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Chapter 2: Problem 72

Graph each equation \(y=0\)

Short Answer

Expert verified
The graph of \( y = 0 \) is a horizontal line on the x-axis.

Step by step solution

01

Identify the Type of Equation

The equation given is \( y = 0 \). This is a linear equation in two variables: \( x \) and \( y \). The form of the equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this case, the slope \( m = 0 \) and y-intercept \( b = 0 \).
02

Determine the Graph Characteristics

Since \( y = 0 \) is a horizontal line, its slope \( m = 0 \) means it is parallel to the x-axis. The line crosses the y-axis at the point \((0,0)\).
03

Plot the Y-Intercept

Begin by plotting the y-intercept of the line on a coordinate plane. The y-intercept is the point where the line crosses the y-axis. For \( y = 0 \), this point is \((0, 0)\).
04

Draw the Line

Since it is a horizontal line, draw a straight line across the plane that goes through the point \((0,0)\) and remains parallel to the x-axis. The line should extend infinitely in both directions along the x-axis while staying at a constant y-value of 0.

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Key Concepts

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

Understanding the Y-Intercept
In the context of a linear equation, the y-intercept is a critical element. It represents the point where the line crosses the y-axis on a graph. To find the y-intercept of a given equation, you can look at the constant term of the equation when it is in the form of \( y = mx + b \). Here, \( b \) is the y-intercept.
For the equation \( y = 0 \), the y-intercept is \( 0 \), as there is no constant or 'b' term visible other than zero. This means that the line will pass through the origin鈥攖he point (0,0).
  • Y-intercept is where the line hits the y-axis.
  • In \( y = 0 \), the y-intercept is 0.
  • This implies the line passes through (0,0).
Recognizing the y-intercept helps you in pinpointing the line accurately on a graph.
Unraveling the Slope
The slope of a line tells us how steep the line is and the direction in which it tilts. It's expressed as the 'rise over run' 鈥� essentially, how much 'y' increases or decreases as 'x' increases. Mathematically, slope is represented by \( m \) in the equation format \( y = mx + b \).
For the equation \( y = 0 \), the slope \( m \) is 0. This is because there is no increase or decrease in the y-values as x changes, leading to a slope of zero. A slope of 0 indicates a perfectly horizontal line.
  • Slope is denoted by \( m \).
  • A slope of 0 indicates no incline鈥攋ust flatness.
  • For \( y = 0 \), the slope is 0.
Understanding the slope can give you insights into the line's angle and orientation, providing essential clues for graphing.
Characteristics of a Horizontal Line
A horizontal line is a type of linear graph where all the y-values remain constant. It's flat and parallel to the x-axis, which means it never rises or falls as you move along it. It's the visual representation of having a slope of zero.
In our example, the equation \( y = 0 \) creates a horizontal line. This particular line lies directly on the x-axis because its y-value is consistently zero. It spans across the graph infinitely in both directions on the x-axis.
  • Horizontal lines are flat and consistent.
  • They have a slope of 0.
  • For \( y = 0 \), the line runs along the x-axis.
Grasping the concept of horizontal lines is crucial as it helps in understanding basic principles of graphing linear equations.

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Most popular questions from this chapter

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. $$ f(x)=x^{3}+2 $$

Explain why we can think of a function as a machine.

A student was asked to determine the slope of the graph of the line \(y=6 x-4 .\) If his answer is \(m=6 x,\) explain his error.

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function g using trans/ations and/or a reflection. See Examples 7 and 8 . $$ g(x)=|x+4|+3 $$

Write an equation for a linear function whose graph has the given characteristics. Passes through ( \(3,0\) ), parallel to the graph of \(g(x)=-\frac{2}{3} x-4\)
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