To graph the circle given by the equation [tex]\((x-2)^2 + (y+5)^2 = 4\)[/tex], follow these steps:
1. Identify the center and radius of the circle:
- The standard form of a circle's equation is [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex].
- Comparing [tex]\((x-2)^2 + (y+5)^2 = 4\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can determine that:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((2, -5)\)[/tex].
- The radius [tex]\(r\)[/tex] is the square root of 4, which is 2.
2. Plot the center:
- Locate the point [tex]\((2, -5)\)[/tex] on the coordinate plane. This is the center of your circle.
3. Draw the circle:
- From the center [tex]\((2, -5)\)[/tex], measure a distance of 2 units in all directions (up, down, left, and right), marking these points on the coordinate plane:
- Right: [tex]\((2 + 2, -5) = (4, -5)\)[/tex]
- Left: [tex]\((2 - 2, -5) = (0, -5)\)[/tex]
- Up: [tex]\((2, -5 + 2) = (2, -3)\)[/tex]
- Down: [tex]\((2, -5 - 2) = (2, -7)\)[/tex]
- These four points are on the circle.
- To draw the circle, sketch a smooth, rounded curve connecting these four points, ensuring the curve is equidistant from the center at all points.
Here is an illustrative step-by-step guide to help visualize it:
1. Plot the center (2, -5):
- Mark the point [tex]\((2, -5)\)[/tex] on your graph.
2. Mark points 2 units from the center:
- Right to (4, -5)
- Left to (0, -5)
- Up to (2, -3)
- Down to (2, -7)
3. Draw the circle:
- Use a compass set to 2 units if you have one, or sketch freehand.
- Draw a rounded curve that passes through the points (4, -5), (0, -5), (2, -3), and (2, -7), ensuring the circle remains consistent in radius throughout.
By following these steps, you will have successfully graphed the circle defined by the equation [tex]\((x-2)^2 + (y+5)^2 = 4\)[/tex].
Here's a quick coordinate guide:
- Center: (2, -5)
- Rightmost point on the circle: (4, -5)
- Leftmost point on the circle: (0, -5)
- Uppermost point on the circle: (2, -3)
- Lowermost point on the circle: (2, -7)
