To derive the exponential function that best fits the given data, we can follow these steps:
1. Transform the Exponential Relationship to a Linear One:
Given the exponential model [tex]\( y = ab^x \)[/tex], we take the natural logarithm of both sides to linearize it:
[tex]\[
\ln(y) = \ln(a) + x \ln(b)
\][/tex]
Here, we let [tex]\( Y = \ln(y) \)[/tex], [tex]\( A = \ln(a) \)[/tex], and [tex]\( B = \ln(b) \)[/tex]:
[tex]\[
Y = A + Bx
\][/tex]
This equation represents a linear relationship between [tex]\( Y \)[/tex] and [tex]\( x \)[/tex].
2. Construct the Transformed Dataset:
Using the given data, we compute [tex]\( Y \)[/tex] as [tex]\( \ln(y) \)[/tex]:
[tex]\[
x: -3, \quad y: \frac{3}{64} \quad \Rightarrow \quad \ln\left(\frac{3}{64}\right)
\][/tex]
[tex]\[
x: -2, \quad y: \frac{3}{16} \quad \Rightarrow \quad \ln\left(\frac{3}{16}\right)
\][/tex]
[tex]\[
x: -1, \quad y: \frac{3}{4} \quad \Rightarrow \quad \ln\left(\frac{3}{4}\right)
\][/tex]
[tex]\[
x: 0, \quad y: 3 \quad \Rightarrow \quad \ln(3)
\][/tex]
[tex]\[
x: 1, \quad y: 12 \quad \Rightarrow \quad \ln(12)
\][/tex]
[tex]\[
x: 2, \quad y: 48 \quad \Rightarrow \quad \ln(48)
\][/tex]
[tex]\[
x: 3, \quad y: 192 \quad \Rightarrow \quad \ln(192)
\][/tex]
[tex]\[
x: 4, \quad y: 768 \quad \Rightarrow \quad \ln(768)
\][/tex]
3. Perform Linear Regression on the Transformed Data:
We fit the transformed data [tex]\((x, \ln(y))\)[/tex] to the linear model [tex]\( Y = A + Bx \)[/tex] to find the coefficients [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
4. Interpret the Regression Coefficients:
After fitting the linear model, we obtain the coefficients:
[tex]\[
A = 1.09861229, \quad B = 1.38629436
\][/tex]
To revert to the exponential form, we need to transform the coefficients back:
[tex]\[
a = e^A = e^{1.09861229} \approx 3.0000000000000004
\][/tex]
[tex]\[
b = e^B = e^{1.38629436} \approx 4.000000000000003
\][/tex]
Therefore, the exponential function that models the given data is:
[tex]\[
y = 3 \cdot 4^x
\][/tex]
