as np import matplotlib.pyplot as plt

Define the function

def f(x): return -(x - 4)**2 + 1

Generate x values

x = np.linspace(0, 8, 500) # Chosen range for better visualization

Compute y values

y = f(x)

Plot the graph

plt.figure(figsize=(8, 6)) plt.plot(x, y, label=r"$y = -(x-4)^2 + 1$") plt.title("Graph of the equation y = -(x-4)^2 + 1") plt.xlabel("x") plt.ylabel("y") plt.axhline(0, color='black', linewidth=0.5, linestyle='--') plt.axvline(0, color='black', linewidth=0.5, linestyle='--') plt.grid(color='gray', linestyle='--', linewidth=0.5, alpha=0.7) plt.legend() plt.show()Here is the graph of the equation $y = -(x-4)^2 + 1$. The parabola is inverted (opens downward) with its vertex at $(4, 1)$.

Would you like a deeper analysis or have additional questions? Here are some relative follow-up questions:

1. What is the vertex and axis of symmetry for this parabola?
2. How does the graph change if the equation is $y = (x-4)^2 + 1$?
3. What are the x-intercepts of this parabola, if any?
4. How would the graph look if a coefficient multiplied the squared term (e.g., $y = -2(x-4)^2 + 1$)?
5. Can this equation be rewritten in standard form $y = ax^2 + bx + c$? How does the graph relate?

Tip: When graphing quadratic equations, identify the vertex, direction (up/down), and intercepts for a clearer understanding.