Final answer:
The quadratic function with roots at 6 and 2, and a vertex at (4, -4/3) should be written in the vertex form using the roots and vertex information, which results in an equation that does not match any of the provided options.
Step-by-step explanation:
The student is asked to write the equation of a quadratic function with given roots and a vertex. The roots of the quadratic equation are 6 and 2, and the vertex is at the point (4, -4/3). Since a quadratic function with these characteristics must have a vertical axis of symmetry, the graph of the function is symmetric about the line x = 4.
Knowing the roots and the vertex makes it possible to use the vertex form of a quadratic function which is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. Since the vertex is (4, -4/3), we can substitute h and k to get: y = a(x - 4)^2 - 4/3. Now, we need to find the value of 'a'. Because the roots are 2 and 6, the axis of symmetry (x = 4) is exactly halfway between the roots. This means that the parabola opens downwards, and the value of 'a' should be negative for the vertex to be at a maximum point.
Given that x = 2 is a root, we can substitute it into the equation to solve for 'a'. When x = 2, y = 0, which gives us the equation 0 = a(2 - 4)^2 - 4/3. By solving this, we get a = -1/3. Substituting this value back into the vertex form, we obtain the equation y = -1/3(x - 4)^2 - 4/3. Multiplying through by 3 to clear the fraction, we get the equation in standard form: y = -(x - 4)^2 - 4 = -x^2 + 8x - 16 - 4 = -x^2 + 8x - 20. However, none of the provided options match this equation exactly, indicating a possible error in the question or the answer choices.