Problem 10 - Algebra 2

Question

Graph each equation. Identify the conic section and describe the graph and its lines of symmetry. Then find the domain and range. $$ x^{2}+y^{2}-4=0 $$

Step-by-Step Solution

Answer

The equation represents a circle with radius 2, centered at the origin. It has infinite lines of symmetry, especially along \(y=x\), \(y=-x\), and the axes. The domain and range are both \([-2, 2]\).

Step 1: Identify the conic section

The given equation is in the form of \(x^{2}+y^{2}=r^{2}\), which represents a circle with center at the origin (0,0) and radius \(r\). In our case, \(x^{2}+y^{2}=4\), therefore the radius \(r\) is equal to the square root of 4, which is 2.

Step 2: Graph the circle

Plot the center of the circle at the origin (0,0). Then, mark a point to the right of the center at a distance equal to the radius, 2 units. Similarly, mark a point to the left, 2 units away. Repeat these steps for points above and below the center. Connect these points with a smooth curve to form a circle.

Step 3: Find lines of symmetry

A circle has infinite lines of symmetry, the most notable ones being the line \(y=x\), the line \(y=-x\), and the lines \(y=0\) and \(x=0\), which are the x-axis and the y-axis respectively.

Step 4: Determine the domain and range

The domain of a function is the set of all possible x-values and the range is the set of all possible y-values. For this circle, the x-values (domain) and the y-values (range) both vary from -2 to 2, inclusive. So, the domain is \([-2, 2]\) and the range is \([-2, 2]\).

Key Concepts

Circle

A circle is a simple and perfectly symmetrical shape, often encountered in geometry. It can be described as the set of all points in a plane that are equidistant from a central point called the center. In mathematical terms, a circle is often expressed with the equation \((x-h)^2 + (y-k)^2 = r^2\), where

• \((h, k)\) is the center of the circle,
• \(r\) is the radius, representing the distance from the center to any point on the circle.

For example, the equation \(x^2 + y^2 = 4\) describes a circle centered at the origin with a radius of 2. To visualize this, you start at the origin, move 2 units to any direction, and you will find a point on the circle. This pattern holds true in every direction, forming the complete circle.

Domain and Range

Understanding domain and range is crucial when dealing with functions or graphs.- **Domain** refers to all possible input values (x-values) that the function can take. For a circle such as the one given by \(x^2 + y^2 = 4\), the domain consists of all x-values that lie within and include the circle.- **Range** denotes all possible output values (y-values) derived from the function. Similarly, for this circle, the range includes all y-values touched by the circle.Given the circle with a radius of 2 centered at the origin, both domain and range are influenced by the extent of the circle.Thus, for the circle \(x^2 + y^2 = 4\):

• The domain is \([-2, 2]\).
• The range is \([-2, 2]\).

Both x and y cover the broad span of -2 to 2, encapsulating the entire area where the circle resides.

Lines of Symmetry

Lines of symmetry in geometry refer to lines that divide shapes into two equal and mirrored parts.For circles, an infinite number of symmetry lines exist due to their inherent symmetrical nature. However, some are more prominent:

• The x-axis (line \(y = 0\)) and y-axis (line \(x = 0\)) are the most notable symmetry lines for our circle centered at the origin.
• Besides, the lines \(y = x\) and \(y = -x\), which are diagonal across the origin, serve as additional symmetry lines.

Each line of symmetry splits the circle into two identical halves. This property illustrates why circles are considered perfect symmetrical shapes.