Answer
a) The graph is shown below
b) Factors like technology, number of shows, natural calamities can affect the accuracy.
c) The maximum value is $242.32.$
Work Step by Step
(a)
The quadratic function modelling the given data is $y=-5.25{{x}^{2}}+26.079x+209.93$
The data is for the number of tickets sold every year.
Now right click on the dots and click on add trending option; then in trending option, select polynomial, and then show quadratic equation and value of R.
Thus, the obtained graph is shown above.
The equation hence obtained is $y=-5.25{{x}^{2}}+26.079x+209.93$.
(b)
We know that the model is a quadratic function; it can be observed that the number of tickets sold increase until a certain year and then decrease.
The factors affecting the change can be attributed to the technology. As the technology is increasing, there are more chances of entertaining shows to be organized in schools for children, which can affect the predictions.
Thus, if the number of shows decreases, it will also affect the accuracy of the prediction. Natural calamities can also affect the sale of tickets.
(c)
Let us consider the given equation:
$y=-5.25{{x}^{2}}+26.079x+209.93$
Where $a=-5.25,\ b=26.079,\ c=209.93.$
Calculate $-\frac{b}{2a}$ and if $a<0$ , the function has maxima at the vertex.
Now, the value of $-\frac{b}{2a}$ is calculated as
$\begin{align}
& x=-\frac{b}{2a} \\
& =-\frac{26.079}{2\left( -5.25 \right)} \\
& =\frac{26.079}{10.5} \\
& =2.48
\end{align}$
Now, find the maximum value of the function at $x=2.48$ as
$\begin{align}
& y=-5.25{{\left( 2.48 \right)}^{2}}+26.079\left( 2.48 \right)+209.93 \\
& =-32.28+64.67+209.93 \\
& =242.32
\end{align}$
Hence, the maximum value of the function is $242.32$.
