To determine which type of function best models the data, let's analyze the table provided:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & y \\
\hline
-6 & 324 \\
\hline
-5 & 225 \\
\hline
-4 & 144 \\
\hline
-3 & 81 \\
\hline
-2 & 36 \\
\hline
\end{tabular}
\][/tex]
We need to determine whether the data fits a linear, quadratic, or exponential model. Let's start testing the quadratic function [tex]\( y = ax^2 \)[/tex].
Consider the first data point [tex]\( (-6, 324) \)[/tex]:
[tex]\[
y = ax^2 \implies 324 = a(-6)^2 \implies 324 = 36a \implies a = 9
\][/tex]
Next, let's verify this [tex]\( a \)[/tex] value with other points:
For [tex]\( x = -5 \)[/tex]:
[tex]\[
y = 9(-5)^2 = 9 \times 25 = 225
\][/tex]
For [tex]\( x = -4 \)[/tex]:
[tex]\[
y = 9(-4)^2 = 9 \times 16 = 144
\][/tex]
For [tex]\( x = -3 \)[/tex]:
[tex]\[
y = 9(-3)^2 = 9 \times 9 = 81
\][/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
y = 9(-2)^2 = 9 \times 4 = 36
\][/tex]
All data points perfectly fit the quadratic equation [tex]\( y = 9x^2 \)[/tex]. Thus, the function that models the data is:
[tex]\[
y = 9x^2
\][/tex]
