To find the exponential equation that models the given data, we use the general form of an exponential function, [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial value (when [tex]\( x = 0 \)[/tex]), and [tex]\( b \)[/tex] is the base representing the growth (or decay) factor.
Given the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 24 \\
\hline
-1 & 16.8 \\
\hline
0 & 11.76 \\
\hline
1 & 8.232 \\
\hline
2 & 5.7624 \\
\hline
\end{array}
\][/tex]
1. Determine [tex]\( a \)[/tex]:
- The initial value [tex]\( a \)[/tex] can be directly obtained from the table when [tex]\( x = 0 \)[/tex]. From the table, it is given:
[tex]\[
a = 11.76
\][/tex]
2. Calculate [tex]\( b \)[/tex]:
- To find [tex]\( b \)[/tex], we can use the ratio of two successive [tex]\( y \)[/tex] values and then average the factor if desired. We use the following pair of points: [tex]\( (x_1, y_1) = (-2, 24) \)[/tex] and [tex]\( (x_2, y_2) = (-1, 16.8) \)[/tex].
The formula to determine [tex]\( b \)[/tex] is:
[tex]\[
b = \left( \frac{y_2}{y_1} \right)^{\frac{1}{x_2 - x_1}}
\][/tex]
Plugging in the values:
[tex]\[
b = \left( \frac{16.8}{24} \right)^{\frac{1}{-1 - (-2)}} = \left( \frac{16.8}{24} \right)^{\frac{1}{1}} = \left( \frac{16.8}{24} \right) \approx 0.7
\][/tex]
Thus, the exponential equation that models the given data is [tex]\( y = 11.76 \cdot (0.7)^x \)[/tex].
From the given choices, the best answer is:
c. [tex]\( y = 11.76 (0.7)^x \)[/tex]
