Problem 15 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$x^{2}+(y-1)^{2}=25$$

Step-by-Step Solution

Answer

The center of the circle is \((0, 1)\), and the radius is \(5\).

Step 1: Identify the center and radius

Comparing the given equation \(x^{2}+(y-1)^{2}=25\) with the standard equation \((x - a)^{2} + (y - b)^{2} = r^{2}\), we can see that: - There is no subtraction in the \(x^{2}\) term, which means \(a = 0\). - In the \((y - 1)^{2}\) term, we can see that \(b = 1\). - The given equation equals 25, so \(r^{2} = 25\). To find the radius, we take the square root: \(r = \pm \sqrt{25} = \pm 5\). Since the radius is always positive, \(r = 5\). Now we have the center of the circle at \((a, b) = (0, 1)\) and the radius is \(r = 5\).

Step 2: Sketch the graph

To sketch the graph of the circle, follow these steps: 1. Plot the center of the circle at point \((0, 1)\). 2. With the center as the reference, draw a circle with a radius of 5 units in all directions. Your graph should now display the circle with center at \((0,1)\) and radius \(5\).

Key Concepts

Center of a Circle

In geometry, the center of a circle is a crucial point within the circle, equidistant from all points along the circle's boundary. The standard circle equation is \((x - a)^2 + (y - b)^2 = r^2\), where \((a, b)\) represents the center.
In this context, the term \((x-a)\) shifts the circle horizontally, while \((y-b)\) adjusts it vertically. When no subtraction occurs, as seen in \(x^2\), it implies a shift of zero, indicating the center is at the origin \((0, y)\).
For the exercise’s equation \(x^2 + (y-1)^2 = 25\), the center comes out to be \((0, 1)\). Here:

• The \(a = 0\) denotes the x-coordinate of the center.


• The \(b = 1\) shows the y-coordinate of the center.

Knowing how to determine the center helps in the placement of the circle on a graph, ensuring precision when drawing or interpreting circles.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle's edge. It is a vital characteristic that defines the size of the circle. In our standard circle equation, \((x - a)^2 + (y - b)^2 = r^2\), the \(r\) is the radius.
In the example equation \(x^2 + (y-1)^2 = 25\), the side of the equation that equals 25 represents \(r^2\). To find the radius, you perform the square root operation on 25.

• Performing \(\sqrt{25}\) gives \(r = 5\).


• It's essential to remember that radius values are always positive, hence we use \(r = 5\) rather than \(-5\).

Understanding the radius is fundamental when it comes to defining how large a circle is on a graph and understanding its scale.

Graphing Circles

Graphing a circle involves mapping its center and exploiting its radius to ensure balance and accuracy when sketching or using graphing technologies. Starting with the fundamental understanding of the circle’s equation helps ensure you graph it correctly.
The first step in graphing is:

• Identify the center \((a, b)\). For the given problem, the center is \((0, 1)\).


• Then locate this point on the coordinate plane as your starting position.

Next, use the radius:

• From your center, measure the radius outwards in all directions — up, down, left, and right.


• For this exercise, the circle’s radius is 5 units, so you would mark points that are 5 units away from \((0, 1)\) in every direction.

This method ensures a perfect circle that can be traced through these points. Graphing is not just about the practice but also about visually understanding how the equation translates into a shape on a grid.