The equation of the circle you're working with is:

$(x - 1)^2 + (y + 2)^2 = 25$

This equation is in the standard form of a circle equation:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle,
• $r$ is the radius of the circle.

Step 1: Identify the center and radius

From your equation $(x - 1)^2 + (y + 2)^2 = 25$, we can extract:

• The center of the circle is $(1, -2)$ (since $h = 1$ and $k = -2$).
• The radius $r$ is $\sqrt{25} = 5$.

Step 2: Graph the circle

• Plot the center: First, mark the center of the circle at $(1, -2)$ on a coordinate plane.
• Draw the circle: With a radius of 5, draw a circle that extends 5 units in all directions from the center.

Step 3: Mark key points

To help sketch the circle, you can mark key points at a distance of 5 units from the center:

• Right: $(1 + 5, -2) = (6, -2)$
• Left: $(1 - 5, -2) = (-4, -2)$
• Up: $(1, -2 + 5) = (1, 3)$
• Down: $(1, -2 - 5) = (1, -7)$

Using these points and the center, you can draw the circle.

Would you like an image of this graph, or any further clarifications?

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Related Questions:

1. How do you derive the equation of a circle given its center and a point on the circle?
2. How can we generalize the equation of a circle when it's not centered at the origin?
3. What happens to the graph if you change the radius of the circle?
4. How does shifting the circle’s center affect its equation?
5. Can we express the equation of a circle in parametric form?

Tip: When graphing circles, it's helpful to identify the center and radius before plotting to ensure accuracy.