To find the best exponential equation that models the given data, we need to compare the possible equations with the calculated coefficients for the exponential model. The given data points for the values of [tex]\( x \)[/tex] and corresponding values of [tex]\( y \)[/tex] are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 22 \\
-1 & 15.4 \\
0 & 10.78 \\
1 & 7.546 \\
2 & 5.2822 \\
\hline
\end{array}
\][/tex]
An exponential function can be written in the form: [tex]\( y = a \cdot b^x \)[/tex].
From the calculated results:
- The coefficient [tex]\( a \)[/tex] is approximately 10.78.
- The base [tex]\( b \)[/tex] is approximately 0.7.
We will compare these values with the options provided:
a. [tex]\( y = 15.4 \cdot (1.7)^x \)[/tex]
- Here, [tex]\( a = 15.4 \)[/tex] and [tex]\( b = 1.7 \)[/tex], which do not match our calculated values.
b. [tex]\( y = 10.78 \cdot (0.7)^x \)[/tex]
- Here, [tex]\( a = 10.78 \)[/tex] and [tex]\( b = 0.7 \)[/tex], which closely match our calculated values.
c. [tex]\( y = 22 \cdot (0.7)^x \)[/tex]
- Here, [tex]\( a = 22 \)[/tex] and [tex]\( b = 0.7 \)[/tex], where [tex]\( a \)[/tex] does not match our calculated coefficient [tex]\( a \)[/tex].
d. [tex]\( y = 10.78 \cdot (1.3)^x \)[/tex]
- Here, [tex]\( a = 10.78 \)[/tex] matches our calculated coefficient, but [tex]\( b = 1.3 \)[/tex] does not match our calculated base.
Thus, the best match for the exponential model given is:
[tex]\[
\boxed{b. \; y = 10.78 \cdot (0.7)^x}
\][/tex]
