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Chapter 5: Problem 6

Identify the quadrant in which each point lies. $$(1,5)$$

Short Answer

Expert verified
The point (1,5) lies in Quadrant I.

Step by step solution

01

Understanding Quadrants

In the Cartesian plane, quadrant I is where both x and y coordinates are positive. Quadrant II is where x is negative and y is positive. Quadrant III is where both x and y are negative. Quadrant IV is where x is positive and y is negative.
02

Identify the Quadrant for the Given Point

Given point has coordinates (1,5). Both the x and y coordinates are positive. Thus, the point lies in the quadrant where both x and y are positive, which is Quadrant I.

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Key Concepts

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

Quadrants
The Cartesian coordinate system divides the plane into four distinct regions known as quadrants. These quadrants help to describe the position of points in the plane. Each quadrant corresponds to a specific combination of signs for the x and y coordinates.

  • **Quadrant I**: Here, both x and y coordinates are positive \( (x > 0, y > 0) \). This is the top-right section of the plane.
  • **Quadrant II**: The x coordinate is negative while the y coordinate is positive \( (x < 0, y > 0) \). This region lies at the top-left of the plane.
  • **Quadrant III**: Both x and y coordinates are negative \( (x < 0, y < 0) \). You will find this quadrant at the bottom-left of the plane.
  • **Quadrant IV**: Here, the x coordinate is positive but the y coordinate is negative \( (x > 0, y < 0) \). This region is located at the bottom-right of the plane.
Each of these quadrants provides a useful way to determine the location of any point on the Cartesian plane.
x-coordinate
The x-coordinate is the first number in the ordered pair representing a point. It signifies how far the point is horizontally from the origin on the Cartesian plane. In the ordered pair \( (x, y) \), the x-coordinate is the value of x.

  • A positive x-coordinate means the point lies to the right of the y-axis.
  • A negative x-coordinate indicates that the point is located to the left of the y-axis.
  • If the x-coordinate is zero, the point lies exactly on the y-axis.
The x-coordinate plays a vital role in determining which quadrant a point belongs to because only certain combinations of positive and negative x and y coordinates will place a point in a given quadrant.
y-coordinate
The y-coordinate, also known as the ordinate, is the second number in an ordered pair, and it indicates the vertical position of a point relative to the origin. In the ordered pair \( (x, y) \), the y-coordinate is represented by y.

  • A positive y-coordinate shows that the point lies above the x-axis.
  • A negative y-coordinate means the point is located below the x-axis.
  • If the y-coordinate is zero, this indicates the point is exactly on the x-axis.
Understanding the y-coordinate is crucial for identifying a point's quadrant location, especially when paired with its respective x-coordinate.

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