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Chapter 3: Problem 19

Answer the given questions. Where are all points whose \(x\) -coordinates are \(1 ?\)

Short Answer

Expert verified
All points are on the vertical line \( x = 1 \).

Step by step solution

01

Understand the Problem Statement

The problem asks us to find the location of all points where the x-coordinates are equal to 1. This means for any point on this specific line, the x-coordinate will always be 1, regardless of the y-coordinate.
02

Identify the Equation

Since the x-coordinate is fixed at 1 for all points, the equation that represents this set of points is a simple vertical line given by the equation \( x = 1 \). This signifies that no matter what the y-coordinate is, the x-coordinate remains constant as 1.
03

Visualize the Solution

Imagine a Cartesian coordinate plane. The line \( x = 1 \) is a vertical line that crosses the x-axis at 1 and extends infinitely up and down parallel to the y-axis. Points on this line have coordinates such as (1,0), (1,2), (1,-3), etc.

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Key Concepts

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

Understanding the Vertical Line Equation
In coordinate geometry, a vertical line equation represents a line that runs up and down parallel to the y-axis on a Cartesian coordinate plane. Unlike horizontal lines, where the y-coordinate remains constant, vertical lines have a constant x-coordinate. The general form of a vertical line is represented by the equation \( x = a \), where \( a \) is a fixed number.For instance, if we鈥檙e given \( x = 1 \), this equation tells us that for any point on this line, the x-coordinate remains at 1, no matter what the y-coordinate value is. This is why we often see it extending infinitely upwards and downwards without ever deviating horizontally.
  • Vertical lines are straight and do not slant, as they only extend in the vertical direction.
  • The equation like \( x = 1 \) suggests a fixed position along the x-axis and all possible variations in the y-coordinates are acceptable.
  • Such equations are straightforward and require no calculation of slope. They simply define the x-position of the entire line.
Exploring the Cartesian Coordinate Plane
The Cartesian coordinate plane is an essential concept in coordinate geometry. It is a two-dimensional plane divided into four quadrants by a horizontal axis (x-axis) and a vertical axis (y-axis). These axes intersect at the origin, denoted as \((0, 0)\).Each point on the plane can be identified by a pair of numbers (coordinates), representing the horizontal and vertical distances from the origin.
  • The x-coordinate indicates how far to move horizontally from the origin, while the y-coordinate determines the vertical movement.
  • This plane helps to visually represent mathematical equations and geometric properties like lines and shapes.
  • Understanding the Cartesian plane is fundamental for plotting equations, as each point has an exact position defined by its coordinates.
Imagine the line \(x = 1\) drawn on this plane. It aligns parallel to the y-axis, passing through the point (1,0) and carrying through all points with x-coordinate 1, such as (1, 1), (1, -2). It is a prime example of how the plane accommodates vertical line equations.
The Role of the X-Coordinate
The x-coordinate plays a crucial role in defining the position of a point on the Cartesian coordinate plane. It specifies the horizontal distance from the y-axis. Understanding how the x-coordinate functions is key to grasping many geometric concepts.
  • The x-coordinate is the first number in an ordered pair, guiding how far right or left a point lies from the y-axis.
  • In vertical lines, where equations are of the form \( x = a \), the x-coordinate consistently returns the same value across all corresponding points on that line.
  • A fixed x-coordinate means the line will cross the x-axis at exactly one point, with all other points in its path vertically aligned with this intersection point.
Whether we are working with equations or visualizing geometric problems, the x-coordinate is a fundamental element that determines the line or point's position along the horizontal axis, dictating its relationship with other mathematical components.

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Most popular questions from this chapter

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\)

Solve the given problems. A jet is traveling directly between Calgary, Alberta, and Portland, Oregon, which are \(550 \mathrm{mi}\) apart. If the jet is \(x\) mi from Calgary and \(y\) mi from Portland, find the domain of \(y=f(x)\)

Graph the indicated functions. For a certain model of truck, its resale value \(V\) (in dollars) as a function of its mileage \(m\) is \(V=50,000-0.2 m .\) Plot \(V\) as a function of \(m\) for \(m \leq 100,000\) mi.

Graph the given functions. $$y=\sqrt{x^{2}-16}$$

Graph the indicated functions. An airline requires that any carry-on bag has total dimensions (length \(+\) width \(+\) height) that do not exceed 45 in. For a carryon that just meets this requirement and \(h\) as a length that is twice the width, express the volume \(V\) as a function of the width \(w\). Draw the graph of \(V=f(w)\)
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