To determine the exponential equation that models the given data, we start by understanding the nature of exponential functions. An exponential function can generally be written in the form:
[tex]\[ y = a \cdot b^x \][/tex]
Where:
- [tex]\(a\)[/tex] is the initial value (the [tex]\(y\)[/tex]-intercept when [tex]\(x=0\)[/tex]).
- [tex]\(b\)[/tex] is the base of the exponential (the common ratio between successive [tex]\(y\)[/tex]-values).
Given the data in the table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
-2 & 24 \\
\hline
-1 & 16.8 \\
\hline
0 & 11.76 \\
\hline
1 & 8.232 \\
\hline
2 & 5.7624 \\
\hline
\end{tabular}
\][/tex]
First, identify the value of [tex]\(a\)[/tex] which is the [tex]\(y\)[/tex]-value when [tex]\(x = 0\)[/tex]. From the table, when [tex]\(x = 0\)[/tex], [tex]\(y = 11.76\)[/tex], thus:
[tex]\[ a = 11.76 \][/tex]
Next, we determine the base [tex]\(b\)[/tex] of the exponential function by finding the ratio of successive [tex]\(y\)[/tex]-values:
[tex]\[ b \approx \frac{y_{i+1}}{y_i} \][/tex]
For instance:
[tex]\[
\frac{16.8}{24} = 0.7, \quad \frac{11.76}{16.8} = 0.7, \quad \frac{8.232}{11.76} = 0.7, \quad \frac{5.7624}{8.232} = 0.7
\][/tex]
Observing these ratios, we see that the common ratio [tex]\(b\)[/tex] between successive [tex]\(y\)[/tex]-values is [tex]\(0.7\)[/tex].
Therefore, we can write the exponential equation that models the data as:
[tex]\[ y = 11.76 \cdot (0.7)^x \][/tex]
Comparing this equation with the provided answer choices:
a. [tex]\(y = 11.76(1.3)^x\)[/tex]
c. [tex]\(y = 11.76(0.7)^x\)[/tex]
b. [tex]\(y = 24(0.7)^x\)[/tex]
d. [tex]\(y = 16.8(1.7)^x\)[/tex]
The equation [tex]\(y = 11.76(0.7)^x\)[/tex] matches our derived equation.
Thus, the best answer to the question is:
C
