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Chapter 7: Problem 16

Plot the given point in a rectangular coordinate system. \((-5,-2.5)\)

Short Answer

Expert verified
The point \((-5,-2.5)\) is found by moving 5 units to the left and 2.5 units down from the origin of the coordinate plane. After plotting, the point is expected to be situated in the third quadrant because both x and y coordinates are negative.

Step by step solution

01

Understanding the Coordinate System

In a two-dimensional plane, the rectangular (also called Cartesian) coordinate system is formed by two perpendicular lines called axes. The horizontal line is called the x-axis and the vertical line is called the y-axis. The point at which they intersect is called the origin, and it has coordinates \((0,0)\). The coordinates of any point in the system are written as an ordered pair \((x,y)\).
02

Identify the Coordinates

The point we need to plot is \((-5,-2.5)\). The first number \(-5\) represents the x-coordinate which tells us the horizontal location of the point. Negative value tells us to move left from the origin. The second number \(-2.5\) is the y-coordinate which tells the vertical location of the point. Negative value tells us to move down from the origin.
03

Plotting the Point

Start at the origin \((0,0)\). First, move \(-5\) units to the left along the x-axis. Then from there, move \(-2.5\) units down along the y-axis. This locates the position of the point \((-5,-2.5)\) on the Cartesian coordinate plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Plotting Points on a Cartesian Coordinate System
When plotting points in the Cartesian coordinate system, you start by locating the origin. The origin is the point where the x-axis meets the y-axis, specifically at \((0,0)\).
To successfully plot a point, follow these steps:
  • Identify the ordered pair that specifies the point's location. This pair, \((x, y)\), includes an x-coordinate and a y-coordinate.
  • The first number in the ordered pair represents the x-coordinate. It affects your movement along the x-axis, while the second number is the y-coordinate, dictating movement along the y-axis.
For instance, to plot the point \((-5, -2.5)\):
  • Begin at the origin \((0,0)\).
  • Move left 5 units to account for the negative x-coordinate of -5.
  • Then from this new position, move down 2.5 units to account for the negative y-coordinate of -2.5.
This process helps you accurately locate and place points on a graph accurately.
Understanding the X-Axis and Y-Axis
The Cartesian coordinate plane consists of two crucial lines, the x-axis and the y-axis, forming a two-dimensional space.
The x-axis is a horizontal line that extends infinitely in both directions, providing a sense of left-right movement. Numbers increase as you move right from the origin and decrease as you move left.
The y-axis, on the other hand, is a vertical line that also extends infinitely. When moving upwards from the origin, numbers increase, while they decrease as you move downwards.
These axes divide the plane into four sections called quadrants:
  • Quadrant I: Both x and y coordinates are positive.
  • Quadrant II: The x-coordinate is negative, and the y-coordinate is positive.
  • Quadrant III: Both x and y coordinates are negative, like our point \((-5,-2.5)\).
  • Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative.
Understanding how these axes work is vital for plotting and interpreting graphs accurately.
The Role of Ordered Pairs
Ordered pairs \((x, y)\) are fundamental in describing the exact location of points within the Cartesian coordinate system.
An ordered pair consists of two elements:
  • x-coordinate: This indicates how far along the horizontal x-axis you need to move.
  • y-coordinate: This shows how far along the vertical y-axis you need to move.
The order of these numbers is significant. Changing their order, such as using \((y, x)\) instead of \((x, y)\), would lead to a different position on the plane altogether.
With \((-5, -2.5)\), the x-coordinate of -5 means a movement 5 units left from the origin, while the y-coordinate of -2.5 requires a 2.5 unit movement downward.
Familiarizing yourself with ordered pairs allows you to navigate and understand the coordinate plane and is essential for more complex mathematical concepts.

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