Problem 25 - 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. $$(2,3)$$

Step-by-Step Solution

Answer

The point (2, 3) is in the First Quadrant.

Step 1: Identify Point Coordinates

We are given the point (2, 3) where '2' is the x-coordinate and '3' is the y-coordinate. We will use these to locate the point on the coordinate system.

Step 2: Locating the Point

On a rectangular (Cartesian) coordinate plane, locate the x-coordinate first. Move 2 units to the right from the origin along the x-axis. Then, move 3 units upward along the y-axis.

Step 3: Determine Quadrant

Once the point is located at (2, 3) , observe which quadrant it falls into. A point where both x and y coordinates are positive is in the First Quadrant.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where we can plot points using a pair of numbers called coordinates. It resembles a large sheet of graph paper with a horizontal line called the x-axis and a vertical line known as the y-axis intersecting at the origin. The origin is the point where the two axes meet, designated as \((0, 0)\). The coordinate plane allows us to visually represent relationships and positions in space. To locate a point, you start from the origin. First, move along the x-axis according to the point's x-coordinate. Then, move parallel to the y-axis, dictated by the y-coordinate. This system makes it incredibly functional for illustrating mathematical concepts, patterns, and geometrical shapes.

Cartesian Coordinate

Cartesian coordinates are a mathematical concept used to determine a point's precise location on the coordinate plane. They are represented in the format \((x, y)\). The first number or component, known as the x-coordinate, indicates how far to move horizontally from the origin. If the number is positive, you move to the right; if negative, to the left.
The second number, called the y-coordinate, shows how far to travel vertically. A positive y-coordinate means moving upwards, while a negative y-coordinate points downwards.

• For example, in the point \((2, 3)\), 2 is the x-coordinate, and 3 is the y-coordinate.
• The x-coordinate tells us to move 2 units to the right from the origin.
• The y-coordinate means moving 3 units upwards.

This process ensures that each point on the plane has a unique set of coordinates, making it easy to identify and work with them. It forms the basis for more complex mathematical and scientific studies, as able to mathematically explore positions and movements in space.

Quadrants

The coordinate plane is divided into four sections called quadrants. These quadrants help in identifying the position of any point based on the sign of its coordinates. Each quadrant is numbered in a counter-clockwise manner starting from the upper right.

• First Quadrant: Both x and y coordinates are positive. For example, the point \((2, 3)\) lies in this quadrant.
• Second Quadrant: x-coordinate is negative, and y-coordinate is positive.
• Third Quadrant: Both x and y coordinates are negative.
• Fourth Quadrant: x-coordinate is positive, and y-coordinate is negative.

These quadrants are fundamental in identifying which section of the plane a point resides, enabling easy analysis of its attributes and relative positions. The quadrant system also plays a critical role in defining functions and in the study of trigonometry, statistics, and geometry.