Problem 51 - Introductory Algebra for College Students

Question

Graph equation. \(x=2\)

Step-by-Step Solution

Answer

The graph of the equation \(x = 2\) is a straight, vertical line that passes through the point (2,0) on the x-axis and runs parallel to the y-axis.

Step 1: Understanding the equation

Firstly, acknowledge that the equation \(x = 2\) represents a vertical line. This line crosses the x-axis at x = 2.

Step 2: Setting up the graph

Spread a pair of perpendicular number lines, the vertical line called the y-axis and the horizontal is called the x-axis. Collectively, they are called coordinate axes.

Step 3: Plotting the line

Sketch a vertical line that passes through the point (2,0) on the x-axis. Since the equation is \(x = 2\), this line will not touch the y-axis, unless the value of y is 0. This line should be straight and it should go upwards and downwards indefinitely.

Key Concepts

Vertical Line Equation

The equation of a vertical line is one of the simplest forms of a linear equation you'll come across in algebra. It takes the form of \(x = a\), where \(a\) is the value at which the line crosses the x-axis. For example, the equation \(x = 2\) defines a vertical line that crosses the x-axis at the point \((2,0)\).

Each point on this line has 2 as its x-coordinate. The important thing to remember about vertical lines is that they do not have a y-coordinate in their equations, which means they go through all y-values. Remember, vertical lines have an undefined slope because their rise (change in y) over run (change in x) would require division by zero, which is not possible.

Coordinate Axes

The coordinate axes are fundamental for graphing lines and plotting points. They consist of two perpendicular lines: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, labeled as (0,0).

The x-axis typically represents the independent variable, while the y-axis represents the dependent variable in a function. When you graph an equation, the x-values determine the position of a point along the x-axis, and the y-values determine the position along the y-axis. When plotting a vertical line, you'll mainly be concerned with the x-axis since the line's x-value is constant.

Plotting Points on a Graph

Plotting points is crucial when creating a visual representation of an equation. To plot a point, you need an ordered pair, which is written in the form (x, y). Start at the origin of the coordinate axes for each point you plot. Move right if the x-value is positive and left if it's negative. Then, move up if the y-value is positive or down if it's negative.

When graphing the equation \(x = 2\), you only need to find points where the x-value is 2. These points could be (2,1), (2,0), (2,-1), etc. You'll mark these positions on the graph, and then draw the vertical line that connects these points, which extends infinitely up and down.

Algebraic Graphs

Algebraic graphs provide a visual representation of different equations. A graph might represent equations ranging from simple lines to complex curves and shapes. Linear equations, like the vertical line equation, are among the simplest to graph because they form straight lines.

When working on algebraic graphs, it's important to identify the type of equation you have—whether it's a linear equation, like \(y = mx + b\), a quadratic equation, like \(y = ax^2 + bx + c\), or in this case, the special instance of a vertical line, like \(x = a\). Understanding the general shape and position of the graph will help you plot it correctly and interpret its meaning in the context of the problem.