Problem 25 - A Graphical Approach to College Algebra

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 \((2, 3)\) lies in Quadrant I.

Step 1: Understand the Coordinate System

A rectangular coordinate system is made up of two axes: the x-axis (horizontal) and the y-axis (vertical). Points in the plane are represented as ordered pairs \((x, y)\).

Step 2: Plot the Point

To plot the point \((2, 3)\), start at the origin \((0,0)\). Move 2 units to the right along the x-axis (since the x-coordinate is 2), and then move 3 units up along the y-axis (since the y-coordinate is 3). The point \((2, 3)\) is now plotted on the coordinate plane.

Step 3: Identify the Quadrant

The coordinate system is divided into four quadrants:- Quadrant I: where both x and y coordinates are positive.- Quadrant II: where x is negative and y is positive.- Quadrant III: where both x and y are negative.- Quadrant IV: where x is positive and y is negative.Since the point \((2, 3)\) has both positive x and y coordinates, it lies in Quadrant I.

Key Concepts

Coordinate Plane

In a rectangular coordinate system, the coordinate plane is a flat, two-dimensional surface. This plane is essential for graphing and analyzing points in mathematics and science, providing a visual platform for understanding relationships between numbers. It is composed of:

• Two perpendicular lines: the x-axis and the y-axis.
• The intersection of these axes is called the origin, denoted as \( (0,0) \).
• Everything to the right of the origin on the x-axis is positive, and to the left is negative.
• Everything above the origin on the y-axis is positive, and below is negative.

These axes divide the plane into four sections known as quadrants. The coordinate plane is a foundational concept for locating points and graphing equations effectively. Understanding it allows you to visualize abstract numerical data and relationships.

Quadrants

The concept of quadrants is critical in the coordinate plane, demarcating distinct sections where points can lie depending on the signs of their coordinates. Each quadrant provides information about the sign of the x and y coordinates:

• Quadrant I: Both x and y coordinates are positive (e.g., (2, 3)).
• Quadrant II: x is negative, and y is positive (e.g., (-2, 3)).
• Quadrant III: Both x and y coordinates are negative (e.g., (-2, -3)).
• Quadrant IV: x is positive, and y is negative (e.g., (2, -3)).

By understanding which quadrant a point lies in, you can quickly assess the sign of the x and y coordinates. This insight is essential not only for basic graphing but also in fields such as physics and engineering, where direction and position matter.

Ordered Pairs

Ordered pairs are the foundation of locating points on a coordinate plane. They are written in a form \( (x, y) \), representing the horizontal and vertical placements, respectively. Understanding ordered pairs involves:

• The x-coordinate: indicates the point's location along the horizontal axis.
• The y-coordinate: indicates the point's position along the vertical axis.

Ordered pairs are vital for determining exact positions in mathematical and practical applications, such as GPS technology, computer graphics, and more. Recognizing how to read and interpret these ordered pairs allows you to navigate and utilize the coordinate plane efficiently.

Plotting Points

Plotting points on a coordinate plane involves understanding and applying the concepts of ordered pairs and the axes. The process can be broken down as follows:

• Start at the origin \( (0,0) \).
• Move horizontally according to the x-coordinate. If positive, move right; if negative, move left.
• From the new x-position, move vertically according to the y-coordinate. If positive, move up; if negative, move down.
• Mark the location where these movements converge as the point \( (x, y) \).

For instance, to plot \( (2, 3) \), move 2 units right, then 3 units up from the origin. Plotting points accurately is crucial for creating graphs and solving geometrical problems. It visually brings mathematics to life, making it easier to solve and understand complex problems.