Problem 18 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((2.5,3.5)\)

Step-by-Step Solution

Answer

The given point (2.5, 3.5) is located 2.5 units to the right of the origin on the x-axis and 3.5 units above the origin on the y-axis.

Step 1: Understand the Coordinate System

A rectangular (or Cartesian) coordinate system represents each point in the plane by an ordered pair of numbers. The first number, or x-coordinate, corresponds to the horizontal position, and the second number, or y-coordinate, corresponds to the vertical position.

Step 2: Locate the X-Coordinate

Starting from the origin (0,0), move 2.5 units to the right along the x-axis. This is because the sign of the x-coordinate is positive.

Step 3: Locate the Y-Coordinate

Starting from the point on the x-axis that we found in step 2, move 3.5 units up along the y-axis. This is because the sign of the y-coordinate is positive.

Step 4: Mark the Point

The intersection of the vertical line from the x-coordinate and the horizontal line from the y-coordinate give the point (2.5, 3.5). Draw or plot this point on the coordinate system.

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in the rectangular coordinate system. It consists of two numbers written in a specific order and enclosed in parentheses, such as \(2.5, 3.5\). The first number in this pair is called the \(x\)-coordinate, and the second number is the \(y\)-coordinate. The order in which these numbers appear is essential because it defines the unique position of a point within the coordinate system.

For example, in the ordered pair \(2.5, 3.5\), \(2.5\) is always associated with the \(x\)-axis, and \(3.5\) is linked with the \(y\)-axis. This structuring is what allows us to pinpoint exact locations in a two-dimensional space.

Exploring the X-Coordinate

The \(x\)-coordinate is the first number in an ordered pair and it tells us how far a point is positioned horizontally from the origin. The origin is the point \(0,0\) where the \(x\) and \(y\) axes of a coordinate system intersect.

To determine the location of an \(x\)-coordinate, we can follow these steps:

• Start at the origin, which is \(0,0\).
• Move right if the \(x\)-coordinate is positive, and left if it's negative.
• For an \(x\)-coordinate of \(2.5\), as in our example, you'll move 2.5 units to the right.

This movement is along the horizontal axis, and it sets the stage for locating the exact position of a point.

Insights into the Y-Coordinate

Like its counterpart, the \(x\)-coordinate, the \(y\)-coordinate provides information about a point's vertical position in the rectangular coordinate system. This is the second number in the ordered pair.

Utilizing the \(y\)-coordinate involves these steps:

• Begin at the position on the \(x\)-axis found previously.
• Move up if the \(y\)-coordinate is positive, or down if it's negative.
• In our example with a \(y\)-coordinate of \(3.5\), move 3.5 units up.

The \(y\)-coordinate determines how high or low the point will be relative to the \(x\)-axis.

Plotting Points in the Coordinate System

Plotting points in a rectangular coordinate system means marking the specific location of an ordered pair on the graph. This process visually represents the components of the ordered pair on the coordinate plane.

Here is how to effectively plot a point:

• First, use the \(x\)-coordinate to mark a point horizontally on the \(x\)-axis.
• Next, from this point, use the \(y\)-coordinate to mark vertically along the \(y\)-axis.
• The point of intersection between the \(x\) and \(y\) movements is where the ordered pair, such as \(2.5, 3.5\), is plotted.

Plotting points accurately helps in analyzing data and understanding geometric relationships in two-dimensional spaces.