Let's work on plotting the data points on a graph and determining the best fit exponential function.
1. Data Points:
We have the following data points:
[tex]\[
(0, 1.4), (1, 2.49), (2, 4.44), (3, 7.9), (4, 14.05)
\][/tex]
2. Plotting the Data:
Plot these points on a graph with [tex]\(x\)[/tex]-axis representing the [tex]\(x\)[/tex] values and [tex]\(y\)[/tex]-axis representing the [tex]\(y\)[/tex] values.
3. Exponential Model:
We need to find an exponential function of the form [tex]\(y = a \cdot b^x\)[/tex] that best fits the data.
4. Determining the Parameters:
By finding the best fit curve, we identify the parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for our exponential function. Here are the steps you would typically follow:
- Initial Guess: Start with an initial guess of parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
- Curve Fitting: Use a curve fitting method to adjust [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to minimize the error between the observed and predicted values.
After performing these steps, the best fit parameters are:
[tex]\[
a \approx 1.4, \quad b \approx 1.78
\][/tex]
5. Writing the Best Fit Exponential Function:
Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the exponential function format:
[tex]\[
y = 1.4 \cdot 1.78^x
\][/tex]
Therefore, the best fit exponential function for the given data, rounded to 2 decimal places, is:
[tex]\[
y = 1.4 \cdot 1.78^x
\][/tex]
This function describes the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] based on the given data points.
