Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$x=0$$

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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).

Step 3: Draw the graph

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

Key Concepts

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.