You are right that since the second differences are constant, a quadratic function will fit the points.
There are ways to determine the function's coefficients from the table of differences, but apparently you haven't been taught that part, as they are leading you in a different direction.
But why can't you plot the graph? Just label the axes appropriately (0 to 6 or so for x, 0 to 150 or so for y) and plot the points. Are you trying to use some particular software or other method, and don't know how to set the scale, perhaps? We'll need to see your work for the parts you are struggling with, so we can help your specific struggles. The parabola will be tall, not wide.
Once you have a graph (or even just visualize it in your head, as I am doing!) it should be easy to answer (c). Possibly before answering (e) you are expected to write the equation for the function, but I am surprised they didn't explicitly ask for it. That would be the best way to answer (e).
I am handicapped by not knowing what specific techniques you have been taught. Perhaps, if you need more help, you can tell us what was covered before this exercise.
I should also comment on the fact that this is called a "unit exam". Are you allowed to have help on it?