Problem 20 - Algebra 2

Question

Find the center and radius of the circle with the given equation. Then graph the circle. $$ x^{2}+(y+2)^{2}=4 $$

Step-by-Step Solution

Answer

The center is (0, -2) and the radius is 2.

Step 1: Identify the Circle Equation Format

The given equation is \(x^2 + (y+2)^2 = 4\). This resembles the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Determine the Center of the Circle

In the equation \((x-h)^2 + (y-k)^2 = r^2\), the center \((h, k)\) is found by identifying the values of \(h\) and \(k\). In this case, \(x-h = x\) and \(y-k = y+2\). Thus, \(h = 0\) and \(k = -2\). Therefore, the center of the circle is \((0, -2)\).

Step 3: Determine the Radius of the Circle

For this equation, \(r^2 = 4\), as shown on the right side of the equation. Solving for \(r\) gives \(r = \sqrt{4} = 2\). So the radius of the circle is 2.

Step 4: Graphing the Circle

To graph the circle, plot the center at the coordinate \((0, -2)\) on a coordinate plane. From there, plot points that are 2 units away in all directions (up, down, left, right, and diagonally). Draw a smooth curve connecting these points to form the circle.

Key Concepts

Center of the Circle

Understanding the center of a circle in its equation is crucial for properly graphing and analyzing properties of the circle. The standard format for a circle's equation is:

• \((x-h)^2 + (y-k)^2 = r^2\)

where \((h, k)\) represents the center.

In the given equation \(x^2 + (y+2)^2 = 4\), we identify the parts that make up \((h, k)\):

• \(x-h = x\)
• \(y-k = y+2\)

By comparing, we identify:

• \(h = 0\)
• \(k = -2\)

Thus, the center is \((0, -2)\).

Knowing the center allows us to easily plot the circle on a graph, as it serves as the circle's anchor point.

Radius of the Circle

The radius of a circle describes its size and how far its edge is from the center. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the term \(r\) is the radius.

For our specific equation, \(x^2 + (y+2)^2 = 4\), we see:

• \(r^2 = 4\)

To find \(r\), we take the square root:

• \(r = \sqrt{4} = 2\)

This tells us the circle extends 2 units in every direction from the center.

In geometry, the radius is a consistent distance from the center to any point on the circle's edge. Knowing this helps us define the circle's boundary when plotting.

Graphing Circles

Graphing circles involves placing the center and then using the radius to draw the curve. Here’s a simple way to do it:

• Start with the center: \((0, -2)\).The center is the starting point for plotting.
• Use the radius: With \(r = 2\), mark points that are 2 units from the center.
• Plot points: Visualize the circle by marking points in all cardinal and diagonal directions, ensuring all lie 2 units away from \((0, -2)\).
• Draw: Connect these points smoothly to form the circle.

The process ensures an accurate representation of the circle in a coordinate plane.

Graphing helps visualize abstract equations and translate mathematical concepts into geometric shapes, providing a tangible form to work with and understand.