Problem 51 - College Algebra

Question

Graph each equation in the rectangular coordinate system. $$y=0$$

Step-by-Step Solution

Answer

The graph of the equation \(y = 0\) is a horizontal line passing through the origin (0,0) on the graph.

Step 1: Understand the Equation

The equation given is \(y = 0\). This is a simple linear equation, where y is always equal to zero.

Step 2: Set Up The Graph

Draw a rectangular coordinate system, otherwise known as a Cartesian grid, which consists of a horizontal x-axis and a vertical y-axis. The intersection of these axes is called the origin, where both x and y are equal to zero.

Step 3: Plot the Equation

For the equation \(y = 0\), y stays at 0 for all values of x. Therefore, the line corresponding to this equation is a horizontal line passing directly through the origin (0,0) on the y-axis.

Key Concepts

Graphing Linear Equations

Linear equations are a central concept in math that describe straight lines on a graph. A linear equation can be represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. However, sometimes these equations vary slightly, as seen with \(y = 0\). Here, the equation describes a line where the y-value is always zero no matter what x-value you choose. This creates a unique horizontal line, representing a special case of linear equations.
This highlights the versatility of linear equations, showing how they can represent different types of lines, even those parallel to the x-axis. When graphing, it's crucial to understand how the format of the equation, combined with plotting points on the graph, results in a line. This forms the basis for making sense of various systems of equations and their graphical interpretations.

• Understand the equation's structure
• Recognize the type of line it represents
• Use graphing to visualize its properties

Horizontal Line

A horizontal line is one where all points on the line have the same y-value. In the equation \(y = 0\), every point along this line has a y-coordinate of 0, regardless of the x-coordinate value. This creates a line that runs parallel to the x-axis. Such lines are simple to graph because once you know the y-value, it stays constant across the x-axis.
Horizontal lines are unique because:

• They have no slope, which is mathematically stated as \(m = 0\).
• They visually separate the graph into two halves; one above and one below.
• They represent equations where y is constant.

To graph a horizontal line like \(y = 0\), simply draw a straight line that passes through the origin along the x-axis. This understanding helps identify the impact of horizontal lines within broader mathematical contexts, where they may serve as boundaries or represent constraints in applied problems.

Cartesian Grid

The Cartesian grid, or rectangular coordinate system, is a foundational tool in graphing equations. It is composed of two axes: the x-axis, which is horizontal, and the y-axis, which is vertical. These axes divide the plane into four quadrants and intersect at a point called the origin, which is labeled (0, 0).
The Cartesian grid makes it possible to visually represent mathematical equations. Each point on the graph corresponds to a pair of values (x, y), showcasing the relationship between variables. When graphing, the grid is crucial because:

• It helps identify where equations intersect.
• It provides a clear structure for plotting points and lines accurately.
• It allows for a precise visual representation of mathematical relationships.

In the context of \(y = 0\), the Cartesian grid aids in quickly seeing that this equation generates a horizontal line straight across the origin. Using this grid, students can easily plot and understand various types of linear, quadratic, and more complex equations.