Problem 66 - Algebra 2

Question

Graph each equation. Describe each graph and its lines of symmetry. Give the domain and range for each graph. $$ x^{2}+y^{2}=4 $$

Step-by-Step Solution

Answer

The graph of the equation \(x^{2}+y^{2}=4\) is a circle centered at the origin (0,0) with a radius of 2. The lines of symmetry are the x-axis and y-axis. The domain and range are both from -2 to +2.

Step 1: Identify the given equation

The given equation is \(x^{2}+y^{2}=4\). This is the equation of a circle. The standard form of a circle equation is \(x^{2}+y^{2}=r^{2}\), where (0,0) is the center of the circle and r is the radius. Hence, in this equation, r equals to the square root of 4, which is 2.

Step 2: Graph the equation

Draw a circle with the center at point (0,0) and a radius equal to 2 units. Plot this on a graph.

Step 3: Identify the lines of symmetry

The lines of symmetry of a circle are its diameters. In this case, since the circle is centered at the origin, the x-axis and y-axis act as the lines of symmetry, dividing the circle into four equal parts.

Step 4: Identify the domain and range

The domain (possible values of x) and range (possible values of y) are determined by the radius of the circle. In this case, as the radius is 2, the domain and range are both from -2 to 2, or mathematically represented as \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\).

Key Concepts

Graphing the Equation of a Circle

Graphing a circle starts with understanding its equation. The given equation is \(x^2 + y^2 = 4\), which is in the standard form \(x^2 + y^2 = r^2\). This tells us it’s a circle, centered at the origin \((0,0)\) with a radius squared equal to 4.
To find the radius, take the square root of 4, which is 2. Now, plot the center point on the coordinate plane. From the center \((0,0)\), measure 2 units in all directions: up, down, left, and right.
Connect these points smoothly to draw the circle. Ensure your circle is round, keeping each point equidistant from the center. This symmetry is key to accurately representing the circle in a graph.

Lines of Symmetry in a Circle

A circle is a very symmetric shape. Its lines of symmetry aren’t limited to just a few. In fact, any line that passes through the center creates symmetry.
For the circle given by \(x^2 + y^2 = 4\), the most notable lines of symmetry are the x-axis and the y-axis. These lines divide the circle into perfect halves, creating four equal segments.
Since the center is at the origin \((0,0)\), every diameter of the circle is also a line of symmetry. This property makes circles unique compared to other geometric shapes, as they have infinite lines of symmetry all passing through the center.

Understanding Domain and Range for a Circle

In mathematics, domain refers to all possible x-values, while range refers to all possible y-values of a function or shape. When graphing a circle like \(x^2 + y^2 = 4\), these values are determined by the radius.
Because the center is at \((0,0)\) and the radius is 2, the circle stretches 2 units outward in all directions. Thus, the domain, or the x-values, ranges from -2 to 2. Similarly, the range, or the y-values, also spans from -2 to 2.
Expressed mathematically, this is written as \(-2 \leq x \leq 2\) for the domain, and \(-2 \leq y \leq 2\) for the range. This concise representation helps in understanding the extent of the circle on a graph.