Problem 99 - A Graphical Approach to College Algebra

Question

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x-2)^{2}+(y+2)^{2}=4$$

Step-by-Step Solution

Answer

Circle centered at (2, -2) with radius 2.

Step 1: Identify the Type of Equation

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

Step 2: Determine the Center of the Circle

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

Step 3: Determine the Radius of the Circle

In the equation \((x-2)^2 + (y+2)^2 = 4\), the right-hand side (\(4\)) represents \(r^2\). By taking the square root, \(r = \sqrt{4} = 2\). So, the radius is 2.

Step 4: Sketch the Circle on the Coordinate Plane

To sketch the circle, plot the center at \((2, -2)\). From the center, measure 2 units in all directions (up, down, left, and right) to mark the boundary points of the circle. Connect these boundary points smoothly to complete the sketch of the circle.

Key Concepts

Equation of a Circle

Understanding the equation of a circle is crucial for graphing it accurately. The general form of a circle's equation is \[ (x-h)^2 + (y-k)^2 = r^2 \] Here,

• \((h, k)\) represents the center of the circle.
• \(r\) represents the radius, and \(r^2\) will always be a positive number.

In our problem, the equation given is \((x-2)^2 + (y+2)^2 = 4\). This fits perfectly into the standard form. To identify the constants for this problem:

• \(h = 2\)
• \(k = -2\)
• \(r^2 = 4\), so \(r = 2\)

By recognizing this structure, you can easily determine both the center and radius of the circle.

Center and Radius

The center and radius are key elements when dealing with circles in geometry.For the equation \((x-2)^2 + (y+2)^2 = 4\), identifying the center involves comparing it with our standard circle equation form, \((x-h)^2 + (y-k)^2 = r^2\). From this equation:

• The center \((h, k)\) can be directly read as \((2, -2)\).

The radius \(r\) refers to the distance from the center to any point on the circle:

• With \(r = \sqrt{r^2} = \sqrt{4} = 2\).

Using these values, every point on this circle will be exactly 2 units away from its center, \((2, -2)\), ensuring symmetry and balance around the center point.

Coordinate Plane Sketching

Sketching a circle on the coordinate plane is all about careful plotting and measuring. Given our circle's center at \((2, -2)\) and a radius of 2, the process involves:

• Start by plotting the center at point \((2, -2)\) on the plane.
• Count 2 units outwards from the center in all cardinal directions — up, down, left, and right.
• These points will help form the boundary of the circle.

Next, carefully connect these boundary points in a smooth, rounded fashion to resemble a circle. The precision in each step ensures that the circle is both accurate and visually pleasing.By consistently following these steps, you can graph circles of various sizes and positions on the coordinate plane, confidently and correctly.