as np import matplotlib.pyplot as plt

Define the range for x and y values

x = np.linspace(-3, 5, 400) y = np.linspace(-5, 5, 400) X, Y = np.meshgrid(x, y)

Define the inequalities

ineq1 = Y >= X2 - 4 ineq2 = Y < -X2 + 4*X + 4

Plotting the inequalities

plt.figure(figsize=(8, 8)) plt.title("System of Quadratic Inequalities", fontsize=14) plt.xlabel("x-axis", fontsize=12) plt.ylabel("y-axis", fontsize=12) plt.xlim(-3, 5) plt.ylim(-5, 5)

Plot the regions satisfying the inequalities

plt.contourf(X, Y, ineq1 & ineq2, levels=[0, 1], colors=['lightblue'], alpha=0.7)

Plot the boundary lines

plt.plot(x, x2 - 4, color="blue", label="y = x^2 - 4") plt.plot(x, -x2 + 4*x + 4, color="red", label="y = -x^2 + 4x + 4")

Add legend and grid

plt.legend() plt.grid(alpha=0.5) pltThe graph above illustrates the system of quadratic inequalities:

1. y ≥ x² - 4: The region above or on the parabola $y = x^2 - 4$, shown in blue.
2. y < -x² + 4x + 4: The region below the parabola $y = -x^2 + 4x + 4$, shown in red.

The overlapping region, shaded in light blue, represents the solution set that satisfies both inequalities.

Would you like more details or have any specific questions?
Here are five related questions to explore further:

1. What is the exact vertex of each parabola?
2. How do we determine the intersection points of these two parabolas?
3. Can this method extend to 3D surfaces for more complex inequalities?
4. What is the significance of boundary conditions in inequalities?
5. How can systems of inequalities be used in optimization problems?

Tip: Always check for boundary inclusion (e.g., ≥ or <) when shading regions for inequalities!