Problem 54 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The graph of the equation \(x=0\) is a vertical line passing through the origin in the Cartesian coordinate system.

Step 1: Understand the equation

The given equation is \(x = 0\). It means for all values of \(y\), the \(x\) coordinate will remain zero. This represents a vertical line going through the \(x\) origin.

Step 2: Plot the points

On a Cartesian grid, plot a few points where x is always zero. For example, you could use the following points: (0, -2), (0, 0), and (0, 2).

Step 3: Sketch the Line

After plotting the points, connect them using a straight line. This will create a vertical line passing through the origin (0,0).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, often referred to as the Cartesian coordinate system, is a method of plotting points on a plane using two intersecting number lines called the axes. The horizontal line is known as the x-axis, while the vertical line is known as the y-axis. These axes divide the plane into four quadrants, with each quadrant allowing us to accurately locate any point using a pair of numerical coordinates.

• The x-coordinate tells us the horizontal position of the point.
• The y-coordinate tells us the vertical position of the point.

When graphing equations, it's crucial to understand these axes as they provide the framework within which points are placed. The intersection of the x-axis and y-axis is called the origin, denoted as (0,0). Each point on the plane can be described by an ordered pair ewline \(x, y\).
Whether you're graphing complex lines or simple straight lines, familiarizing yourself with this system will help you accurately represent equations visually.

Vertical Line

A vertical line is a straight line that runs up and down parallel to the y-axis of a graph. In graphing terms, a vertical line means that the x-coordinate remains constant while the y-coordinate can take any value. Therefore, it can be expressed in an equation form as \(x = c\), where \(c\) is a constant number.
When a line is described by an equation like \(x = 0\), it means that for every value of y, the value of x remains at zero. This line will always align perfectly with the y-axis, reflecting the concept of a "vertical line."
Vertical lines are unique in that they cannot be described using a slope, as they have an undefined slope. This is because the change in y can be anything, but there is no change in x, making it impossible to define a slope using the traditional formula \(m = \frac{\Delta y}{\Delta x}\).Listing out the properties of a vertical line:

• Constant x-coordinate value.
• Undefined slope.
• Infinite length extending through all values of y.

Cartesian Grid

A Cartesian grid is essentially the graph paper background where we plot points according to their coordinates in the rectangular coordinate system. This grid consists of numerous evenly spaced vertical and horizontal lines that cross each other to form squares or rectangles.
It's important to use a Cartesian grid when plotting equations because:

• It allows for precise placement of points.
• It visually assists in identifying the slope and intercepts of lines.
• It provides an orderly structure to help check the accuracy of graphed points.

When working with and plotting points on a Cartesian grid, you start at the origin \(0,0\) and move horizontally and vertically to plot each point. This grid is a staple tool in graphing as it simplifies the process of visualizing mathematical equations and their solutions. It helps in clearly seeing how changes in equations influence the shape and position of their graphs on the plane.