Problem 32 - 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. $$(3,-3)$$

Step-by-Step Solution

Answer

The point (3,-3) lies in Quadrant IV.

Step 1: Understanding the Rectangular Coordinate System

A rectangular coordinate system, also known as the Cartesian plane, consists of an x-axis (horizontal) and a y-axis (vertical) that intersect at a point known as the origin (0,0). The plane is divided into four quadrants: 1. Quadrant I: Top right, where both x and y are positive. 2. Quadrant II: Top left, where x is negative and y is positive. 3. Quadrant III: Bottom left, where both x and y are negative. 4. Quadrant IV: Bottom right, where x is positive and y is negative.

Step 2: Locate the Point (3,-3)

In the coordinate pair (3,-3), '3' represents the x-coordinate and '-3' represents the y-coordinate. To locate the point on the coordinate plane, start at the origin (0,0). Move 3 units to the right along the x-axis and then 3 units down along the y-axis to reach the point (3,-3).

Step 3: Identify the Quadrant

To identify the quadrant, observe the signs of the coordinates. The x-coordinate is positive and the y-coordinate is negative, placing the point in Quadrant IV of the Cartesian plane.

Key Concepts

Cartesian Plane

The Cartesian Plane is a fundamental concept in coordinate geometry. It consists of two perpendicular number lines that intersect at a point called the origin. These number lines are known as the x-axis (horizontal) and the y-axis (vertical). Together, they divide the plane into four sections, called quadrants. The position of points on the plane is described by pairs of numbers known as coordinates, often written in the form \(x, y\).

• The x-coordinate represents the horizontal position, with values increasing from left to right.
• The y-coordinate represents the vertical position, with values increasing from bottom to top.

Understanding the Cartesian plane is crucial because it provides a unified way to study geometric shapes and algebraic functions. Remember, both axes are number lines extending infinitely in both directions. This knowledge allows for a wide range of mathematical applications, such as graphing equations and visualizing geometric transformations.

Quadrants

The Cartesian Plane is divided into four regions called quadrants. Each quadrant corresponds to a unique combination of sign for the coordinates, providing a way to categorize points based on their positions relative to the origin. Moving counterclockwise:

• **Quadrant I:** Located in the top right, where both x and y coordinates are positive.
• **Quadrant II:** Positioned in the top left, where the x coordinate is negative and the y coordinate is positive.
• **Quadrant III:** Found in the bottom left, with both x and y coordinates being negative.
• **Quadrant IV:** Situated in the bottom right, where the x coordinate is positive and the y coordinate is negative.

These quadrants guide us in identifying the location of a point based on the sign of its coordinates. For example, the point (3, -3) lies in Quadrant IV since the x-coordinate is positive and the y-coordinate is negative. Understanding quadrants reinforces our ability to navigate the Cartesian Plane effectively.

Coordinate Points

Coordinate Points specify a location on the Cartesian Plane using a pair of values. Written as \(x, y\), these coordinates uniquely identify each point's position relative to the origin.The x-coordinate denotes how far a point is along the horizontal axis, while the y-coordinate indicates its position along the vertical axis. To plot a point like (3, -3):

• Begin at the origin, (0,0).
• Move 3 units to the right, corresponding to the x-coordinate 3.
• Then, move 3 units downward, corresponding to the y-coordinate -3.

Once plotted, this location becomes a reference for many types of analyses and graph interpretations in math. Knowing how to read and plot coordinate points is an essential skill that enhances your understanding of graphing and geometry tasks. Recognizing this pattern lets you process mathematical concepts in a clear and organized manner.