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Chapter 1: Problem 36

Graph the given line. $$ y=0 $$

Short Answer

Expert verified
The line is horizontal along the x-axis, passing through (0,0).

Step by step solution

01

Identify the Type of Line

The equation given is \( y = 0 \). This equation represents a horizontal line where the value of \( y \) is constant at 0.
02

Determine the Y-Intercept

Since \( y = 0 \), the y-intercept is at the point on the y-axis where \( y = 0 \), which is the origin (0, 0).
03

Plot Key Points

Plot the point (0,0) on the graph. Then, identify additional points. For a horizontal line \( y = 0 \), the line will include points such as (-2, 0), (1, 0), and (3, 0). These points confirm the horizontal nature of the line.
04

Draw the Line

Connect the plotted points with a straight line that extends infinitely in both directions along the horizontal axis (x-axis), ensuring it remains at \( y = 0 \) for all values of \( x \).
05

Label the Line

Label the line on the graph as \( y = 0 \) to indicate clearly which equation it represents.

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Key Concepts

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

Graphing Lines
Graphing a line is as simple as connecting dots - each dot representing a specific point on the coordinate plane. When graphing lines, understanding the equation you're working with is crucial. In this exercise, we are given the equation \( y = 0 \), which is a specific type of line known as a horizontal line. What makes this line special is that every point on it shares the same \( y \)-coordinate; in this case, 0. The process involves:
  • Identifying the equation: This tells you the nature and direction of the line.
  • Understanding the line's orientation: Horizontal lines run parallel to the x-axis because the y-value remains constant.
  • Marking some points: This helps visualize and accurately draw the line.
Graphing is about seeing these relationships on paper, which helps in understanding how equations translate visually.
Y-Intercept
The y-intercept of a line is the point where it crosses the y-axis, where the x-coordinate is zero. This is a vital piece of information because it tells you exactly where the line will hit the vertical axis, providing a starting point for graphing.In our equation \( y = 0 \), the y-intercept is straightforward. It happens right at the origin, which is the point (0, 0). The point (0, 0) is the y-intercept for this horizontal line. This means that the line starts at the origin and will move horizontally to the left and right.Understanding the y-intercept:
  • Provides an anchoring point for graphing the line.
  • Helps understand if and where two lines might intersect.
  • Is critical in determining the line's placement on a graph.
Remember, the y-intercept is always one of the first characteristics you need to identify when dealing with linear equations.
Plotting Points
Plotting points is akin to setting the foundation for building a graphical representation of mathematical ideas. Each point you plot on a graph carries specific coordinates \((x, y)\). For the line \( y = 0 \), plotting is simple:
  • Start with the y-intercept, (0, 0).
  • Since the line is horizontal, choose several points along the x-axis, like (-2, 0), (1, 0), (3, 0).
Ensure these points all have the y-coordinate of 0, reflecting the line's horizontal nature. Once you have these points, draw a straight line through them horizontally, ensuring it spans the graph from left to right.Key aspects of plotting:
  • Confirms the line鈥檚 orientation on the graph (horizontal, in this case).
  • Visualizes the consistency of the y-coordinates for all chosen x-values.
  • Highlights the relationship between algebraic equations and their graphical outputs.
Plotting accurately allows for a true visual representation and helps avoid errors in graph interpretation.

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Most popular questions from this chapter

In Exercises \(13-20\), sketch the graph of the given piecewise-defined function. $$ f(x)=\left\\{\begin{array}{rll} -2 x-4 & \text { if } & x<0 \\ 3 x & \text { if } & x \geq 0 \end{array}\right. $$

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=\frac{\sqrt[3]{x^{3}+x}}{5 x} $$

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=\sqrt{1-x^{2}} $$

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=x^{6}-x^{4}+x^{2}+9 $$

In Exercises \(21-45,\) find and simplify the difference quotient \(\frac{f(x+h)-f(x)}{h}\) for the given function. $$ f(x)=\frac{x^{2}}{2 x+1} $$
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