Let a be a non-zero real constant. Consider the family of parabolas defined by y2=4ax.
a. (i) Prove that the sign of a determines whether the parabola opens to the right or to the left.
b. (ii) For a=3 and a=−3, write the equations of the directrices and calculate the distance between the corresponding foci.
c. (iii) On a common set of axes, sketch the parabolas for a=3 and a=−3, clearly marking their vertices, foci, and directrices.