Problem 74 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the equation of a circle with the given center and radius. Graph the circle. Center \((-3,0),\) radius 2

Step-by-Step Solution

Answer

Equation: \((x + 3)^2 + y^2 = 4\). Center: \((-3,0)\), Radius: 2.

Step 1: Understanding the Problem

To find the center-radius form of a circle's equation, we need to understand that it takes the form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Substitute Center Coordinates

Substitute the center of the circle \((-3, 0)\) into the equation. This gives us \((x - (-3))^2 + (y - 0)^2 = r^2\) or \((x + 3)^2 + y^2 = r^2\).

Step 3: Substitute the Radius

Now substitute the radius \(r = 2\) into the equation. This gives us \((x + 3)^2 + y^2 = 2^2\) or \((x + 3)^2 + y^2 = 4\).

Step 4: Final Equation Review

The equation \((x + 3)^2 + y^2 = 4\) represents a circle with center \((-3,0)\) and radius 2. The equation is now in the center-radius form.

Step 5: Graphing the Circle

To graph the circle, plot the center at \((-3, 0)\) on the coordinate plane. Then, draw a circle with a radius of 2 units around this center. Ensure the circle touches points at \(-1, 0\), \(-5, 0\), \(-3, 2\), and \(-3, -2\) for accuracy.

Key Concepts

Center-Radius Form

To express the equation of a circle, we use what's known as the center-radius form. This mathematical representation makes it easy to identify key characteristics of the circle: its center and radius. The standard formula is \((x-h)^2 + (y-k)^2 = r^2\). Here's what each part represents:

• \((h, k)\) is the center of the circle. It tells you where the center of the circle is located on the coordinate plane.
• \(r\) stands for the radius, the distance from the center to any point on the circle.

To convert a general circle equation into center-radius form, identify the center and radius and substitute these values into the formula. For the given problem:

• The center is at \((-3, 0)\), so \(h = -3\) and \(k = 0\).
• The radius is 2, so \(r = 2\).

By substituting these values, we obtain the equation \((x + 3)^2 + (y - 0)^2 = 4\). This represents our circle in center-radius form.

Circle Graphing

Graphing a circle built from the center-radius form equation is straightforward. Begin by identifying the circle's center on the coordinate plane from the values \((h, k)\). Plot this point first, which serves as the anchor point for the rest of the circle.
Next, use the radius \(r\). In our problem, that is 2 units. Imagine or mark the points that are 2 units away from the center in all directions. These points include locations directly up, down, left, and right from the center:

• \((-3 + 2, 0)\) or \((-1, 0)\)
• \((-3 - 2, 0)\) or \((-5, 0)\)
• \((-3, 0 + 2)\) or \((-3, 2)\)
• \((-3, 0 - 2)\) or \((-3, -2)\)

Ensure a smooth curve connects these edge points, creating the circle’s outline. A compass or a round object can assist in sketching an accurate circle or use graphing software for precision.

Coordinate Geometry

Coordinate Geometry, or Cartesian Geometry, allows the description of geometric shapes like circles using coordinates on a plane. Here, each point is defined by an \(x\) and \(y\) coordinate, simplifying visual and analytical comprehension of geometrical figures.
In the context of circles:

• The circle's center provides a fixed reference point, \((h, k)\), using coordinate values.
• The radius \(r\) is a fixed distance that defines every point along the circle in relation to the center.

The coordinate plane itself is a grid divided into quadrants by the \(x\) and \(y\) axes. This framework enables easy plotting and measurement.
Through coordinate geometry, not only can we plot the circle visually, but we can also easily perform calculations and transformations, such as translating the circle to a new center point or altering its radius, while maintaining its mathematical description clear and precise.