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Chapter 3: Problem 71

Given that \(x\) is a positive number and that \(y\) is a positive number, determine the quadrant or axis in which each point lies. $$ (x, 0) $$

Short Answer

Expert verified
The point \((x, 0)\) lies on the positive x-axis.

Step by step solution

01

Understanding the Cartesian Plane

The Cartesian coordinate plane is divided into four quadrants. Quadrant I has both positive x and y, Quadrant II has a negative x and positive y, Quadrant III has both negative x and y, and Quadrant IV has a positive x and negative y. Points can also lie on axes, specifically the x-axis or y-axis.
02

Identifying given point attributes

The given point is \((x, 0)\). From the coordinates, we see that the x-value is positive and the y-value is zero.
03

Analyzing the location on the coordinate plane

If a point has a y-coordinate of zero, it lies on the x-axis. Since the x-coordinate \(x\) is positive, the point \((x, 0)\) lies on the positive x-axis.

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Key Concepts

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

Quadrants in the Cartesian Plane
The Cartesian coordinate plane is a two-dimensional graph divided into four sections. These sections are called quadrants. To understand how points are positioned, it's important to become familiar with these quadrants.

  • Quadrant I: Both the x and y coordinates are positive. This is the upper right section of the plane.
  • Quadrant II: The x coordinate is negative, and the y coordinate is positive. This is the upper left section.
  • Quadrant III: Both x and y coordinates are negative. This quadrant is situated in the lower left.
  • Quadrant IV: The x coordinate is positive, while the y coordinate is negative. This is found in the lower right.
The quadrants help determine where a point is located based on its coordinates. Understanding which numbers are positive or negative guides you to the correct quadrant. A crucial aspect is that if either the x or y value is zero, the point will not lie within any quadrant but rather on an axis. This is particularly relevant when analyzing points that lie directly on the x-axis or y-axis.
Understanding Cartesian Coordinates
Cartesian coordinates are a system used to numerically determine positions on a plane through two intersecting, perpendicular lines. These lines are the x-axis and y-axis.

  • X-coordinate: This represents the horizontal position of a point. Positive values are to the right of the origin, while negative values are to the left.
  • Y-coordinate: This represents the vertical position of a point. Positive values go above the origin, negative values go below.
  • Origin: This is where the x-axis and y-axis intersect. Its coordinates are (0,0), functioning as the central reference point on the plane.
Coordinates are presented as ordered pairs, like ewline (x, y), where x defines the horizontal position and y defines the vertical position. It's important to remember that the order matters鈥攂egin with the x information! These coordinates let us efficiently describe where a point is situated on the plane.
Role of the x-axis on the Coordinate Plane
The x-axis is one of the two foundational lines on a Cartesian plane, serving as the horizontal line where values for the x-coordinate are plotted. It's a crucial reference for organizing the plane and determining the positions of points.

  • Definition of the x-axis: The x-axis runs horizontally and crosses the y-axis at the origin (0,0).
  • Positive and Negative Divisions: On the x-axis, numbers to the right of the origin are positive, while those to the left are negative.
  • Zero Values: Points directly on the x-axis have a y-coordinate of zero. For instance, a point like (x, 0) sits directly along this axis.
If a point has coordinates where the y value is zero, like (x, 0), it lies on the x-axis. This means that while the point is not in a specific quadrant, it still has a defined place on the plane. Understanding the x-axis is essential for interpreting and graphing coordinates accurately.

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