Problem 29 - College Algebra

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 for points on the x-axis.

Step 1: Understanding the x-axis

The x-axis in a coordinate plane is the horizontal line that runs left to right. Any point that lies on the x-axis will have coordinates in the form \(x, y\).

Step 2: Identifying Points on the x-axis

For a point to be on the x-axis, it must not move up or down from the axis itself. This means that the point does not have any vertical component.

Step 3: Determining the y-coordinate

Since any point on the x-axis does not move up or down, the y-coordinate of these points is always zero.

Key Concepts

x-axis

The **x-axis** is an essential part of the coordinate plane. It is the horizontal line that extends infinitely from left to right. This axis, along with the y-axis, helps form the coordinate grid, which is a crucial tool in mathematics for graphing and locating points in two-dimensional space.

On the coordinate plane, the x-axis is often used as a reference line. It is notable because it serves as the baseline from which vertical distances are measured. A point on this axis has a y-coordinate of zero, indicating no vertical movement. Essentially, any point that lies on the x-axis has the form \( (x, 0) \).

When plotting or reading points on a graph, remembering the function of the x-axis can make it much easier to understand relationships between coordinates and make quick assessments of positions.

y-coordinate

The **y-coordinate** is a vital part of any point's location on the coordinate plane. It tells us how far up or down a point is from the x-axis. In the coordinate system, each point is defined by a pair of values: \( (x, y) \).

While the x-coordinate indicates horizontal positioning, the y-coordinate reveals vertical displacement. This can be thought of as measuring height from the x-axis. It's important because it shows the elevation or depth concerning the baseline.

• If the y-coordinate is positive, the point lies above the x-axis.
• If it is negative, the point resides below the x-axis.
• A zero y-coordinate means the point is directly on the x-axis without any vertical movement.

Understanding these distinctions can assist greatly in both graph reading and creating as it affects where points are plotted and how they are analyzed.

Coordinate Plane

The **Coordinate Plane** is a two-dimensional surface used to determine the position of points. It consists of two perpendicular number lines: the x-axis and the y-axis, intersecting at a point called the origin. This intersection point has coordinates \( (0, 0) \).

The coordinate plane is divided into four quadrants, which help in specifying the location of any point through pairs \( (x, y) \):

• Quadrant I (both x and y are positive)
• Quadrant II (x is negative, y is positive)
• Quadrant III (both x and y are negative)
• Quadrant IV (x is positive, y is negative)

Each quadrant provides a unique environment where points are plotted based on their value characteristics. By understanding how the coordinate plane works, one can effectively read and interpret graphs and the functions they represent across various mathematical contexts.