Question

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

Answer 1

The degree of the function is odd and the leading coefficient is positive – so the function goes to negative infinity as x goes to negative infinity and to positive infinity as x goes to positive infinity.

Answer 2

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, f(x) > 0

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

x > 2, f(x) > 0