Problem 44 - Algebra 1

Question

Use a table of values to graph the equation. \(x=9\)

Step-by-Step Solution

Answer

The graph of the equation \(x=9\) is a vertical line that crosses the x-axis at 9 and is unaffected by the y-coordinate on the plane.

Step 1: Creating a Table of Values for y and x

Make a table of values and choose any 3 (or more) numbers for \(y\), let's say -5, 0, and 5. Now, regardless of what \(y\) is, \(x\) is always 9. Therefore, the three points we get from these values are (9,-5), (9, 0), and (9, 5).

Step 2: Graph the Equation

Plot the points (9,-5), (9, 0), and (9, 5) on a graph. Once plotted, you'll notice that they all line up perfectly. Draw a straight line through these points. This is the line \(x=9\).

Step 3: Verifying the Graph

The line on the graph should be vertical and pass through the point (9,0) on the x-axis. Every point on this line will have an x-coordinate of 9 regardless of the y-value. Thereby verifying the graph.

Key Concepts

Using a Table of Values to Assist Graphing

One of the fundamental ways to start plotting an equation is by utilizing a table of values. This method involves picking several values for one variable and determining the corresponding values for the other variable based on the equation. Let's take a closer look at how this works.

• First, decide which variable(s) in the equation to assign arbitrary values to. In many instances, this will be the dependent variable, but in our exercise, it was the variable \(y\).
• By choosing, say -5, 0, and 5 for \(y\), we've set our stage. Next, calculate the corresponding \(x\) values using the equation provided, \(x = 9\).
• Because \(x\) is simply 9 regardless of \(y\), we quickly create ordered pairs or points like (9, -5), (9, 0), and (9, 5).

This approach allows us to see what the graph might look like even before plotting, enabling a smoother transition into graphing.

Understanding Vertical Lines on a Graph

A vertical line on a graph represents a constant value of \(x\). Unlike horizontal lines, where \(y\) is constant, vertical lines have a unique property that often surprises students.

• Initially, it's essential to recognize that all points on a vertical line share the same \(x\)-coordinate. In our case, for the line \(x = 9\), every point such as (9, -3), (9, 0), (9, 4), and so forth have \(x = 9\).
• This means they span vertically on a graph, effectively creating a line parallel to the \(y\)-axis.
• In graph theory terms, this line is not a function, as it doesn't pass the vertical line test (one \(x\) value can have multiple \(y\) values here).

A deep understanding of vertical lines will help you with more complex graph interpretations, as they hold unique attributes compared to other line equations.

Navigating the Coordinate Plane

The coordinate plane is fundamental to understanding graphing techniques, as it provides a framework for positioning points defined by ordered pairs \((x, y)\).

• The plane is divided into four quadrants, each representing a combination of positive and negative values for \(x\) and \(y\).
• The center point, called the origin, is where \(x = 0\) and \(y = 0\). This is the reference point for plotting other locations on the plane.
• When graphing, it's crucial to accurately plot points by locating their \(x\)-coordinate first, then moving vertically to their corresponding \(y\)-coordinate.

Mastering the coordinate plane is vital for translating equations into comprehensible visual data, such as the vertical line \(x = 9\), to help visualize mathematical relationships clearly.