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Chapter 2: Problem 14

A student incorrectly claimed that the point (0,-4) lies on the \(x\) -axis because the \(x\) -coordinate is \(0 .\)

Short Answer

Expert verified
The point (0, -4) lies on the y-axis, not the x-axis.

Step by step solution

01

Understand the Definition

The point (0, -4) is given as having coordinates (x, y), where x=0 and y=-4. Recall that a point lies on the x-axis if and only if its y-coordinate is zero.
02

Analyze the Given Point

Examine the coordinates of the point (0, -4). Here, the x-coordinate is indeed 0, but the y-coordinate is -4.
03

Compare to the Definition

Since the y-coordinate of the given point is -4, not 0, the point (0, -4) does not lie on the x-axis. Instead, this point lies on the y-axis because its x-coordinate is 0.
04

Correct the Claim

Based on these steps, the correct claim is that the point (0, -4) lies on the y-axis as its x-coordinate is 0. The student's statement that it lies on the x-axis because the x-coordinate is 0 is incorrect.

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Key Concepts

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

x-axis
The x-axis is one of the two main axes in the coordinate plane. It is the horizontal line that runs from left to right across the plane. Points located on the x-axis have a y-coordinate of zero. This means that if you have a point like (3, 0) or (-2, 0), these points lie directly on the x-axis because the second number, representing the y-coordinate, is zero. The x-axis serves as a reference line for determining the position of points along the horizontal direction.
  • For example, the point (0, 0) is where the x-axis and y-axis intersect.
  • A point (4, 0) lies 4 units to the right of the origin on the x-axis.
  • Conversely, a point (-3, 0) lies 3 units to the left of the origin on the x-axis.
Understanding the x-axis is crucial for correctly plotting points and for interpreting their positions on the coordinate plane.
y-axis
The y-axis is the other main axis on the coordinate plane and is perpendicular to the x-axis. This vertical line runs up and down. Any point located on the y-axis has an x-coordinate of zero. In other words, the first number in the coordinate pair of such points will always be zero. For instance, (0, 5) and (0, -3) both lie on the y-axis because their x-coordinate is zero. The y-axis helps in identifying the vertical position of points.
  • The point (0, 0) where the x-axis and y-axis intersect is known as the origin.
  • A point (0, 7) is 7 units above the origin on the y-axis.
  • Similarly, (0, -4) is 4 units below the origin on the y-axis.
Proper understanding of the y-axis is essential for pinpointing the upward or downward location of points on the coordinate plane.
coordinate points
Coordinate points are pairs of numbers that define the exact position of a point on the coordinate plane. These points are written in the form (x, y), where 'x' represents the horizontal position and 'y' represents the vertical position. Each pair indicates a unique location on the grid formed by the x-axis and y-axis.
  • For instance, the point (4, 5) means the point is 4 units to the right and 5 units up from the origin.
  • Conversely, (-3, -2) indicates the point is 3 units to the left and 2 units down from the origin.
Coordinate points are fundamental to graphing functions and understanding spatial relationships on the plane. Remember, (x, y) tells you exactly where to find the point in terms of 'x' units along the x-axis and 'y' units along the y-axis.
definition of axes
The axes on a coordinate plane are defined based on their orientation and role in plotting points. There are two primary axes:
  • The x-axis: Horizontal line that helps in determining the horizontal position of points.
  • The y-axis: Vertical line that helps in determining the vertical position of points.
The point where the x-axis and y-axis intersect is called the origin, denoted as (0, 0). This origin serves as the reference point when plotting any other points on the plane. Understanding the definitions and differences between the x-axis and y-axis is crucial for accurate navigation and calculation in the coordinate plane.

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Most popular questions from this chapter

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ x+3 y=12 $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(2 x=y+3\) and \(2 y+x=3\)

Graph each line passing through the given point and having the given slope. (-3,1)\(;\) undefined slope

Graph each linear or constant function. Give the domain and range. $$ f(x)=0 $$

Graph each line passing through the given point and having the given slope. (0,-4)\(; m=-\frac{3}{2}\)
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