A parabola intersects the x-axis at the points (1,0) and (7,0) and passes through the point (4,12).
a. (i) Express the polynomial representing the parabola in factored form with an unknown leading coefficient a.
b. (ii) Determine the value of a using the point (4,12).
c. (iii) Write the polynomial in standard form and find its y-intercept.
d. (iv) Show that the axis of symmetry of the parabola is x=4 and state the coordinates of the vertex.