Problem 53 - 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. $$x^{2}+y^{2}=1$$

Step-by-Step Solution

Answer

The graph is a circle with center at (0, 0) and radius 1.

Step 1: Recognize the Equation Type

The equation given is \(x^2 + y^2 = 1\). This is a standard form for the equation of a circle.

Step 2: Identify Circle Parameters

The general form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius. For \(x^2 + y^2 = 1\), it can be rewritten as \((x - 0)^2 + (y - 0)^2 = 1^2\), indicating the center is \((0, 0)\) and the radius is \(1\).

Step 3: Sketch the Circle

Since the center of the circle is at \((0, 0)\) and the radius is \(1\), plot the center on the coordinate plane. From the center, draw a circle with a radius of \(1\) unit, touching the points \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\) on the coordinate axes.

Key Concepts

Understanding the Circle Equation

The equation of a circle is a mathematical representation that helps us understand and visualize the shape of a circle on the coordinate plane. A circle in a coordinate plane is defined by an equation of the form \((x-h)^2 + (y-k)^2 = r^2\). This equation tells us several important things:

• The terms \((x-h)^2\) and \((y-k)^2\) represent the movement from the center of the circle to any point \((x, y)\) on the circle.
• The number \(r^2\) on the right-hand side represents the square of the radius, which is the distance from the center to any point on the circle.

Thus, the circle equation is crucial as it defines both the position of the circle's center and its radius. Recognizing this equation is the first step in graphing a circle.

Center and Radius of a Circle

When dealing with the circle equation \((x-h)^2 + (y-k)^2 = r^2\), the parameters \(h\), \(k\), and \(r\) have specific meanings:

• \((h, k)\) is called the center of the circle. It is the point from which every point on the circle is equidistant.
• \(r\) is the radius, indicating how far from the center the circle extends.

For the equation \(x^2 + y^2 = 1\), it simplifies to \((x-0)^2 + (y-0)^2 = 1^2\), giving a center of \((0, 0)\) and a radius of \(1\). Understanding these elements is crucial for drawing the circle accurately and gauging its size in the coordinate space.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves, such as circles. It is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is represented by a pair of numbers called coordinates. For a circle, the center's coordinates will tell us exactly where to place it on this plane.
To graph a circle such as \(x^2 + y^2 = 1\), we plot the center, which is \((0, 0)\), at the origin of the plane. Once the center is established, you can draw points at a distance equal to the radius in all directions to form the circle shape.

Sketching a Circle on the Plane

Graphing a circle accurately is a simple process once you identify its center and radius. Here's how you can do it step by step:

• First, mark the center of the circle on the coordinate plane. In our example, the center is at \((0, 0)\).
• Next, use the radius to determine the boundary of the circle. For \(x^2 + y^2 = 1\), the radius is \(1\).
• From the center, measure one unit in all cardinal directions (up, down, left, and right) and make marks at these points.
• Finally, connect these points with a smooth, curved line to form the circle.

With these steps, you will have an accurate sketch of the circle on the plane. Remember, practice makes perfect, and using this method will become second nature as you work with more examples.