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To begin, we will use Python with libraries such as NumPy, SciPy, and Matplotlib to perform nonlinear regression and plot the data. Import the required libraries and prepare the dataset as follows: 1. Import necessary libraries: NumPy, SciPy.optimize, and Matplotlib.pyplot 2. Define the x and y values as separate arrays. Here's a Python code snippet to achieve the above steps: ```python import numpy as np from scipy.optimize import curve_fit import matplotlib.pyplot as plt x_data = np.array([0.1, 0.2, 0.4, 0.6, 0.9, 1.3, 1.5, 1.7, 1.8]) y_data = np.array([0.75, 1.25, 1.45, 1.25, 0.85, 0.55, 0.35, 0.28, 0.18]) ```

Define the nonlinear model function \(y = \alpha_{4} xe^{\beta_{4} x}\) that accepts x, \(\alpha_{4}\), and \(\beta_{4}\) as arguments and returns the evaluated y value. In Python: ```python def nonlinear_model(x, alpha_4, beta_4): return alpha_4 * x * np.exp(beta_4 * x) ```

The curve_fit function from SciPy's optimize module will be used to find the optimal parameters \(\alpha_{4}\) and \(\beta_{4}\). Pass our nonlinear_model function, x_data, and y_data to curve_fit to find the optimized parameters: ```python params_opt, params_cov = curve_fit(nonlinear_model, x_data, y_data) alpha_4_opt, beta_4_opt = params_opt print(f"Optimal alpha_4: {alpha_4_opt}\nOptimal beta_4: {beta_4_opt}") ```

Finally, we need to plot both the fitted nonlinear curve and the given data points. To achieve this, we will use Matplotlib.pyplot: 1. Plot the given data points using scatter function. 2. Generate a set of x values to evaluate our nonlinear_model with our optimal parameters. 3. Plot the fitted nonlinear curve using the plot function. 4. Add necessary labels, legend, and display the plot. Here's the Python code to plot the data and the fitted curve: ```python plt.scatter(x_data, y_data, label='Data points', color='red') x_fit = np.linspace(min(x_data), max(x_data), 100) y_fit = nonlinear_model(x_fit, alpha_4_opt, beta_4_opt) plt.plot(x_fit, y_fit, label='Fitted curve', color='blue') plt.xlabel('x') plt.ylabel('y') plt.legend() plt.show() ``` The output plot should now display the given data points in red and the fitted nonlinear curve in blue. We have successfully used nonlinear regression to estimate the parameters \(\alpha_{4}\) and \(\beta_{4}\) and plotted the fitted model alongside the data points.