Question

which of the following is the graph of ((x - 1)^2 + (y + 2)^2 = 4)?

Explanation:

Step1: Recall the circle equation formula

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify center and radius from the given equation

For the equation \((x - 1)^2 + (y + 2)^2 = 4\), we can rewrite \(y + 2\) as \(y - (-2)\). So, comparing with the standard form:

• The center \((h, k)\) is \((1, -2)\).
• The radius \(r\) is \(\sqrt{4}=2\).

Step3: Analyze the graphs based on center and radius

• The center is at \((1, -2)\) (1 unit to the right of the origin on the x - axis and 2 units down on the y - axis) and radius 2.
• We check the three graphs:
• The first graph: The center seems to be around (1,2) (positive y - coordinate), which does not match \((1, -2)\).
• The second graph: The center is at (1, -2) (1 on x - axis, - 2 on y - axis) and radius 2. The circle is drawn such that it is in the fourth - quadrant - like area (since center is (1, - 2)) with radius 2, which matches our center and radius.
• The third graph: The center seems to be around (- 1,2), which does not match \((1, -2)\).

Answer:

The second graph (the one with center at (1, - 2) and radius 2, located such that it is below the x - axis and to the right of the y - axis with radius 2) is the graph of \((x - 1)^2+(y + 2)^2 = 4\).