x→-∞) f (x) = - ∞

lim (x→∞) f (x) = ∞

Next, factor to find the zeros.

f (x) = x (x² + 2x - 8)

f (x) = x (x + 4) (x - 2)

The zeros are (-4, 0), (0, 0), and (2, 0).

Therefore, the intervals are:

x < - 4, f (x) < 0

-4 < x 0

0 < x < 2, f (x) < 0

x > 2, f (x) > 0