Problem 24 - Introductory and Intermediate Algebra for College Students 4th

Question

Plot the given point in a rectangular coordinate system. $$\left(0, \frac{7}{2}\right)$$

Step-by-Step Solution

Answer

The given point (0, 7/2) indicates that no move would occur in the x-direction and a move of 7/2 units will occur in the positive y-direction from the graph's origin. Therefore, the point will be plotted on the y-axis, 7/2 units above the origin.

Step 1: Understand the Point

The given point is (0, 7/2). In the Cartesian coordinates, the first value (0 in this case) represents the x-coordinate and the second value (7/2 in this case) represents the y-coordinate. This means the position of the point is 0 units towards the right (positive x-direction) from origin on the x-axis and 7/2 units up from the origin (positive y-direction) on the y-axis. Since x is 0, the point lies on the y-axis.

Step 2: Set Up the Coordinate System

Draw two perpendicular lines that intersect at point called the origin (0,0). Label the horizontal line as x-axis and the vertical line as y-axis.

Step 3: Plot the Point

Start at the origin, move 0 unit along the x-axis (as x=0). Then move 7/2 units upwards along the y-axis (as y=7/2). Plot the point at this location.

Key Concepts

Cartesian Coordinates

Understanding Cartesian coordinates is a foundational skill in graphing and mathematics. A Cartesian coordinate system consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Every point in this system can be described using a pair of numbers called coordinates, expressed as \( (x, y) \).

For example, consider the point \( (0, \frac{7}{2}) \). In this pair, the first number, 0, is the x-coordinate. It tells you how far to move horizontally from the origin, which is the point where the two axes intersect. The second number, \( \frac{7}{2} \), is the y-coordinate. It represents the vertical movement from the origin. If the x-coordinate is zero, as in our example, it indicates that the point is on the y-axis itself, exactly \( \frac{7}{2} \) units away from the origin. Knowing this, you can precisely locate the point on the coordinate plane.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane defined by two perpendicular lines called axes. These intersect at a point called the origin, which has the coordinates \( (0, 0) \).

To set up your coordinate system, start by drawing a horizontal line, which is the x-axis, and a vertical line, which is the y-axis. Ensure they intersect at right angles, creating four distinct quadrants. Each quadrant is numbered counter-clockwise starting from the positive x-axis and the positive y-axis quadrant which is known as Quadrant I. The axes are labeled with positive and negative numbers, with positive numbers on the right side of the y-axis and above the x-axis, and negative numbers on the left side of the y-axis and below the x-axis. Points are plotted using these values as guides, with the origin being the central point of reference for the entire system.

X-axis and Y-axis

In the rectangular coordinate system, the x-axis and y-axis are the backbone for locating points. The x-axis is the horizontal axis, running left to right. Points on the right side of the origin have positive x-coordinates, while those on the left have negative x-coordinates. Conversely, the y-axis is vertical, stretching from bottom to top. Points above the origin have positive y-coordinates, and those below have negative y-coordinates.

When plotting a point such as \( (0, \frac{7}{2}) \), you begin at the origin. Since the x-coordinate is 0, you do not move left or right. Instead, you move straight up along the y-axis because of the positive y-coordinate, which in this case is \( \frac{7}{2} \). Plotting along the y-axis when the x-coordinate is zero simplifies to a vertical movement, indicating the point lies directly on the y-axis itself.