Problem 10 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$(x+8)^{2}+(y-4)^{2}=4$$

Step-by-Step Solution

Answer

The center of the circle is at point (-8, 4) and the radius is 2. To graph the circle, plot the center at (-8, 4), draw a circle with a radius of 2, and label the center point and equation \((x+8)^2+(y-4)^2=4\).

Step 1: Identify the center of the circle

The given equation is \((x+8)^2+(y-4)^2=4\). Comparing this with the standard form \((x-a)^2+(y-b)^2=r^2\), we see that the center of the circle (a, b) is at point (-8, 4).

Step 2: Identify the radius of the circle

Now, let's determine the radius of the circle. The given equation is \((x+8)^2+(y-4)^2=4\). From the standard form of the circle equation \((x-a)^2+(y-b)^2=r^2\), we identify r^2 as 4. Taking the square root of both sides, we get \(r = 2\). So, the radius of the circle is 2.

Step 3: Graph the circle

To graph this circle, follow the steps below: 1. Plot the center of the circle, which is the point (-8, 4) on the coordinate plane. 2. From the center, draw a circle with a radius of 2. 3. Finally, label the center point and write the equation of the circle. Now, you have successfully graphed the circle with the equation \((x+8)^2+(y-4)^2=4\).

Key Concepts

Center of Circle

In the equation of a circle, the center plays a vital role in defining its position on the coordinate plane. The general form of the equation for a circle is \((x-a)^2 + (y-b)^2 = r^2\). Here,

• \(a\) and \(b\) are the coordinates of the center of the circle, \((a, b)\).
• In this form, x and y are variables that represent points on the circle's outline.

To find the center from the given equation \((x+8)^2 + (y-4)^2 = 4\), you compare it with the standard form. Here:

• \(a = -8\)
• \(b = 4\)

Thus, the center of the circle is at point \((-8, 4)\). This means all other points on the circle are equidistant from this center point. Understanding the center's role helps in graphing the circle correctly, as you begin plotting from this fixed position.

Radius of Circle

The radius of a circle is the distance from its center to any point on its perimeter. It is a crucial measure as it defines the size of the circle. From the equation \((x-a)^2 + (y-b)^2 = r^2\),

• \(r\) represents the radius.

To find the radius in our equation \((x+8)^2 + (y-4)^2 = 4\), you first recognize that the right side, 4, equates to \(r^2\). Thus,

• \(r^2 = 4\)
• \(r = \sqrt{4} = 2\)

The radius is 2 units. This means every point on the circle is exactly 2 units away from the center point \((-8, 4)\). Knowing the radius is essential for accurate circle graphing and calculations involving the circle's area and circumference.

Graphing Circles

Graphing a circle involves positioning it accurately on the coordinate plane based on its center and radius. Here's a simple guide to graphing a circle using the equation \((x+8)^2 + (y-4)^2 = 4\):1. **Locate the Center**: Start by plotting the center of the circle, which we identified as \((-8, 4)\). This is the anchor point from where the circle extends outward.
2. **Draw the Radius**: From the center, use the radius, which is 2. Measure 2 units in all directions (up, down, left, right) to outline the boundary of the circle.
3. **Sketch the Circle**: Use the marked points to draw a smooth, rounded shape connecting them, ensuring all points are equidistant from the center.
4. **Label Important Points**: Clearly label the center and any crucial points for reference. Writing the circle's equation nearby helps identify what you've graphed.Graphing a circle with these steps highlights how the center and radius determine its precise location and size on the graph, forming the basis for understanding more complex graphical transformations.