Problem 36 - Algebra 2

Question

Graph each circle so that the center is at the origin. Then write the equation. radius \(\frac{1}{2}\)

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = \frac{1}{4}\). You would graph this circle by drawing a circle with a radius of \(\frac{1}{2}\) units around the origin (0,0).

Step 1: Understand the equation of a circle

The standard equation of a circle with radius \(r\) centered at the origin (0,0) is \(x^2 + y^2 = r^2\). In this case, the radius is known, so it can be plugged into the equation.

Step 2: Insert the given radius into the equation

The given radius is \(\frac{1}{2}\). Plugging this value into the equation \(x^2 + y^2 = r^2\) results in the equation \(x^2 + y^2 = \left(\frac{1}{2}\right)^2\). Squaring \(\frac{1}{2}\) yields \(\frac{1}{4}\), so our equation becomes \(x^2 + y^2 = \frac{1}{4}\).

Step 3: Graph the circle based on the equation

To graph this on a Cartesian plane, remember that this is a circle centered at the origin (0,0). The circle extends \(\frac{1}{2}\) units in all directions from the origin. Draw a circle around the origin with this radius. The points on this graph fulfill the equation \(x^2 + y^2 = \frac{1}{4}\).

Key Concepts

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its edge. It is a key component of the equation of a circle. In mathematical terms, this is represented by the symbol \(r\). If you know the radius, you can easily define the size of a circle. For example, a radius of \(\frac{1}{2}\) can be visualized as a very short line extending from the center of the circle. This not only determines the diameter, which is twice the radius, but also the area of the circle.

• Diameter: The full width of the circle, measured through the center. Calculated as \(2r\).
• Area: Using the formula \(\pi r^2\), you can determine how much space the circle covers.

Remember, when the radius changes, both the diameter and the area change accordingly.

Graphing Circles

Graphing a circle involves drawing it onto a coordinate system or graph. The circle is defined by its radius and its center point, which guides its placement on the grid. To visualize a circle on a graph, you use its equation. For a circle with center at the origin

• The equation format is \(x^2 + y^2 = r^2\).
• The circle extends \(r\) units in all directions from the center point.
• In our case, where \(r = \frac{1}{2}\), that means the circle stretches \(\frac{1}{2}\) units right, left, up, and down.

Follow these steps on graph paper to accurately represent the circle:1. Locate the origin, which is your circle's center.2. Using your compass or a steady hand, mark points \(\frac{1}{2}\) units from the origin in all directions.3. Connect these points smoothly to form the circle.

Cartesian Plane

The Cartesian plane is a two-dimensional graph or grid composed of two perpendicular lines that intersect at a point called the origin. This plane is used to locate points defined by pairs of numbers, known as coordinates, which describe positions along the horizontal (x-axis) and vertical (y-axis) lines. These axes help to position shapes like circles accurately on the graph.

• The x-axis runs horizontally.
• The y-axis runs vertically.
• The intersection of the x-axis and y-axis is known as the origin, represented by (0,0).

It is a crucial tool in mathematics, especially when analyzing geometric shapes such as circles. When you plot a circle centered at the origin, as in our example, you rely heavily on the Cartesian plane.

Origin

The origin is a central point on the Cartesian plane where the x-axis and y-axis intersect. It holds a coordinate of \((0, 0)\). Serving as the starting point for graphing, the origin is often the center for symmetrical shapes like circles. When graphing circles specifically, choosing the origin as the center simplifies calculations and ensures that any movement along the grid corresponds directly to changes in x and y values.

• When the center of a circle is at the origin, the circle's equation simplifies to \(x^2 + y^2 = r^2\).
• All quadrants have equal access to the circle as it radiates evenly from this point.
• Graphically, it is the perfect symmetry point for tests and transformations in geometry.

Having the origin as the reference point helps to easily visualize the circle's interaction with the axes and simplifies many calculations.