Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 23-34: Sketch the graph of the circle or semicircle. $$ (x-4)^{2}+(y+2)^{2}=4 $$

Step-by-Step Solution

Answer

The circle has center (4, -2) and radius 2, sketch it based on these parameters.

Step 1: Identify the Components of the Circle Equation

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

Step 2: Determine the Center and Radius

From the equation \((x-4)^2 + (y+2)^2 = 4\), we can see that the center of the circle is \((h, k) = (4, -2)\) and the radius \(r\) is \(\sqrt{4} = 2\).

Step 3: Plot the Center of the Circle

Plot the point \((4, -2)\) on the Cartesian plane. This is the center of the circle.

Step 4: Sketch the Circle

Using the center \((4, -2)\) and radius \(2\), draw a circle by marking points 2 units away from the center in all directions: right, left, up, and down from \((4, -2)\). Connect these points smoothly to form the circle.

Key Concepts

Graphing Circles

Graphing a circle on a coordinate plane might seem challenging, but it becomes straightforward once you understand the equation and its components. The main goal is to visualize how a circle is positioned and defined by its center and radius.
Here's how you can graph a circle:

• First, identify the circle's center coordinates from the equation. This tells you where the circle is situated on the plane.
• Next, determine the radius, which shows how large the circle is.
• Plot the center point on the graph.
• Using the radius, measure out from the center in all directions to mark points that are equidistant from the center.
• Connect these points to sketch the circle. A perfect circle won't necessarily look perfect if drawn by hand, but aiming for an even distance around the center is key.

Graphing a circle by these steps allows you to accurately represent its equation visually.

Standard Form of Circle Equation

The standard form of a circle equation is an essential concept for understanding and graphing circles. This form is expressed as \((x-h)^2 + (y-k)^2 = r^2\).

• \((h, k)\) represents the center of the circle. By substituting in these values, you can identify where the circle sits on the coordinate grid.
• \(r\) is the radius. In the equation, \(r^2\) is what you see on the right side. Always remember to take the square root to find the actual radius when plotting.

The convenience of the standard form is that it clearly outlines both the center and radius, which simplifies the task of sketching the circle. When the equation is presented in this form, interpreting the characteristics of the circle becomes much more manageable.

Radius and Center of a Circle

Understanding the radius and center of a circle is key to both interpreting and graphing it efficiently.
The center of the circle is the fixed point that defines its position on the graph. In the equation \((x-h)^2 + (y-k)^2 = r^2\), the values \((h, k)\) indicate the center coordinates.
The radius is the distance between the center and any point on the circle. It is a constant for any given circle, and in the equation, you find it as \(r = \sqrt{r^2}\).

Key Points About Radius and Center:

• The center \((h, k)\) ensures the circle's symmetry, meaning the circle looks the same in all directions from this point.
• The radius \(r\) dictates the size of the circle. A larger radius results in a bigger circle.
• To graph a circle correctly, accurately locate its center and consistently measure outwards using the radius. This method will help you draw an even and proportional circle.

By mastering these concepts, you make working with circle equations and graphs a more intuitive and rewarding experience.