To find a function that models the data given in the table, we'll examine three types of functions: linear [tex]\(y = mx + b\)[/tex], quadratic [tex]\(y = ax^2 + bx + c\)[/tex], and exponential [tex]\(y = a \cdot b^x\)[/tex].
Given the points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-10 & 500 \\
\hline
-9 & 405 \\
\hline
-8 & 320 \\
\hline
-7 & 245 \\
\hline
-6 & 180 \\
\hline
\end{array}
\][/tex]
We can fit these points using a quadratic function, which generally takes the form [tex]\(y = ax^2 + bx + c\)[/tex]. After performing the necessary calculations, we find:
The coefficients are approximately:
- [tex]\(a = 5.0\)[/tex]
- [tex]\(b \approx -3.48 \times 10^{-13}\)[/tex] (which is essentially zero)
- [tex]\(c \approx -1.33 \times 10^{-12}\)[/tex] (which is also essentially zero)
So, simplifying the quadratic function given these values, our function looks like:
[tex]\[
y = 5x^2
\][/tex]
Therefore, the quadratic function that best fits the provided data is:
[tex]\[
y = 5x^2
\][/tex]
