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Chapter 15: Problem 5

In which quadrant is each point located? \((3,-2.6)\)

Short Answer

Expert verified
The point (3, -2.6) is located in Quadrant IV.

Step by step solution

01

Identify the Coordinates

The point provided is \(3, -2.6\). Identify the x-coordinate and the y-coordinate. Here, the x-coordinate is \3\, and the y-coordinate is \ -2.6\.
02

Understand the Quadrants

The Cartesian plane is divided into four quadrants: - 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)
03

Determine the Sign of Each Coordinate

For the point \(3, -2.6\), the x-coordinate (3) is positive and the y-coordinate (-2.6) is negative.
04

Locate the Quadrant

According to the sign of the coordinates: the x-coordinate is positive and the y-coordinate is negative. This means that the point \(3, -2.6\) is located in Quadrant IV.

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Key Concepts

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

Cartesian coordinates
In mathematics, a system called the Cartesian coordinate system helps us locate points in a plane. Each point is identified by an ordered pair of numbers written in the form \((x, y)\).
The first number, called the x-coordinate, tells us how far to move horizontally from the origin (the center of the plane).
The second number, the y-coordinate, shows us how far to move vertically.
For example, in the point \((3, -2.6)\), the x-coordinate is 3, indicating a move 3 units to the right. The y-coordinate is -2.6, meaning a move 2.6 units downward. To sum up: each point in the Cartesian plane is uniquely determined by its x and y coordinates.
coordinate plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can plot points, lines, and curves.
It consists of a horizontal number line called the x-axis and a vertical number line called the y-axis, which intersect at a point known as the origin (0,0). Important features of the coordinate plane include:
  • The x-axis runs left to right.
  • The y-axis runs up and down.
  • The plane is divided into four quadrants by these axes.
  • Each point on the plane is described by an ordered pair \((x, y)\).
This system not only allows us to visualize concepts in mathematics but also helps us solve problems related to geometry and algebra.
quadrants
The Cartesian plane is divided into four regions called quadrants. Each quadrant represents a unique combination of the signs of x and y coordinates.
Here's a closer look at each:
  • **Quadrant I**: Both x and y are positive \((x > 0, y > 0)\). Examples: \((2, 3)\)
  • **Quadrant II**: x is negative, but y is positive \((x < 0, y > 0)\). Examples: \((-4, 5)\)
  • **Quadrant III**: Both x and y are negative \((x < 0, y < 0)\). Examples: \((-3, -1)\)
  • **Quadrant IV**: x is positive, but y is negative \((x > 0, y < 0)\). Examples: \((3, -2.6)\)
By examining the signs of the coordinates of any point, we can easily determine which quadrant it lies in. For instance, the point \((3, -2.6)\) has a positive x-coordinate and a negative y-coordinate, putting it in Quadrant IV.

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