To determine the end behavior of the function f(x) = x^5 – 8x^4 + 16x^3, we focus on the leading term of the polynomial, which is x^5. The leading term is crucial because it dominates the growth of the function as x approaches positive or negative infinity.
Since the leading term is x^5, we know that:
- As x approaches positive infinity (x → ∞), f(x) will also approach positive infinity. In simpler terms, the graph will rise to the right.
- As x approaches negative infinity (x → -∞), f(x) will approach negative infinity. This means the graph will fall to the left.
In summary, the end behavior of the graph of the function can be described as follows:
- The graph rises to the right (as x → ∞, f(x) → ∞)
- The graph falls to the left (as x → -∞, f(x) → -∞)
Understanding this behavior helps in sketching the graph and analyzing its overall shape.