Question

bell ringer
name: giovanni navia date: period:
identify the center and radius of each. then sketch the graph.

1. ((x - 1)^2 + (y + 3)^2 = 4)
2. ((x - 2)^2 + (y + 1)^2 = 16)

Explanation:

Problem 1: \((x - 1)^2 + (y + 3)^2 = 4\)

Step 1: Recall the standard form of a circle's equation

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.

Step 2: Identify \(h\), \(k\), and \(r\) for the first equation

For \((x - 1)^2 + (y + 3)^2 = 4\), we can rewrite \(y + 3\) as \(y - (-3)\). So comparing with \((x - h)^2 + (y - k)^2 = r^2\), we have \(h = 1\), \(k = -3\), and \(r^2 = 4\). To find \(r\), we take the square root of \(4\), so \(r = \sqrt{4} = 2\).

Problem 2: \((x - 2)^2 + (y + 1)^2 = 16\)

Step 1: Recall the standard form of a circle's equation

Again, use the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify \(h\), \(k\), and \(r\) for the second equation

For \((x - 2)^2 + (y + 1)^2 = 16\), rewrite \(y + 1\) as \(y - (-1)\). Comparing with the standard form, we get \(h = 2\), \(k = -1\), and \(r^2 = 16\). Taking the square root of \(16\), we find \(r = \sqrt{16} = 4\).

Answer:

s:

1. For \((x - 1)^2 + (y + 3)^2 = 4\):

• Center: \((1, -3)\)
• Radius: \(2\)

1. For \((x - 2)^2 + (y + 1)^2 = 16\):

• Center: \((2, -1)\)
• Radius: \(4\)

(For sketching the graphs:

• For the first circle, plot the center \((1, -3)\) and draw a circle with radius \(2\) units.
• For the second circle, plot the center \((2, -1)\) and draw a circle with radius \(4\) units.)