To find the quadratic function [tex]\(f(x) = ax^2 + bx + c\)[/tex] that models the given data, we start by identifying the key characteristics of a quadratic function:
1. The quadratic function is represented as [tex]\( f(x) = ax^2 + bx + c \)[/tex].
2. We have the values of [tex]\( x \)[/tex] and [tex]\( f(x) \)[/tex] from the table. Here, [tex]\( x \)[/tex] represents time in months, and [tex]\( f(x) \)[/tex] represents the number of guests (in hundreds).
Given data points:
[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]
Using these points, we aim to find the quadratic coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]. After analysis, we find that the quadratic equation which fits these points is:
[tex]\[ f(x) = -0.25x^2 + 3x + 10 \][/tex]
Here’s the breakdown of the identified coefficients:
- The coefficient [tex]\(a\)[/tex] (the quadratic term coefficient) is [tex]\(-0.25\)[/tex].
- The coefficient [tex]\(b\)[/tex] (the linear term coefficient) is [tex]\(3.0000000000000004\)[/tex]. For practical purposes and in context, this can be approximated to [tex]\(3\)[/tex].
- The coefficient [tex]\(c\)[/tex] (the constant term) is [tex]\(10\)[/tex].
Thus, the quadratic function that models the number of guests per month at the resort is:
[tex]\[ f(x) = -0.25x^2 + 3x + 10 \][/tex]
