Problem 29 - College Algebra with Corequisite Support

Question

For each of the following exercises, identify the information requested. If a point is located on the \(x\) -axis, what is the \(y\) -coordinate?

Step-by-Step Solution

Answer

The y-coordinate is 0.

Step 1: Understanding the Coordinate System

In a standard Cartesian coordinate system, a point is represented as \(x,y\). The \(x\) value tells how far the point is left or right along the \(x\)-axis, while the \(y\) value tells how far it is up or down along the \(y\)-axis. Points on the axes have special properties.

Step 2: Identifying Points on the x-axis

Any point on the \(x\)-axis has the form \(x, 0\). This is because the \(y\)-coordinate is zero for all points on the \(x\)-axis. The \(x\)-axis is the line where \(y = 0\). This means the point lies horizontally along the \(x\)-axis without any vertical displacement.

Step 3: Concluding the y-coordinate

Thus, if a point is located on the \(x\)-axis, its \(y\)-coordinate is strictly \(0\). The \(x\)-coordinate can be any real number depending on the location along the \(x\)-axis.

Key Concepts

x-axis

The x-axis is a fundamental component of the Cartesian coordinate system. In this system, the x-axis is the horizontal line used to measure how far a point is from a designated origin in a left or right direction.
The Cartesian plane is divided by two lines: the x-axis and y-axis. Any movement along the x-axis does not change the vertical positioning of a point; it solely influences the horizontal location.

When describing a point on the x-axis, the coordinates are often expressed in the form \(x, 0\). This means that regardless of where a point stands on the x-axis, the y-coordinate remains zero, signifying no vertical movement.

• The x-axis is also known as the horizontal axis.
• It runs left to right across the plane.
• Every point directly on the x-axis has a y-coordinate of zero.

Understanding the properties of the x-axis is crucial for solving many coordination problems in mathematics and geometry, as it represents an essential part of the coordinate plane.

y-coordinate

The y-coordinate is a crucial parameter in the Cartesian coordinate system, denoting the vertical position of a point within this two-dimensional plane. It is the second element in any ordered pair (x, y), indicating how far up or down the point is from the origin.
The significance of the y-coordinate lies in its ability to determine vertical positioning independently of horizontal movement.

The y-coordinate tells you how high above or below the horizontal x-axis a particular point is. Points can have positive y-coordinates (above the x-axis) or negative y-coordinates (below the x-axis).

• Positive y-coordinates indicate positions above the x-axis.
• Negative y-coordinates indicate positions below the x-axis.
• A y-coordinate of zero places the point exactly on the x-axis.

When a point lies directly on the x-axis, its y-coordinate is zero. This means there is no vertical displacement from this baseline horizontal line.

points on the axis

Points on the axis in the Cartesian coordinate plane exhibit unique and simple properties. These points are located either directly on the x-axis or the y-axis, making them easy to identify and analyze

Any point specifically on the x-axis has a y-coordinate of zero. Conversely, any point on the y-axis has an x-coordinate of zero.

• Points on the x-axis are in the form \(x, 0\)
• Points on the y-axis are in the form \(0, y\)

Understanding these properties helps simplify many geometry problems. You just need to identify whether the point lies horizontally or vertically, thus knowing which y or x coordinate is zero.

coordinate geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry using the Cartesian plane. This field focuses on representing geometric shapes using a coordinate system, offering a powerful tool for analyzing and solving complex geometric problems.

The Cartesian plane is the workbench for coordinate geometry, allowing for precise plotting and manipulation of points, lines, and shapes. Key aspects include:

• Points are expressed as ordered pairs \(x, y\).
• Lines and curves can be represented through equations.
• Enables computation of distances, slopes, and areas.

Coordinate geometry simplifies the study of geometry by turning it into numeric calculations, allowing for easy visualization and solving of spatial problems.