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Chapter 3: Problem 44

In which quadrant or on which axis is each point located? $$ (1,-5) $$

Short Answer

Expert verified
Quadrant IV.

Step by step solution

01

- Understand the Cartesian Plane

The Cartesian plane is divided into four quadrants. Quadrant I (+, +), Quadrant II (-, +), Quadrant III (-, -), and Quadrant IV (+, -). Points can also lie on the axes.
02

- Identify Coordinates

Consider the point (1, -5). The first coordinate is the x-coordinate (1), and the second coordinate is the y-coordinate (-5).
03

- Determine the Sign of Coordinates

For the point (1, -5), the x-coordinate is positive (1 > 0), and the y-coordinate is negative (-5 < 0).
04

- Locate the Quadrant

A point with a positive x-coordinate and a negative y-coordinate 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.

Quadrants
The Cartesian plane is a valuable tool for graphing and understanding the relationship between numbers.
It divides the plane into four sections called quadrants. Each quadrant has a specific combination of positive and negative values for the x and y coordinates.

These quadrants are defined as follows:

  • Quadrant I: Both x and y are positive \((+, +)\)
  • Quadrant II: x is negative and y is positive \((- , +)\)
  • Quadrant III: Both x and y are negative \((- , -)\)
  • Quadrant IV: x is positive and y is negative \((+, -)\)

A point can also be on one of the axes. If it is on the x-axis, its y-coordinate is zero, \((x, 0)\). If it is on the y-axis, its x-coordinate is zero, \(( 0, y)\). Understanding these quadrants helps in precisely locating points on the plane.
Coordinates
Coordinates are a set of values that show an exact position on the Cartesian plane. Each point on the plane is described by a pair of numbers written as \((x, y)\). The first number is the x-coordinate, and the second number is the y-coordinate.

For example, if we have the point \((1, -5)\), 1 is the x-coordinate, and -5 is the y-coordinate.
The x-coordinate tells us the position along the horizontal axis, while the y-coordinate gives us the position along the vertical axis.
The correct interpretation of coordinates is crucial for placing points accurately on the graph and understanding their relationships.
Positive and Negative Values
Positive and negative values in the coordinates indicate the direction in which the point lies concerning the origin \((0, 0)\). Points to the right of the origin have positive x-values; points to the left have negative x-values.
Similarly, points above the origin have positive y-values, and points below have negative y-values.

For instance, the point \((1, -5)\) has a positive x-value (right of the origin) and a negative y-value (below the origin). This means the point is in Quadrant IV.
Recognizing positive and negative values helps in understanding the geometric arrangement and aids in calculations involving these points.

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