Problem 55 - Algebra 2

Question

The graph of the equation \(x^{2}+y^{2}=121\) is a circle. Describe the graph and its lines of symmetry. Find the domain and the range.

Step-by-Step Solution

Answer

The graph is a circle centered at the origin with a radius of 11. The lines of symmetry are the x and y axes. The domain and range are both \(-11 \leq x, y \leq 11\).

Step 1: Identify the center and radius of the circle

The standard equation of a circle is given by \(x^{2}+y^{2}=r^{2}\), where (0, 0) is the center of the circle and \(r\) is the radius. In this case, \(x^{2}+y^{2}=121\), therefore \(r = \sqrt{121} = 11\). Thus, the circle is centered at the origin, (0, 0), and has a radius of 11.

Step 2: Describe the lines of symmetry

The lines of symmetry of a circle are the vertical and horizontal lines passing through its center. Since the circle is centered at the origin, the x-axis and the y-axis are its lines of symmetry.

Step 3: Identify the domain and range

The domain and range of a function or a relation represent the set of possible x-values (domain) and y-values (range) respectively. In the case of a circle, the domain and range are defined based on its radius. Because the circle's radius is 11 and it is centered at the origin, any x-value or y-value is valid as long as it is no more than 11 units away from the origin. Therefore, the domain is \(-11 \leq x \leq 11\) and the range is \(-11 \leq y \leq 11\).

Key Concepts

Lines of Symmetry

In geometry, lines of symmetry refer to imaginary lines where you can fold a shape, and both halves will match exactly. For a circle, which is a perfectly symmetrical shape, lines of symmetry run through its center. This means that any line that traces through the circle's central point is a line of symmetry; however, we most commonly recognize the x-axis and y-axis. These axes equally divide the circle into two mirror-image halves.

Since the problem involves a circle centered at the origin (0,0), both the x-axis and y-axis are its quiet lines of symmetry.

• The x-axis creates a horizontal mirror line.
• The y-axis creates a vertical mirror line.

This inherent symmetry is special to circles because no matter how you rotate or look at a circle from its central point, it will always appear the same. This property makes circles unique and allows for easy symmetry identification.

Domain and Range

The terms domain and range are essential in understanding the extents of a circle in a coordinate system. The domain refers to the set of all possible x-values that the circle can reach, while the range refers to all possible y-values.

For the circle described by the equation \(x^2 + y^2 = 121\), its center is at the origin (0, 0) and the radius is 11. This tells us that the circle stretches 11 units left, right, up, and down from the center. Thus, the range of values for \(x\) is from -11 to 11, inclusive. Similarly, the range of values for \(y\) is also from -11 to 11. Therefore, the domain of the circle is

• \(-11 \leq x \leq 11\)

and the range is

• \(-11 \leq y \leq 11\)

Understanding the domain and range concerning a circle could be visualized as drawing a large box that neatly snug-fits the circle, touching it at the furthest extents in each direction.

Radius of a Circle

The radius of a circle is a fundamental concept and is simply the distance from the center of the circle to any point on its edge. It is always consistent, meaning that no matter which direction you measure, the radius will be the same.

For the circle in this exercise, the equation is given as \(x^2 + y^2 = 121\). This equation is in the standard form \(x^2 + y^2 = r^2\), where \(r\) is the radius. Simply taking the square root of 121 gives us the radius, which is 11.

Summing this up:

• A radius defines the size of the circle.
• The radius for this particular circle is 11.
• The center from which we measure the radius is the origin (0, 0) in this case.

The concept of the radius is foundational in understanding not just the size of a circle, but how the circle will sit within any given space on a graph or in real-world applications.