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Chapter 3: Problem 90

Think of the points on the graph of the horizontal line \(y=4\) What do the points have in common? How do they differ?

Short Answer

Expert verified
Points on the line share the y-coordinate 4, but differ in x-coordinates which vary.

Step by step solution

01

Understand the Line Equation

The equation of the line given is \( y = 4 \). This means that it is a horizontal line that intersects the y-axis at 4. Every point on this line has a y-coordinate of 4.
02

Identify Commonality in Points

All the points on this line share the common characteristic that their y-coordinate is always 4. This means that, regardless of what x-coordinate a point has, if it is on the line \( y = 4 \), its y-coordinate must be 4.
03

Identify the Difference in Points

The points on the line differ in their x-coordinates. Unlike the y-coordinate, which is constant, the x-coordinate can be any real number. This means that the points (x, 4) have the same y-value but can have any possible x-value.
04

Conclusion

In summary, all points on the line have the y-coordinate 4 in common, but they differ in their x-coordinates, which can vary freely across all real numbers.

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Key Concepts

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

Line Equation
A line equation represents a set of points that satisfy a certain condition or relationship described by the equation. In the context of the horizontal line stated in the problem, the line equation is given as \( y = 4 \). This equation tells us that every point on this line has the same y-coordinate, and it equals 4.
Horizontal lines are unique because they do not slope upwards or downwards. Instead, they run parallel to the x-axis.
  • For a line equation in the form \( y = c \) (where \( c \) is a constant), the line is horizontal.
  • This means that no matter where you move along the line, the y-value remains constant.
  • All points on this line can be described as \( (x, 4) \), where \( x \) can be any real number.
Understanding the nature of the line equation helps in visualizing the line on a coordinate plane, with the stability in y-values ensuring it's perfectly horizontal.
Y-Coordinate
The y-coordinate in a two-dimensional coordinate system specifies the position of a point along the vertical y-axis. For the horizontal line given by \( y = 4 \), the y-coordinate of every point is 4. This means that, regardless of how far left or right you go along the line, this vertical position doesn't change.
The consistency of the y-coordinate is the defining feature of a horizontal line. It's what sets every point on such a line apart from any other line on the plane.
  • The y-coordinate is shared by all points on the line \( y = 4 \), illustrating the characteristic shared property of this line.
  • When drawing or plotting points, the fixed y-coordinate allows us to establish a predictable pattern: moving side to side on the graph, but never up or down.
In essence, this unchanging y-value ensures the line remains perfectly flat, creating a clear line along the plane.
X-Coordinate
While the y-coordinate remains fixed for points on a horizontal line, the x-coordinate provides flexibility. It specifies the horizontal position of a point and can vary freely, which is why the line extends indefinitely along the x-axis.
In simple terms, while the y-coordinate keeps you on a straight path, the x-coordinate decides how far and wide that path stretches.
  • The x-coordinate in equations like \( y = 4 \) can be any real number.
  • Each unique x-coordinate maps out a different point on the line, which means points can spread infinitely far in both directions along the x-axis.
  • This variation in the x-coordinate is what allows the line to form a continuous and endless path horizontally.
By understanding the role of the x-coordinate, one appreciates the line's infinite reach and the potential for different points, all sharing the same y-value yet differing in horizontal placement.

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