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Chapter 3: Problem 26

Answer the given questions. What is the \(y\)-coordinate of all points on the \(x\)-axis?

Short Answer

Expert verified
The \(y\)-coordinate is 0.

Step by step solution

01

Understanding the Axis

The origin of a two-dimensional Cartesian coordinate system is \((0, 0)\), where the \(x\)-axis is the horizontal line.
02

Identifying Coordinates on the Axis

For any point on the \(x\)-axis, the coordinate takes the form \((x, 0)\), where the \(y\)-coordinate is always zero.
03

Conclusion

The \(y\)-coordinate of any point on the \(x\)-axis is consistently zero, as shown by the general form \((x, 0)\).

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Key Concepts

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

Understanding the Cartesian Coordinate System
The Cartesian Coordinate System is a way to pinpoint a location on a plane using an ordered pair of numbers. This system was named after the French mathematician Ren茅 Descartes. It provides a simple yet powerful way to describe an exact location in a two-dimensional space.
  • The plane consists of two number lines that intersect at a right angle. These are known as the x-axis and y-axis.
  • Where these axes intersect is called the origin, marked by the point \( (0,0) \), which is essentially the center of the system.
  • Points on the plane are described by coordinates, written as \( (x, y) \), where \( x \) represents the position along the x-axis and \( y \) along the y-axis.
This method of mapping points is essential for graphing equations, analyzing geometric shapes, and understanding spatial relationships.
The Role of the x-axis in Coordinate Geometry
The x-axis is a horizontal line that forms part of the two-dimensional Cartesian coordinate plane. Its main role is to represent the horizontal distance of any given point from the origin.
  • The x-axis is denoted by the equation \( y = 0 \), implying any point located on this axis has a y-coordinate of zero.
  • This axis helps in determining how far a point is to the right or left from the vertical y-axis.
  • Points can be anywhere along this horizontal line as long as the coordinate format appears as \( (x, 0) \).
Understanding the x-axis allows you to easily locate points strictly based on horizontal movement. This clarity is crucial when plotting graphs and solving engineering and physics problems.
Demystifying the y-coordinate
The y-coordinate is fundamental in determining a point's vertical position in the Cartesian coordinate system.
  • It specifies how far up or down a point is located from the x-axis on a plane.
  • Any point with only an x-coordinate along the x-axis will have a y-coordinate equal to zero. This is because the y-axis dictates vertical placement, and on the x-axis, this is always neutral.
  • The value of the y-coordinate helps define the point's distance from the horizontal x-axis and is crucial for graphing and spatial analysis.
Understanding the y-coordinate gives you insight into vertical alignment in graphs and aids in the precise calculation of intersections and slopes.

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Most popular questions from this chapter

Graph the indicated functions. The number of times \(S\) that a certain computer can perform a computation faster with a multiprocessor than with a uniprocessor is given by \(S=\frac{5 n}{4+n},\) where \(n\) is the number of processors. Plot \(S\) as a function of \(n\)

Solve the given problems. $$\text { For } f(x+2)=|x|, \text { find } f(0)$$

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Graph the indicated functions. A guideline of the maximum affordable monthly mortgage \(M\) on a home is \(M=0.25(I-E),\) where \(I\) is the homeowner's monthly income and \(E\) is the homeowner's monthly expenses. If \(E=\$ 600,\) graph \(M\) as a function of \(I\) for \(I=\$ 2000\) to \(I=\$ 10,000\)

Use the following table that gives the rate R of discharge from a tank of water as a function of the height \(H\) of water in the tank. For Exercises 19 and \(20,\) plot the graph and find the values from the graph. For Exercises 21 and \(22,\) find the indicated values by linear interpolation. $$\begin{array}{l|c|c|c|c|c|c|c}\text {Height}(\mathrm{ft}) & 0 & 1.0 & 2.0 & 4.0 & 6.0 & 8.0 & 12 \\ \hline \text {Rate }\left(\mathrm{ft}^{3} / \mathrm{s}\right) & 0 & 10 & 15 & 22 & 27 & 31 & 35 \end{array}$$. Find \(R\) for \(H=1.7 \mathrm{ft}\).
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