Problem 39 - A Graphical Approach to College Algebra

Question

If the \(x\) -coordinate of a point is \(0,\) the point must lie on which axis?

Step-by-Step Solution

Answer

The point lies on the y-axis.

Step 1: Understanding the Coordinate System

In a 2-dimensional coordinate system, every point is represented by an ordered pair \((x, y)\). The \(x\) coordinate indicates the position along the horizontal axis (x-axis), while the \(y\) coordinate indicates the position along the vertical axis (y-axis). In this exercise, we need to find the position of a point where the \(x\)-coordinate is \(0\).

Step 2: Determining Axis from Coordinate Information

A point that has an \(x\)-coordinate of \(0\) means it does not move left or right from the origin on the x-axis. Thus, the point must lie directly above or below the origin on the y-axis, which is the vertical line where all points have \(x = 0\).

Step 3: Identifying the Axis

Since the \(x\)-coordinate is \(0\), the point must lie on the y-axis. This is because all points on the y-axis have an \(x\) value of \(0\) and vary only along the \(y\) dimension. Therefore, the answer is that the point lies on the y-axis.

Key Concepts

Understanding the X-Coordinate

The x-coordinate is a crucial component of the coordinate system, especially when discussing the position of a point in a 2-dimensional space. This coordinate is the first number in any ordered pair, written as \((x, y)\). It tells us where to place the point along the horizontal line, known as the x-axis.
When the x-coordinate is zero, it is particularly interesting because it indicates that the point is not moving left or right from the origin — the point where the x-axis and y-axis intersect. This behavior aligns the point vertically along the y-axis. In essence, the x-coordinate determines horizontal placement, or lack thereof, and when it is zero, we focus on the vertical position dictated by the y-coordinate.

The Role of the Y-Axis

The y-axis is one of the two fundamental axes in a 2-dimensional coordinate system, serving as the vertical reference line. It intersects the x-axis at the origin \((0, 0)\). This line is crucial for determining the vertical position of any point in this system.
Any point that has an x-coordinate of zero will be somewhere along the y-axis, directly above or below the origin. It is this special characteristic that defines the y-axis: all points along it have their x-coordinate as zero.

• This makes it a unique indicator for one-dimensional vertical movement or position in the coordinate plane.
• The y-axis helps separate the plane into left and right halves, each with positive and negative y-coordinates.

Understanding the position of a point relative to the y-axis helps clarify its vertical journey, strictly guided by its y-coordinate.

The 2-Dimensional Coordinate System

The 2-dimensional coordinate system is a framework for spatial organization. It involves two axes — the x-axis and y-axis — intersecting at the origin \((0, 0)\). Each point in this 2D plane is denoted by an ordered pair \((x, y)\) that specifies its exact location.

• The x-coordinate represents how far the point is from the vertical y-axis and how much it has moved along the horizontal x-axis.
• The y-coordinate tells us the point's distance from the horizontal x-axis and what position it holds along the vertical y-axis.
• This system allows for precise placement and identification of points, lines, and shapes within the plane.

By understanding both coordinates, you can easily locate any point within the plane, visualize geometric figures, and solve problems related to positions and distances. It's fundamental in various branches of mathematics, physics, and engineering.