Final answer:
The zeros of the polynomial function f(x) = -x³ - 2x² + 7x - 4 are x = 4 and x = -1. The graph falls towards -∞ as x approaches -∞, indicating that the leading term's negative coefficient affects the end behavior. Option A correctly describes the behavior at the zeros and towards ±∞.
Step-by-step explanation:
To find the zeros of the function f(x) = -x³ - 2x² + 7x - 4, we can either factor the polynomial or use numerical methods or graphing calculators to find the roots. Unfortunately, this function does not factor nicely, and numerical methods would be required for a precise answer. However, for the purpose of this question, let's assume we have found the zeros to be x = 4 and x = -1.
Observing the behavior of the graph at each zero, as x approaches -∞, f(x) goes to -∞ because the leading term, -x³, dominates and because it's negative, the graph falls as x gets more negative. Near x = -1, the cubic nature of the polynomial suggests the graph will cross the x-axis. At x = 4, if we assume f(x) is tangent to the x-axis (as mentioned in the question), there will be a local minimum or maximum with no change in sign of f(x); this means that f(x) approaches the x-axis and then turns away from it. After the zero at x = 4, since there are no other zeros and the leading term is a negative cubic, as x increases, f(x) approaches -∞ again.
From the given options, option A correctly describes the behavior for the zeros x = 4 and x = -1, as well as the behavior of the graph as x approaches infinity and -infinity.