To find a function that models the data given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-1 & -1 \\
\hline
0 & -3 \\
\hline
1 & -9 \\
\hline
2 & -27 \\
\hline
3 & -81 \\
\hline
\end{array}
\][/tex]
Let's explore if an exponential function of the form [tex]\( y = a(b)^x \)[/tex] fits the data.
### Step-by-Step Solution:
1. Identify the Pattern:
Observing the data points, notice how the [tex]\( y \)[/tex] values change:
[tex]\[
-1 \rightarrow -3 \rightarrow -9 \rightarrow -27 \rightarrow -81
\][/tex]
The ratio between consecutive [tex]\( y \)[/tex]-values is:
[tex]\[
\frac{-3}{-1} = 3, \quad \frac{-9}{-3} = 3, \quad \frac{-27}{-9} = 3, \quad \frac{-81}{-27} = 3
\][/tex]
This suggests that each [tex]\( y \)[/tex]-value is obtained by multiplying the previous [tex]\( y \)[/tex]-value by 3. This is characteristic of an exponential function.
2. Determine [tex]\( b \)[/tex]:
Since each [tex]\( y \)[/tex]-value is a result of multiplying the previous [tex]\( y \)[/tex]-value by 3, we can propose:
[tex]\[
b = 3
\][/tex]
3. Determine [tex]\( a \)[/tex]:
To find the coefficient [tex]\( a \)[/tex], we can use one of the given data points, say [tex]\( (0, -3) \)[/tex]. Using [tex]\( y = a(b)^x \)[/tex] and substituting [tex]\( x = 0 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[
-3 = a(3)^0
\][/tex]
Since [tex]\( 3^0 = 1 \)[/tex], we have:
[tex]\[
a = -3
\][/tex]
4. Formulate the Exponential Function:
Using the calculated values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex], the exponential function that models the data is:
[tex]\[
y = -3(3)^x
\][/tex]
### Final Function:
[tex]\[
y = -3(3)^x
\][/tex]
This exponential function accurately fits the given data points in the table.
