Problem 11 - College Algebra

Question

Plot the given point in a rectangular coordinate system. $$ \left(\frac{7}{2},-\frac{3}{2}\right) $$

Step-by-Step Solution

Answer

The point \((\frac{7}{2},-\frac{3}{2})\) or (3.5, -1.5) has been plotted on the rectangular coordinate system.

Step 1: Identify the x-coordinate

The x-coordinate of the given point is \(\frac{7}{2}\) or 3.5. This point lies on the right side of the origin along the x-axis because it's positive. Mark this point on your x-axis.

Step 2: Identify the y-coordinate

The y-coordinate of the given point is \(-\frac{3}{2}\) or -1.5. This point lies below the origin along the y-axis because it's negative. Starting from the point previously marked on the x-axis, move vertically down to -1.5 on the y-axis.

Step 3: Plot the Point

Now, you'll have the point \((\frac{7}{2},-\frac{3}{2})\) or (3.5, -1.5). This is done by marking an intersection where 3.5 on the x-axis and -1.5 on the y-axis meet. Draw this point on your graph.

Key Concepts

Plotting Points

Plotting points in a rectangular coordinate system is a fundamental skill in mathematics. This system, also known as the Cartesian coordinate system, helps to represent points in two-dimensional space using two numbers. Each point is defined by an ordered pair, such as \((x, y)\).
To plot a point, follow these steps:

• Identify the ordered pair, \((x, y)\).
• Locate the first number on the horizontal axis, which is the x-axis.
• Locate the second number on the vertical axis, which is the y-axis.
• Find the spot where these two values intersect and mark it with a point.

This method creates a visual representation of the point, making it easier to understand its location in the plane.

x-coordinate

The x-coordinate of a point in the Cartesian plane tells us how far the point is horizontally from the origin. The origin is the point where both the x and y axes intersect, specifically at \((0, 0)\).
In the example point \(\left(\frac{7}{2}, -\frac{3}{2}\right)\), the x-coordinate is \(\frac{7}{2}\) or 3.5. This positive value indicates the point is located to the right of the origin.

• If the x-coordinate is positive, the point is to the right of the y-axis.
• If it's negative, the point is to the left of the y-axis.
• If it is zero, the point is on the y-axis.

Understanding the x-coordinate helps in accurately placing the point horizontally on the graph.

y-coordinate

The y-coordinate tells us the vertical distance of the point from the origin. This is crucial for determining the point's exact position on the y-axis. In the point \(\left(\frac{7}{2}, -\frac{3}{2}\right)\), the y-coordinate is \(-\frac{3}{2}\) or -1.5.
This negative value indicates that the point lies below the x-axis.

• If the y-coordinate is positive, the point is above the x-axis.
• If it is negative, the point is below the x-axis.
• If it is zero, the point lies exactly on the x-axis.

By fully understanding the y-coordinate, you're better equipped to determine the vertical positioning of the point on the graph.