Problem 50 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. Use the graph to identify the relation's domain and range. $$x^{2}+(y-2)^{2}=4$$

Step-by-Step Solution

Answer

The center of the circle is at (0,2) and the radius is 2. The domain of the graph is \([-2,2]\) and the range is \([0,4]\).

Step 1: Express in Standard Form

The given equation is already in the standard form of a circle's equation, which is \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h,k)\) are the coordinates of the center and \(r\) is the radius. Therefore, no rewriting is necessary.

Step 2: Identify the Center and Radius

Comparing the given equation, \(x^{2}+(y-2)^{2}=4\), to the standard form, we can identify the center as \((h,k)\)=(0,2) and the radius, \(r\), as \(\sqrt{4}\)=2.

Step 3: Sketch the Graph

Plot the center of the circle at point (0,2) then sketch the circle with a radius of 2.

Step 4: Identify Domain and Range

The domain and range can be identified from the graph. The domain is all the x-values that the graph covers. Looking at the graph, we can tell that the graph covers x-values from -2 to 2. Similarly, the range is all the y-values that the graph covers. Since the center is at y=2 and the radius is 2, the graph on the y-axis ranges from 0 to 4. Hence, the domain is \([-2, 2]\) and the range is \([0, 4]\).

Key Concepts

Domain and Range

The domain and range of a circle describe the set of all possible coordinates that the circle covers on a graph. These concepts are crucial in understanding the limits of a circle's reach within the coordinate plane. The domain relates to the x-values, while the range pertains to the y-values.

In the equation given, \(x^{2} + (y - 2)^{2} = 4\), the center is at \((0, 2)\) and has a radius of 2. This tells us that the circle extends 2 units in every direction from the center. Therefore:

• The domain is the set of x-values from -2 to 2, since the circle extends 2 units left and right from the x-coordinate of the center, which is 0.
• The range is the set of y-values from 0 to 4, as the circle extends 2 units above and below the y-coordinate of the center, which is 2.

By plotting these limits on a graph, it's easier to visualize the entire coverage of the circle and confirms that the domain is \([-2, 2]\), and the range is \([0, 4]\).

Center and Radius

Understanding the center and radius of a circle is fundamental when working with circle equations. These two components help determine the circle's position and size on a graph.

For a circle's equation in standard form \((x-h)^2 + (y-k)^2 = r^2\), the center is described by the coordinates \((h, k)\), and the radius \(r\) is the distance from the center to any point on the circumference.

• In our equation, \(x^2 + (y-2)^2 = 4\), we directly compare this to \((x-h)^2 + (y-k)^2 = r^2\) to find the center at \((0, 2)\).
• The number 4 is the radius squared, so the actual radius \(r\) is the square root of that number, which is 2.

Knowing the center and radius allows us to easily plot the circle on a graph and analyze its domain and range. It also provides insight into how shifting the center or changing the radius will affect the original circle.

Standard Form of a Circle

The standard form of a circle's equation is crucial for quickly identifying a circle’s geometric properties. This form facilitates easy identification of the circle’s center and radius.

A circle's equation is generally expressed as:\[(x-h)^2 + (y-k)^2 = r^2\]

• This equation ensures that any point \((x, y)\) on the circle maintains a constant distance, or radius, \(r\), from the circle's center \((h, k)\).
• Here, \(h\) and \(k\) are the horizontal and vertical shifts from the origin to reach the center, respectively.

In our example equation, \(x^2 + (y - 2)^2 = 4\), it is evident that:

• The center is at \((0, 2)\)
• The radius is determined by taking the square root of the right-hand side, which is 2, since \(r^2 = 4\).

The simplicity of the standard form lies in its ability to provide direct information about a circle’s structure, allowing us to analyze and graph it seamlessly.