Problem 58 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$y=0$$

Step-by-Step Solution

Answer

The graph of the equation \(y = 0\) is a straight line along the x-axis, passing through the origin.

Step 1: Understand the equation

This equation represents a horizontal line that passes through the y-coordinate \(y = 0\).

Step 2: Begin graphing

To graph this equation, create a graph with two perpendicular lines to represent the x and y axes.

Step 3: Draw the line

Draw a straight line along the x-axis (horizontally) that passes through the origin point (0, 0). This line represents the equation \(y = 0\). Since \(y\) always equals 0, no matter what \(x\) equals, the line does not slope up or down and instead lies flat along the x-axis.

Key Concepts

Horizontal line

A horizontal line is a straight line that runs parallel to the x-axis on a graph. Unlike diagonal lines, it does not rise or fall as it extends across the graph. Let's dive deeper into understanding why a horizontal line is unique:

• The key characteristic is that for every point on the line, the y-coordinate remains the same.
• This means there's no change in the vertical direction; the line remains flat.

In the case of our equation, \( y = 0 \), the horizontal line runs through the y-coordinate 0. In essence, it possesses zero slope, indicating that the rise over run—a measure of the steepness of the line—is zero. Therefore, it visually appears as a flat line on the graph.

x-axis

The x-axis is an essential part of a coordinate plane, running horizontally from left to right. It forms one half of the intersection that creates the graph, the other being the y-axis. Here's how it contributes:

• All points on the x-axis have a y-coordinate of 0. Thus, it coincides with horizontal lines like \( y = 0 \).
• The x-axis is often used as a reference line to compare other points or lines on the graph.

In our specific exercise, \( y = 0 \) directly translates to the x-axis itself, making it effortless to graph. As such, any time you are asked to graph \( y = 0 \), it aligns perfectly with the x-axis.

Equation representation

Equations like \( y = 0 \) offer a straightforward representation of lines in algebra. Here's a clear breakdown of this concept:

• This equation form is called a linear equation because it produces a straight line when graphed.
• "\( y = \)" indicates that no matter what x-value is selected, the y-value remains constant.

This constancy is what makes \( y = 0 \) a horizontal line across the graph. By understanding this, you can easily determine its position on the coordinate plane—straight along the x-axis. Such simplicity in equation representation not only aids in visualizing the graph but also in understanding the relationship between variables when depicted graphically.