Problem 44 - 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. $$(x-2)^{2}+(y-3)^{2}=16$$

Step-by-Step Solution

Answer

The center of the circle is at \((2,3)\), its radius is 4. The domain is \([-2,6]\) and the range is \([-1,7]\).

Step 1: Identify the Center and Radius

The equation of a circle in Cartesian coordinates is given by \((x-a)^2+(y-b)^2=r^2\), where \((a,b)\) is the center of the circle and \(r\) is the radius. In the given equation \((x-2)^{2}+(y-3)^{2}=16\), it can be observed that the center of the circle is at \((2,3)\) and the radius is \(\sqrt{16}=4\).

Step 2: Graph the Circle

Begin by marking the center of the circle at point \((2,3)\). Then, use the radius to draw the circle. A point of the circle is obtained by moving 4 units (the radius length) from the center in all directions.

Step 3: Identify the Domain and Range

The domain (all possible x values) of the function is \([2-4,2+4]=[ -2,6]\) and the range (all possible y values) is \([3-4,3+4]=[-1,7]\).

Key Concepts

Center of a Circle

The center of a circle is a crucial part of understanding its geometry. It acts like the "heart" of the circle, determining its exact position on the coordinate plane. In the standard circle equation \((x-a)^2 + (y-b)^2 = r^2\), the center is represented by the coordinates \((a, b)\). In this case, the center is \((2, 3)\). This means the circle is rooted at 2 units along the x-axis and 3 units along the y-axis.

• Keep in mind: the center is the midpoint that the entire circle revolves around.
• To find the center, look at the values next to \(x\) and \(y\), and change their signs.

A circle is perfectly symmetrical around its center. Whether you're graphing, calculating or visualizing, understanding the center helps you pin the circle's position precisely on a graph.

Radius of a Circle

The radius of a circle is like its "arm," reaching from the center to any point on the circle's edge. In our equation \((x-2)^2 + (y-3)^2 = 16\), the term on the right side \(16\) is actually \(r^2\), where \(r\) is the radius. Solving for \(r\), we find that \(r = \sqrt{16} = 4\).

• The radius is always a positive number, representing distance.
• It's constant for a given circle, meaning every point on the circle is 4 units away from the center \((2, 3)\).

Having a fixed radius means you can easily draw the circle by measuring equal distance in all directions from the center. It's like the circle's way of maintaining shape and size.

Domain and Range

Domain and range help us understand the horizontal and vertical reach of the circle on a graph. The domain refers to all possible x-values the circle covers, while the range pertains to the y-values.

For the equation \((x-2)^2 + (y-3)^2 = 16\), the circle's center is \((2, 3)\) and its radius is 4. To find the domain:

• Start from the center's x-coordinate, 2.
• Move 4 units left and right (the length of the radius).
• This gives us the domain: \([-2, 6]\).

For the range:

• Begin with the center’s y-coordinate, 3.
• Move 4 units up and down.
• This results in the range: \([-1, 7]\).

Both domain and range are inclusive, capturing the circle's full spread. They provide a neat way to visualize where the circle lies on the graph, ensuring you grasp both the width and height dimensions.