The graph of a polynomial is shown - HSC - SSCE Mathematics Advanced - Question 4 - 2023 - Paper 1

The polynomial graph is a cubic function that opens downward, as evident from the sketch. We can determine that the leading coefficient must be negative. Let's analyze each equation:

1. Option A: $y = -(x - b)(x - c)^2$ - Here, since $c$ is squared, the graph will touch the x-axis at $c$ and will have an upward-facing point, while $b$ will determine the behavior as we approach it from the left. Thus both $b > 0$ and $c < 0$ are correct.
2. Option B: Similar structure, but $c > 0$ is incorrect since it would make the polynomial rise indefinitely.
3. Option C: This has a different structure altogether and contradicts the graph being a quartic leading to an upward face.
4. Option D: Similar reasoning from option C leads to incorrect conclusions.

According to the examination of these options, Option A is the only valid response.