Problem 20 - Thinking Mathematically

Question

Plot the given point in a rectangular coordinate system. \((2.25,-4.25)\)

Step-by-Step Solution

Answer

The point \((2.25, -4.25)\) is plotted by moving 2.25 units to the right from the origin along the x-axis (since 2.25 is positive), and then moving 4.25 units down (since -4.25 is negative).

Step 1: Identify the x and y coordinates

The given point is \((2.25, -4.25)\). So, the x-coordinate is 2.25 and the y-coordinate is -4.25.

Step 2: Plot the x-coordinate on the x-axis

Start at the origin (where the x and y axis intersect) and move 2.25 units to the right on the x-axis. This is because 2.25 is a positive number.

Step 3: Plot the y-coordinate on the y-axis

From the point on the x-axis, move 4.25 units downward, because -4.25 is a negative number.

Step 4: Mark the point

The final location is the point \((2.25, -4.25)\) on the rectangular coordinate system. Put a dot on this location to mark the point.

Key Concepts

Rectangular Coordinate System

A rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional space defined by two perpendicular axes. The horizontal axis is called the x-axis, and the vertical axis is called the y-axis. These axes divide the plane into four quadrants.
- **Origin**: This is the point where the x-axis and y-axis intersect, represented as (0, 0). - **Quadrants**: The plane is divided into four sections:

• **Quadrant I**: Where both x and y 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.

Understanding this system is crucial as it helps you visualize and analyze the location of points based on their coordinates in the plane. This forms the foundation for solving many problems in mathematics and physics involving planes.

Plotting Points

Plotting points on a rectangular coordinate system is like placing a star on a map. Each coordinate defines a unique location.
Here’s a simple way to plot a point like (2.25, -4.25):
- **Step 1**: **Identify coordinates** - You start by noting which number is for x and which is for y. In our point, 2.25 is the x-coordinate and -4.25 is the y-coordinate.
- **Step 2**: **Move along x-axis** - From the origin (0, 0), move directly to the right by 2.25 units. This is because x is positive.
- **Step 3**: **Adjust on y-axis** - From the spot you reached on the x-axis, move downward since the y value is negative, moving 4.25 units.
- **Step 4**: **Mark your point** - Once you have reached the intersection of these movements, place a point or star. You've now plotted (2.25, -4.25) on your map.

X-Coordinate and Y-Coordinate

The x-coordinate and y-coordinate tell you precisely where to place a point on a grid.
- **X-Coordinate**: Always comes first. It tells you how far to move left or right from the origin on the x-axis. A positive x-coordinate means right of the origin, while a negative one means left.
- **Y-Coordinate**: Comes second. This specifies how far to move up or down the y-axis from your x-location. Positive values mean moving up, whereas negative values direct you down.
Together, these two numbers will guide you right to the spot. Remembering which is x and which is y is half the battle when plotting points. They work like GPS coordinates directing you to an exact map location.