The equation $x^2 + (y+1)^2 = 4$ represents a circle centered at $(0, -1)$ with a radius of 2. Here's a breakdown of the components:

1. The general form of a circle is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center, and $r$ is the radius.
2. In this case:
   • $h = 0$, $k = -1$, and $r = 2$.

Let me draw the graph for you. numpy as np import matplotlib.pyplot as plt

Define the circle parameters

center = (0, -1) radius = 2

Generate the circle

theta = np.linspace(0, 2 * np.pi, 500) x = center[0] + radius * np.cos(theta) y = center[1] + radius * np.sin(theta)

Plot the circle

plt.figure(figsize=(6, 6)) plt.plot(x, y, label=r"$x^2 + (y+1)^2 = 4$", color="blue") plt.axhline(0, color='black',linewidth=0.5, linestyle='--') plt.axvline(0, color='black',linewidth=0.5, linestyle='--') plt.gca().set_aspect('equal', adjustable='box')

Add details to the graph

plt.title("Graph of the Circle $x^2 + (y+1)^2 = 4$", fontsize=14) plt.xlabel("x-axis") plt.ylabel("y-axis") plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5) plt.legend() plt.show()Here is the graph of the circle defined by the equation $x^2 + (y+1)^2 = 4$. The circle is centered at $(0, -1)$ and has a radius of 2.

Would you like further details, have any questions, or need explanations about the graph? Here are some related questions you might consider:

1. How is the equation of a circle derived?
2. How can the radius and center of a circle be identified from its equation?
3. What happens if the equation has additional terms like $kx + my$?
4. How do translations affect the graph of a circle?
5. Can a circle's equation have complex solutions, and what would that mean graphically?

Tip: The general form of a circle's equation can be expanded to $Ax^2 + Ay^2 + Dx + Ey + F = 0$. Completing the square helps rewrite it in standard form!