Problem 66 - Intermediate Algebra

Question

Sketch the graph of each equation. If the graph is a parabola, find its vertex. If the graph is a circle, find its center and radius. $$2 x^{2}+2 y^{2}=\frac{1}{2}$$

Step-by-Step Solution

Answer

A circle with center at (0,0) and radius 1/2.

Step 1: Identify the Type of Equation

The given equation is \( 2x^2 + 2y^2 = \frac{1}{2} \). This is an equation involving both \(x^2\) and \(y^2\) with equal coefficients, suggesting that it could represent a circle.

Step 2: Simplify the Equation

Divide every term by 2 to simplify: \( x^2 + y^2 = \frac{1}{4} \). This confirms the equation is of a circle as it takes the form \( x^2 + y^2 = r^2 \).

Step 3: Determine the Center and Radius

The standard form for a circle centered at the origin is \( x^2 + y^2 = r^2 \). Comparing, we see this circle's center is at the origin \((0,0)\) with a radius \( r = \sqrt{\frac{1}{4}} = \frac{1}{2} \).

Step 4: Sketch the Circle

Plot a circle on a coordinate plane with its center at \((0,0)\) and radius \(\frac{1}{2}\). Draw the circle by marking points equidistant from the center at \(\frac{1}{2}\) units away in all directions.

Key Concepts

Graphing Circles

Graphing a circle on the coordinate plane starts with understanding its equation. When you have a circle's equation such as \( x^2 + y^2 = r^2 \), it indicates a circle centered at the origin. The value of \( r^2 \) is the square of the radius. In our example equation, \( x^2 + y^2 = \frac{1}{4} \), this tells us that our circle's radius is \( \frac{1}{2} \).
To graph this circle:

• Mark the center of the circle at the origin \((0,0)\) on your coordinate plane.
• Using a ruler or a compass, draw a boundary of points which lie exactly \(\frac{1}{2}\) units away from the center in all directions.

Since the radius is \( \frac{1}{2} \), this boundary forms your circle. The concept of constant distance from a point (the center) to the curve defines a circle.

Circle Center and Radius

The precise definition of a circle involves two main components: the center and the radius. A circle's center is a fixed point, from which every point on the circle is equidistant.
In a standard circle equation, \( (x - h)^2 + (y - k)^2 = r^2 \), \( (h, k) \) represents the center of the circle, and \( r \) is the radius.
For our equation \( x^2 + y^2 = \frac{1}{4} \):

• The center \((h, k)\) is at \((0,0)\) because it can be rewritten as \((x - 0)^2 + (y - 0)^2 = (\frac{1}{2})^2\).
• The radius \( r \) is \( \frac{1}{2} \) as the equation equates to \( r^2 \).

Understanding the center and radius is essential for sketching and analyzing a circle's position and size. The center specifies where the circle is located on the plane, and the radius indicates the distance from the center to any point on the circle.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, \( (x, y) \). It consists of two perpendicular lines, known as axes: the horizontal axis (x-axis) and the vertical axis (y-axis).
Every point on this plane corresponds to a unique pair of coordinates. For a circle, the position of its center given by \( (h, k) \) is crucial. In our circle example, the center at \((0,0)\) is right at the intersection of the x-axis and y-axis.
To plot our circle on the coordinate plane:

• Identify the center. Since it’s at the origin for this circle, you start there.
• Using the radius \(r\), move \(\frac{1}{2}\) units up, down, left, and right to help sketch the boundary of the circle.

The coordinate plane assists in visualizing how geometric figures relate in space. By finding intersections and distances, it helps in understanding the geometry and location of shapes like circles, thus facilitating problem-solving and comprehension in mathematics.