Final answer:
The function h(x) = (x³)(x - 1) is a quartic function and does not have a single vertex or axis of symmetry as a quadratic function does. The options provided do not apply to quartic functions. The correct option is A) Vertex: (0,0), Axis of symmetry: x=0
Step-by-step explanation:
The function h(x) = (x³)(x - 1) is not a quadratic function, so it does not have a vertex in the same sense that a parabola does. Instead, this is a quartic function (degree 4), which can have up to three turning points. Because quartic functions do not have a single vertex, none of the listed options (A, B, C, or D) are correct.
To understand which points could be considered as vertices and what kind of symmetry the function has, the function would need to be analyzed for its critical points where the first derivative equals zero.
The concept of an axis of symmetry applies to parabolic functions of the form ax² + bx + c = 0, where the axis of symmetry can be found using the formula x = -b/2a. However, since h(x) is a quartic function, the idea of a single axis of symmetry does not apply in the same way it does to a parabola.
To find the vertex and axis of symmetry of the given function h(x) = x³(x - 1), we can use the formula for the vertex of a quadratic function, which is given by (-b/2a, f(-b/2a)). In this case, the quadratic part of the function is x(x - 1), so a = 1, b = 0, and c = 0. Plugging these values into the formula, we get the vertex as (0, 0). The axis of symmetry is a vertical line passing through the vertex, which is x = 0.
The correct option is A) Vertex: (0,0), Axis of symmetry: x=0