Final answer:
The behavior of the graph at its zeros depends on whether the zeros are simple or repeated; it can resemble a linear or a quadratic function.
Step-by-step explanation:
The behavior of the graph (y = − x³ − 5x² − 3x + 9) at each of its zeros can be determined by analyzing the sign of the derivative (rate of change) near those zeros. For a cubic function like this, the behavior around its zeros can resemble either a linear function or a quadratic function, depending on whether the zero is a simple zero or a repeated zero. To determine the exact nature of each zero, we would need to find the zeros of the function and then analyze the sign of the first and possibly second derivatives at those points.
If at a zero the function's graph crosses the x-axis, the behavior near this zero will look linear, meaning that the graph will pass straight through the x-axis at that point. If the graph just touches the x-axis at the zero and turns around, that is indicative of a repeated zero, and the graph near this point will resemble a quadratic function.