The cubic function with zeros at (2,0), (-1,0), and (3,0) and a point on the graph at (0,6) is f(x) = (x - 2)(x + 1)(x - 3).
To write a cubic function that has zeros at (2,0), (-1,0), and (3,0), we use the fact that a cubic function has the form:
f(x) = a(x - r1)(x - r2)(x - r3)
where 'a' is a constant and r1, r2, r3 are the roots of the equation.
Given the roots 2, -1, and 3, the function takes the form:
f(x) = a(x - 2)(x + 1)(x - 3)
By using the point (0,6), which is not a root but a point on the graph, we can find the value of 'a'.
f(0) = 6 = a(0 - 2)(0 + 1)(0 - 3)
6 = a(-2)(1)(-3)
6 = 6a
a = 1
Therefore, the cubic function is:
f(x) = (x - 2)(x + 1)(x - 3)