Final answer:
To sketch a graph of the polynomial function, one should position zeros at (-4, 0) and (-1, 0), ensuring the function is negative between these points and positive outside these intervals. It should also decrease towards -∞ and from 1 to ∞, and increase from -2 to 1 with an additional zero at 3.
Step-by-step explanation:
To sketch a graph of the polynomial function with the given conditions, we will follow the descriptions provided:
The function is negative on the intervals (-4, -1) and (3, ∞).
The function is positive on the intervals (-∞, -4) and (-1, 3).
The function is decreasing on the interval (-∞, -2] and [1, ∞).
The function is increasing on the intervals [-2, 1] and [3, ∞).
We begin by sketching a point at (-4, 0) and another at (-1, 0), representing the zeros of the function. The graph should start above the x-axis, decreasing towards the first zero at -4, then go below the x-axis to the next zero at -1, making the function negative in between these points.
Then it should increase and become positive from -1 to 3. After the point at 3, the function will once again decrease, becoming negative and continuing to do so towards infinity. To fully comply with the function's behavior, another zero should be placed at 3.
Multiple graphs can satisfy these conditions, as the exact shape may vary depending on the degrees and coefficients of the polynomial. The function could have additional zeros and turns beyond the specified intervals, influencing the graph's appearance.