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The number of guests per month at a large resort is given in the table below, where [tex]$f(x)$[/tex] is the number of guests, in hundreds, [tex]$x$[/tex] months since the beginning of the year.

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline[tex]$x$[/tex] & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline[tex]$f(x)$[/tex] & 10 & 15 & 18 & 19 & 18 & 15 \\
\hline
\end{tabular}

Use the data in the table to create the standard form of the function that models this situation, where [tex]$a, b$[/tex], and [tex]$c$[/tex] are constants.

[tex]$f(x) = ax^2 + bx + c$[/tex]



Answer :

GinnyAnswer
Given the data points of the function [tex]\( f(x) \)[/tex], we want to find the quadratic function in standard form:
[tex]\[ f(x) = ax^2 + bx + c \][/tex]

We are given that:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline f(x) & 10 & 15 & 18 & 19 & 18 & 15 \\ \hline \end{array} \][/tex]

To determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], we can use quadratic regression analysis. From this analysis, we find:

[tex]\[ a = -0.25000000000000033 \][/tex]
[tex]\[ b = 3.0000000000000036 \][/tex]
[tex]\[ c = 9.999999999999996 \][/tex]

Therefore, the quadratic function that models the number of guests per month is:
[tex]\[ f(x) = -0.25x^2 + 3x + 10 \][/tex]