Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Locate each point on a rectangular coordinate system. Identify the quadrant, if any, in which each point lies. $$(-3,2)$$

Step-by-Step Solution

Answer

The point \\((-3, 2)\\) lies in Quadrant II.

Step 1: Understand the Coordinate System

A rectangular coordinate system is a plane with two number lines: the x-axis (horizontal) and the y-axis (vertical). Each point is represented by a pair of numbers \(x, y\), where \(x\) is the coordinate on the x-axis, and \(y\) is the coordinate on the y-axis.

Step 2: Plot the Point

Given the point \((-3, 2)\), locate \(-3\) on the x-axis and \(+2\) on the y-axis. To plot the point, move to the left 3 units from the origin on the x-axis, and move up 2 units from there on the y-axis. Mark this point.

Step 3: Determine the Quadrant

A coordinate plane is divided into four quadrants: 1. Quadrant I (positive x, positive y)2. Quadrant II (negative x, positive y)3. Quadrant III (negative x, negative y)4. Quadrant IV (positive x, negative y)Since \((-3, 2)\) has a negative x-value and a positive y-value, it lies in Quadrant II.

Key Concepts

Quadrants

In the rectangular coordinate system, the plane is divided into four distinct areas known as quadrants. These quadrants are numbered in a counter-clockwise direction starting from the top right:

• Quadrant I: Here, both the x and y coordinates are positive, which means any point located in this quadrant falls to the right of the y-axis and above the x-axis.
• Quadrant II: In this quadrant, the x coordinate is negative while the y coordinate is positive, placing it to the left of the y-axis and above the x-axis.
• Quadrant III: Involves both negative x and y coordinates, indicating points are to the left of the y-axis and below the x-axis.
• Quadrant IV: Contains points with positive x coordinates but negative y coordinates, thus they are to the right of the y-axis and below the x-axis.

Each quadrant's layout helps in determining the position and sign of coordinates for any given point, like (-3, 2), which lies in Quadrant II because its x is negative and y is positive.

Plotting Points

Plotting points on a rectangular coordinate system is like pinpointing an exact location on a map. Each point is identified by an ordered pair (x, y), known as coordinates. To plot a point:

• Start at the origin, where the x-axis and y-axis intersect at (0, 0).
• Read the x-value first. A negative value means moving left from the origin, while a positive value means moving right.
• Then, interpret the y-value. A positive value requires moving upwards, while a negative value means moving downwards along the y-axis.

For the point (-3, 2), you move left 3 units from the origin and then move up 2 units. This systematic approach ensures accuracy in representing the point's location on the coordinate plane.

Rectangular Coordinate System

The rectangular coordinate system, often called the Cartesian coordinate system, is a foundational component in mathematics for plotting points and understanding geometry. It consists of two perpendicular lines, or axes:

• The x-axis: This is the horizontal line that stretches left and right.
• The y-axis: This is the vertical line that runs up and down.

These axes divide the plane into four sections, helping organize where each point will fall based on their respective coordinates. The point where these axes intersect is called the origin, marked as (0, 0).

Each point in this system is written as (x, y), where the first number is the x-coordinate, indicating horizontal movement, and the second is the y-coordinate, showing vertical movement. Whether in mathematics, navigation, or computer graphics, understanding the rectangular coordinate system is essential for accurately locating points and interpreting spatial relationships.