Problem 104 - Precalculus Essentials

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

The circle with center (1,-1) and radius 1 can be graphed by first plotting the center. Then, draw a circle around this center by measuring 1 unit in all directions from this point.

Step 1: Understanding Coordinates and Radius

The center of the circle is given by the point (1,-1) and the radius is 1. This identifies the location of the circle on the coordinate system and its size.

Step 2: Form the Equation of Circle

It is known that the equation of a circle in a plane with center at point \(h, k\) and with radius \(r\) is given by \((x-h)^2 + (y-k)^2 = r^2\). Replace \(h\), \(k\), and \(r\) with the given values to form the equation of the circle which is \((x-1)^2 + (y+1)^2 = 1^2\).

Step 3: Graph the Circle

Plot the center of the circle at point (1,-1). From this point, draw a circle with a radius of 1 unit. This can be done by measuring 1 unit in all directions (left, right, up, down) from the center. Connect these points with a round curve to complete the graph of the circle.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a framework used to graphically represent mathematical figures and equations. This system consists of two perpendicular lines called axes:

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.

These axes divide the plane into four quadrants, which are numbered I to IV in a counter-clockwise direction. Each point in this system is identified by a pair of numerical coordinates \((x, y)\). The x-value determines the position along the horizontal axis, while the y-value determines the position along the vertical axis. This system enables us to plot graphs and visualize functions like circles and lines by providing a clear, structured way to navigate the plane. Remember:

• Positive x-values are to the right of the y-axis.
• Negative x-values are to the left of the y-axis.
• Positive y-values are above the x-axis.
• Negative y-values are below the x-axis.

Graphing Circles

Graphing a circle involves plotting its center and ensuring all points are equidistant from it. The equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h, k\) is the center of the circle, and \r\ is the radius. The radius determines the distance from the center to any point on the circle's boundary.

To graph a circle, start by:

• Plotting the center using its coordinates (h, k).
• Using the radius to measure equal distances from the center in all directions.

Given a circle with center \((1, -1)\) and radius \1\, its equation becomes \((x-1)^2 + (y+1)^2 = 1^2\). This means that from the point \((1, -1)\), you measure one unit away in every direction and connect these outer points to form the circle's outline.

Center and Radius

The concepts of a circle's center and radius are crucial in understanding its equation and graph.

• The center of the circle is the point \(h, k\), which signifies the circle’s midpoint in the coordinate plane. This point does not belong to the circle's boundary but is pivotal in defining its position.
• The radius \(r\) is the distance from the center to any point on the circle's edge. It is a constant value that represents how large or small the circle is.

If you think of the circle as a wheel, the center is its hub, and the radius measures from the hub to the outer rim. Knowing the center and radius allows one accurately to write the circle's equation and graph it on a coordinate plane with precision. For the given exercise, by using a center of \((1, -1)\) and radius \1\, you can easily plot and visualize the size and location of your circle.