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Chapter 8: Problem 30

What is the graph of \(x y=0\) ? Defend your answer.

Short Answer

Expert verified
The graph of \(xy = 0\) is the set of all points on the x-axis and y-axis.

Step by step solution

01

Understand the equation

The equation given is \(xy = 0\). This implies that the product of \(x\) and \(y\) must be zero. For the product of two numbers to be zero, at least one of the numbers must be zero.
02

Analyze conditions for zero

To satisfy the equation \(xy = 0\), either \(x\) must be zero or \(y\) must be zero. This means we need to analyze these two conditions separately to understand their impacts on the graph.
03

Graph \(x = 0\)

If \(x = 0\), we are looking at the set of all points that lie on the y-axis. This is because any point with an x-coordinate of zero lies on the y-axis.
04

Graph \(y = 0\)

If \(y = 0\), we are looking at the set of all points that lie on the x-axis. This is because any point with a y-coordinate of zero lies on the x-axis.
05

Combine the results

The graph of \(xy = 0\) is the combination of the graphs of \(x = 0\) and \(y = 0\). Thus, it consists of both the x-axis and the y-axis.

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Key Concepts

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

Coordinates
Coordinates are vital in understanding how we locate points on a graph. They consist of two numbers written in a pair, like \(x, y\). \(x\) is known as the x-coordinate, and \(y\) is the y-coordinate. These coordinates help us determine the exact position of a point on a two-dimensional plane.

Imagine coordinates like a GPS system for a graph. The x-coordinate tells us how far along (right or left) to go from the origin (0,0), and the y-coordinate tells us how far up or down to go. This system makes it easy to pinpoint the position of any point.

  • The first number in the pair is always the x-coordinate.
  • The second number is always the y-coordinate.
  • Coordinates are written inside parentheses and separated by a comma, like this: \(3, 4\).
x-axis
The x-axis is a crucial part of any graph. It is the horizontal line that runs from left to right, passing through the origin where it meets the y-axis. When graphing an equation like \(y = 0\), all points lie along the x-axis. This is because their y-coordinates are zero, meaning no vertical movement.

The x-axis is like the ground or baseline of the graph, forming one of the central reference lines in the coordinate plane.

  • The x-axis is often used to represent the independent variable in equations.
  • Every point on this axis has a y-coordinate of zero.
  • It extends infinitely in both positive and negative directions.
y-axis
The y-axis is the vertical counterpart to the x-axis. It runs up and down through the origin and is used to determine the second part of a coordinate. In our context of \(x = 0\), points that meet this condition lie on the y-axis, showing no horizontal movement because their x-coordinates are zero.

The y-axis acts like a wall or boundary stretching vertically on the graph.

  • It represents the dependent variable in many equations.
  • Every point on this axis has an x-coordinate of zero.
  • Like the x-axis, it extends infinitely in both upwards and downwards directions.

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Most popular questions from this chapter

\(4 x^{2}-9 y^{2}=16\)

Find the equation of the circle that passes through the origin and has its center at \((8,-15)\).

\(x^{2}+y^{2}-2 x-6 y-6=0\)

How does the graph of \(-y=x^{2}\) compare to the graph of \(y=x^{2}\) ? Explain your answer.

\(x^{2}+y^{2}-4 x-6 y-12=0\)
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