Answer:)
Part 1.) The horizontal coordinate in the case of The Dark Knight and Avatar both represent the constant cumulative box office revenue that both these movies will have obtained once they are out of all the box offices. Then, since no new screenings will be done, the cumulative revenue stays a constant and thus, with time, a horizontal asymptote is achieved.
Once we reach a time when the box offices feel it is no longer profitable to keep screening Titanic, it too shall approach a horizontal asymptote.
Part 2.)
a.)
At the vertex, for any quadratic function, we have the slope to be zero, and this leads us to :
Also, since the vertex would lie on the curve itself, it is a solution to the given equation :
Further, we want (0,0) to also be a point on the curve, since at the start of time (t = 0 weeks), a movie cannot have non-zero cumulative box office revenue, that is y(0) = 0:
Hence, the curve is given by :
b.)
Since y = 600 is a horizontal asymptote to the given curve we want to fit, it must be true that :
Again, we would want (0,0) to be a point on the curve for the same reasons, which leads us to :
Which leads us to the following equation :
c.) It is evident that the exponential curve with the horizontal asymptote would fit the curve better, since it would fit the observed behavior so far well, as well as the long term (asymptotic) behavior as well. On the other hand, the quadratic function will curve downwards with increase in time (x), and thus not model the behavior of the cumulative box office revenue function (y) accurately.
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