Look at this table: \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 1 & -9 \\ \hline 2 & -27 \\ \hline 3 & -81 \\ \hline 4 & -243 \\ \hline 5 & -729 \\ \hline \end{tabular} Write a linear $(y=m x+b)$, quadratic $\left(y=a x^{2}\right)$, or exponential $\left(y=a(b)^{x}\right)$ function that models the data. \[ y= \]

Step 1 ：Given the table, we observe that the y-values are not increasing or decreasing at a constant rate, indicating that this is not a linear function. The rate of change is also not constant, suggesting it's not a quadratic function. However, each y-value is three times the previous y-value, suggesting an exponential function.

Step 2 ：We attempt to write the function in the form of \(y = a(b)^x\).

Step 3 ：Using the point (1, -9), we substitute into the equation to solve for 'a': \(-9 = a(b)^1\) simplifies to \(-9 = a * b\).

Step 4 ：Since each y-value is three times the previous y-value, we set \(b = -3\). Substituting \(b = -3\) into the equation \(-9 = a * b\), we get \(a = -9 / -3\) which simplifies to \(a = 3\).

Step 5 ：We propose the function \(y = 3(-3)^x\).

Step 6 ：To verify, we substitute x = 2 into the function, expecting y = -27. However, we get \(y = 3 * 9\) which simplifies to \(y = 27\). This does not match the table, indicating our function is incorrect.

Step 7 ：To correct the function, we need it to alternate between positive and negative values as x increases. We achieve this by raising -3 to the power of x, and then multiplying by -1.

Step 8 ：The corrected function is \(y = -3(-3)^x\).

Step 9 ：To verify, we substitute x = 2 into the function, expecting y = -27. We get \(y = -3 * 9\) which simplifies to \(y = -27\). This matches the value in the table, confirming our function is correct.