Problem 52 - College Algebra

Question

Graph each equation in the rectangular coordinate system. $$x=0$$

Step-by-Step Solution

Answer

The graph of \(x=0\) is a vertical line along the y-axis, stretching from negative infinity to positive infinity.

Step 1: Identify the constants in the equation

In the equation \(x=0\), '0' is the constant. This is where the line will cross the x-axis.

Step 2: Plot the line

Since \(x=0\), the x-coordinate of every point on the line is 0. Therefore, the line passes through the origin (0,0) and remains on the y-axis

Key Concepts

Rectangular Coordinate System

Imagine a large grid on a piece of paper, with a horizontal line and a vertical line crossing at the center. This setup is known as the rectangular coordinate system or Cartesian plane. It is made up of two perpendicular number lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. Where these two lines intersect is called the origin and is denoted by (0,0).

The space is divided into four sections called quadrants. From top to bottom and left to right, they are numbered counterclockwise: the upper right quadrant is the first, the upper left is the second, the lower left is the third, and the lower right is the fourth.

• Quadrant I: (+x, +y)
• Quadrant II: (-x, +y)
• Quadrant III: (-x, -y)
• Quadrant IV: (+x, -y)

This system helps us locate points on a plane. Knowing how to navigate the rectangular coordinate system is the first step to mastering graphing. Each point is represented by an ordered pair \((x,y)\), where \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance.

Plotting Points

Plotting points is similar to finding treasure on a map. When you have an ordered pair like \((3, 2)\), this means that you start at the origin, move 3 units to the right along the x-axis, and then 2 units up along the y-axis. This pinpoints your exact location, just like X marks the spot.

Each point you plot has as its first number the x-coordinate, which tells you how far left or right to go from the origin, and as its second number, the y-coordinate, which tells you how far up or down to travel on the plane.

• If \(x\) is positive, go right; if negative, go left.
• If \(y\) is positive, go up; if negative, go down.

Practice plotting different points to get comfortable with moving around the plane. Over time, the system will become second nature. The more points you plot, the clearer any graph becomes!

Vertical Lines

Vertical lines in the rectangular coordinate system can seem strange at first, but they're quite simple! When you see an equation like \(x = 0\), this means the line is vertical. All points on this line have the same x-coordinate of 0.

This creates a vertical line passing through the y-axis at the origin. Unlike slopes you've seen in other lines, a vertical line has an undefined slope because it would require division by zero, which isn’t possible.

• Vertical lines are parallel to the y-axis, and the equation \(x = a\) means all points are vertically aligned at \(x = a\).
• The x-coordinate is constant while the y-coordinate can be any value.

Once you visualize the concept of vertical lines, applying it becomes much easier in graphing linear equations. Whether you're graphing \(x = 0\) or \(x = -3\), just remember that all those points line up vertically, making them easy to plot!