Answer
a. See graph; the values increase, then decrease.
b. $y = -0.48x^2 + 6.17x + 9.57$
c. Season $6$: $29.3$ millions
d. underestimates by $1.1$ million.
e. See graph and explanations.
Work Step by Step
a. We can graph the scatter plot of the data as shown in the figure where the smooth curve is a fitted quadratic function. We can see that the data increases initially and then decreases later, so a quadratic function will model the data well.
b. We can find the quadratic function as $y = -0.48x^2 + 6.17x + 9.57$
c. We have $a=-0.48, b=6.17$ and a maximum can be found at $x=-\frac{b}{2a}=-\frac{6.17}{2(-0.48)}\approx6$ and the number of viewers is
$y = -0.48(6)^2 + 6.17(6) + 9.57\approx29.3$ millions
d. Examine the data. We notice that the maximum is $30.4$ million viewers in season $5$. Thus the model in part (c) underestimates the real value by $30.4-29.3=1.1$ million.
e. See graph. We can see that without a shake-up, the number of viewers will drop as time increases.
