Problem 48 - Algebra 1

Question

Use a table of values to graph the equation. \(x=0\)

Step-by-Step Solution

Answer

The graph of the equation \(x=0\) is a vertical line passing through the points (0,-2), (0,0), and (0,2) on the y-axis.

Step 1: Tabulating values

Start by creating a table with two columns, one for 'x' and another for 'y'. At least three points are needed in order to define a line. Since the equation is \(x=0\), all x-coordinates will be zero. Now, you can pick several arbitrary y-values, let's say -2, 0, and 2.

Step 2: Plotting the points

The three points from the table of values are (0,-2), (0,0), and (0,2). Plot these points on a coordinate plane. Remember, each point has the form (x,y).

Step 3: Drawing the line

Now that you have plotted the points, connect them using a straight line. This line is the graph of the equation \(x=0\) and it should be vertical, passing through the y-axis.

Key Concepts

Table of Values

Creating a table of values is a fundamental method for graphing linear equations. It involves choosing a set of input values, computing the output values using the equation, and then plotting these as points on a coordinate plane. For example, to graph the equation \(x=0\), we begin by constructing a table with two columns for \(x\) and \(y\). Since \(x\) is constant at 0, we can choose any values for \(y\).

Let's pick -2, 0, and 2 for our \(y\)-values. We then fill in our table:

• \((x,y) = (0,-2)\)
• \((x,y) = (0, 0)\)
• \((x,y) = (0, 2)\)

The chosen points provide a clear visual representation when plotted, making it easier to understand the resulting graph's shape and position.

Coordinate Plane

The coordinate plane is an essential tool for visualizing equations and understanding their geometrical implications. It consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), which intersect at the origin (0,0). To graph the equation from our table of values, each point, represented as \( (x,y) \), is located on the plane. For the equation \(x=0\), the points are directly on the y-axis, showcasing the concept of a vertical line.

Each axis of the plane represents a different dimension, allowing the plotting of points in two-dimensional space. By plotting and connecting the points from our table, we give a visual dimension to the equation, aiding comprehension and analysis.

Vertical Line Equation

A vertical line equation such as \(x=0\) indicates that for all points on the line, the x-coordinate remains the same—in this case, 0. This equation doesn't involve \(y\), meaning \(y\) can take any value, resulting in a line that extends infinitely in both the positive and negative directions of the y-axis.

A vertical line has some unique properties:

• It never intersects the x-axis (except at infinity).
• It isn't represented by the typical slope-intercept form \(y=mx+b\) since its slope is undefined.
• Its equation is always in the form of \(x=c\), where \(c\) is the x-coordinate of all points on the line.

Understanding that an equation in the form of \(x=c\) represents a vertical line is pivotal for graphing such lines without confusion, as well as for identifying their slopes and points of intersection with other lines or curves on the coordinate plane.