Problem 101 - College Algebra

Question

Will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center \((1,-1)\) and radius 1.

Step-by-Step Solution

Answer

In the Cartesian coordinate system, a circle can be graphed centered at (1, -1) with a radius of 1. Clearly, this would pass through these four points - (1, 0), (2, -1), (1, -2), and (0, -1).

Step 1: Formulate equation of the circle

Apply the equation of the circle using the given parameters to get the equation. So, \( (x-1)^2 + (y+1)^2 = 1^2 \), which results in \( (x-1)^2 + (y+1)^2 = 1 \).

Step 2: Plot the center of the circle

On a Cartesian coordinate system, plot the center of the circle at (1, -1).

Step 3: Graph the circle

From the center, draw a circle of radius 1. Due to the symmetry of the circle, it will pass through the points (1, 0), (2, -1), (1, -2), and (0, -1).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by two perpendicular number lines: the x-axis and the y-axis. The intersection of these axes is called the origin, located at point (0,0).
This system is used to locate points on a plane by using ordered pairs \(x, y\). Each pair represents the point's distance from the origin along the x-axis and y-axis respectively.

• The x-coordinate indicates the horizontal position.
• The y-coordinate indicates the vertical position.

When graphing shapes like circles, the rectangular coordinate system allows precise placement and measurement using these pairs.

Equation of a Circle

The equation of a circle in the rectangular coordinate system is a mathematical expression that describes all the points on the plane that are equidistant from a fixed central point, known as the center of the circle.
This is typically given by the formula: \( (x-h)^2 + (y-k)^2 = r^2 \).

• \(h, k\) represent the coordinates of the circle's center.
• \(r\) is the radius of the circle, or distance from the center to any point on the circle.

So for a circle with the center at \( (1, -1) \) and a radius of \(1\), its equation becomes \( (x-1)^2 + (y+1)^2 = 1 \). This formula helps graph circles accurately by clearly defining all necessary points to plot the shape on a coordinate plane.

Cartesian Coordinates

The term Cartesian coordinates refers to the use of two numbers to specify a location on a plane. This system of coordinates is named after René Descartes, who introduced it in the 17th century.
Each point is given by an ordered pair \(x, y\), where \(x\) signifies the horizontal position relative to the origin, and \(y\) shows the vertical position.

• Points above the x-axis have positive y-coordinates, while points below have negative y-coordinates.
• Points to the right of the y-axis have positive x-coordinates, and those to the left have negative x-coordinates.

In the context of graphing circles, understanding Cartesian coordinates is crucial as it allows the precise plotting of a circle's center and the tracing of its perimeter by using a consistent system for defining point locations.