How do you visualise regression results and residuals using Seaborn and Matplotlib?

import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt

# Simulate some data with a non-linear relationship
rng = np.random.default_rng(42)
x = rng.uniform(0, 10, 200)
y = 2 * x + 0.5 * x**2 + rng.normal(0, 3, 200)
df = pd.DataFrame({'x': x, 'y': y})

# 1. Scatter + regression line (with confidence interval)
sns.regplot(data=df, x='x', y='y', ci=95, scatter_kws={'alpha': 0.4})
plt.title('Scatter with OLS Regression Line')
plt.show()

# 2. Residual plot — built in seaborn
sns.residplot(data=df, x='x', y='y', lowess=True,
              scatter_kws={'alpha': 0.4})
plt.axhline(0, color='red', linestyle='--')
plt.title('Residuals vs x (lowess smoothed trend)')
plt.show()
# A horizontal band around 0 = good; a curve = model is missing non-linearity

# 3. Manual residuals (after sklearn model)
from sklearn.linear_model import LinearRegression
model = LinearRegression().fit(df[['x']], df['y'])
df['predicted'] = model.predict(df[['x']])
df['residual']  = df['y'] - df['predicted']

fig, axes = plt.subplots(1, 2, figsize=(12, 4))
axes[0].scatter(df['predicted'], df['residual'], alpha=0.4)
axes[0].axhline(0, color='red', linestyle='--')
axes[0].set(xlabel='Fitted Values', ylabel='Residuals',
            title='Residuals vs Fitted')
sns.histplot(df['residual'], kde=True, ax=axes[1])
axes[1].set_title('Residual Distribution')
plt.tight_layout(); plt.show()