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Chapter 3: Problem 57

Graph each equation. $$x=0$$

Short Answer

Expert verified
The graph of the equation \(x = 0\) is a vertical line passing through the origin (0,0).

Step by step solution

01

Identify the equation form

The equation falls under the category of vertical lines in the format of \(x = k\), where \(k\) is a constant. In this case, \(k\) equals to 0.
02

Identify points

For all points on this line, the \(x\)-value is 0. The \(y\)-value can be anything. So we pick a few \(y\)-values, and the corresponding \(x\)-value is always 0. Here are the coordinates of a few points: (0, -2), (0, 0), (0, 2).
03

Draw the graph

Plot these points on a graph and draw a vertical line through them.

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Key Concepts

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

Vertical Lines
When we talk about vertical lines in mathematics, we focus on equations of the form \(x = k\), where \(k\) is a constant value. This simply means that every point on the line has the same \(x\)-coordinate, and it doesn't matter what the \(y\)-coordinate is. For example, the equation \(x = 0\) represents a vertical line on the coordinate plane that passes through the origin and extends infinitely in both the positive and negative \(y\)-direction. Vertical lines have unique properties:
  • They are always parallel to the \(y\)-axis.
  • They have an undefined slope because they go straight up and down.
  • The \(x\)-coordinate remains constant, while the \(y\)-coordinate can take any value.
To graph a vertical line, you simply need to know the \(x\)-coordinate, which remains constant, and understand that the line will cut through this point on the \(x\)-axis.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graph points, lines, and curves. It consists of two perpendicular number lines: the horizontal \(x\)-axis and the vertical \(y\)-axis. These axes intersect at a point called the origin, which has the coordinates \((0,0)\). The coordinate plane is divided into four quadrants:
  • Quadrant I: where both \(x\) and \(y\) are positive.
  • Quadrant II: where \(x\) is negative, and \(y\) is positive.
  • Quadrant III: where both \(x\) and \(y\) are negative.
  • Quadrant IV: where \(x\) is positive, and \(y\) is negative.
Each point on the coordinate plane is identified by an ordered pair \((x, y)\). The \(x\)-value tells you how far to move horizontally from the origin, and the \(y\)-value tells you how far to move vertically.
Plotting Points
Plotting points on the coordinate plane is a fundamental skill for graphing linear equations. Each point is represented by an ordered pair \((x, y)\) that shows its position relative to the origin. Plotting points involves just a few simple steps:
  • Start at the origin \((0,0)\).
  • Move horizontally to the \(x\)-coordinate. If \(x\) is positive, move to the right; if it's negative, move to the left.
  • From this new position, move vertically to the \(y\)-coordinate. If \(y\) is positive, move up; if it's negative, move down.
Once you have reached the correct location, mark the point on the graph. For example, the points \((0, -2)\), \((0, 0)\), and \((0, 2)\) lie on the vertical line described by the equation \(x = 0\). Plotting several points accurately allows you to visualize and draw straight lines on the graph.

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Most popular questions from this chapter

If two lines are perpendicular, describe the relationship between their slopes.

Will help you prepare for the material covered in the next section. Remember that a solution of an equation in two variables is an ordered pair. Let \(x=0\) and find a solution of \(3 x-4 y=24\)

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$2 x+y=4$$

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y-30.0=0.265(x-10)$$

A college campus goes beyond the standards described in Exercise \(49 .\) All wheelchair ramps on campus are designed so that every vertical rise of 1 foot is accompanied by a horizontal run of 14 feet. What is the grade of such a ramp? Round to the nearest tenth of a percent.
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