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Chapter 3: Problem 58

Graph each equation. $$y=0$$

Short Answer

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

Step by step solution

01

Understand the equation

This equation represents a horizontal line that passes through the y-coordinate \(y = 0\).
02

Begin graphing

To graph this equation, create a graph with two perpendicular lines to represent the x and y axes.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攁 measure of the steepness of the line鈥攊s 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鈥攕traight 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.

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Most popular questions from this chapter

In Exercises \(74-77\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.

If you are given an equation of the form \(A x+B y=C\) explain how to find the \(x\) -intercept.

Use a graphing utility to graph each equation.Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. $$y=2 x+4$$

How many points are needed to graph a line? How many should actually be used? Explain.

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(1,3) ;\left(x_{2}, y_{2}\right)=(6,13)$$
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