Problem 28 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$(x+3)^{2}+(y+3)^{2}=4$$

Step-by-Step Solution

Answer

Center: (-3, -3); Radius: 2

Step 1: Recognize the Equation of a Circle

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

Step 2: Identify Center Point

By comparing the given equation \((x+3)^2 + (y+3)^2 = 4\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\), identify that \(h = -3\) and \(k = -3\). Therefore, the center of the circle is \((-3, -3)\).

Step 3: Calculate Radius

The term on the right side of the equation is \(r^2=4\). Therefore, the radius \(r\) is obtained by taking the square root: \(r = \sqrt{4} = 2\).

Step 4: Graph the Circle

To graph the circle, start by plotting the center at \((-3, -3)\). From this point, measure a distance of 2 units in all directions (up, down, left, right) to draw the circle. Ensure that the circle passes through these points, maintaining a constant radius of 2 units from the center.

Key Concepts

Center of a Circle

In a circle's equation, the center of a circle is the point from which every point on the circle is equidistant. In mathematical terms, this is often represented in the form

• extbf{Standard form:} \((x-h)^2 + (y-k)^2 = r^2\)

Here,

• extit{h} is the x-coordinate of the center, and
• extit{k} is the y-coordinate of the center.

In our example, the equation is \((x+3)^2 + (y+3)^2 = 4\). Comparing it with the standard form gives you:

• The x-coordinate, \(h\), as -3,
• and the y-coordinate, \(k\), also as -3.

So, the center of our circle is \((-3, -3)\). It's like the home base from which the boundary of the circle extends equally in all directions.

Radius of a Circle

The radius of a circle is simply the distance from its center to any point lying on the edge of the circle. It represents how big or large the circle will be on a graph. This is the value represented by \(r\) in the standard circle equation.

In our case:

• We look at the equation \((x+3)^2 + (y+3)^2 = 4\). The number on the right side, 4, represents \(r^2\), which is the square of the radius.
• To find \(r\), take the square root of 4. This gives you \(r = \sqrt{4} = 2\).

Therefore, the radius of our circle is 2. This means the circle extends 2 units outwards from the center in every direction.

Graphing Circles

Graphing circles on a coordinate plane helps visualize their size and location. It's a straightforward process once you know the center and radius.

Here's how to graph a circle:

• Start by plotting the center. For our circle, that's at \((-3, -3)\).
• Next, use the radius to determine how far from the center the circle extends. Here, our radius is 2 units.

From the center point, measure 2 units in four primary directions: up, down, left, and right. These points show how wide the circle spans. Now, draw a smooth curve to connect these points in a round shape, making sure it curves uniformly. Your circle will encompass all these points, etching a boundary exactly 2 units away from the center in every direction.