Problem 84 - Intermediate Algebra

Question

Graph each equation. $$x=-2$$

Step-by-Step Solution

Answer

Graph a vertical line at \( x = -2 \) on the coordinate plane.

Step 1: Understand the equation

The equation given is \( x = -2 \). This equation represents a vertical line on the coordinate plane where the value of \( x \) is always \( -2 \) for every value of \( y \).

Step 2: Set up the coordinate plane

Draw a pair of perpendicular axes on a graph: the horizontal axis is the x-axis and the vertical axis is the y-axis. Each axis should be marked with evenly spaced intervals in both positive and negative directions.

Step 3: Identify the line location

Since the equation is \( x = -2 \), locate the point where \( x \) is \( -2 \) on the x-axis. This point is directly to the left of the origin on the x-axis.

Step 4: Draw the vertical line

From the point \( x = -2 \) on the x-axis, draw a straight vertical line that extends infinitely in both upward and downward directions. This line should not intersect the x-axis again and should be parallel to the y-axis.

Step 5: Double-check the line's properties

Ensure that the vertical line is straight and that it does not bend. Verify that it crosses the x-axis only at \( x = -2 \) and is consistent for all values of \( y \).

Key Concepts

Vertical Lines

Vertical lines are a unique kind of line in geometry and algebra that run up and down across the coordinate plane. What makes vertical lines distinctive is that they maintain a constant x-coordinate while the y-coordinate can take any value. This kind of line results in an equation of the form \( x = a \), where \( a \) is a constant. In a vertical line, you will notice the following features:

• The line is perfectly straight and aligned with the y-axis.
• It does not slant or slither horizontally.
• The x-values are the same, producing a continuous, unbroken path vertically through any y-value.

For example, the equation \( x = -2 \) describes a vertical line where no matter what y-value you choose, the x-value of the point on this line is always -2.

Coordinate Plane

The coordinate plane is a foundational tool in graphing equations, serving as the stage where all graphing occurs. It is made up of two intersecting lines called axes that cross at right angles. These axes divide the plane into four quadrants. The system is also commonly referred to as the Cartesian plane. Here are some essential features:

• The horizontal line is known as the x-axis.
• The vertical line is known as the y-axis.
• The point where they intersect is called the origin, denoted as \((0, 0)\).
• Coordinates are often written as pairs \((x, y)\) that identify specific points on the plane.

It is important to become familiar with how to plot points and draw lines within this plane to understand graphing better.

X-axis

The x-axis is the horizontal component of the coordinate plane. It allows us to locate the position of various points along the horizontal direction. When engaging with graphing:

• The x-axis is like a number line that stretches both to the left and right of the origin.
• It is marked with numbers to help in locating positions.
• All points on the x-axis have a y-coordinate of zero, meaning they lie directly on the axis itself.

In practice, such as with a vertical line like \( x = -2 \), you would start by finding -2 on the x-axis and use this point to draw your line.

Y-axis

The y-axis complements the coordinate plane by providing a vertical scale. This axis aids in determining the vertical position of points. Just like the x-axis:

• The y-axis runs up and down, parallel to the direction of vertical lines.
• It is used to mark values that can change or vary, represented as the y-coordinate in point pairs.
• Points that reside exactly on the y-axis have an x-coordinate of zero.
• The y-axis helps in understanding how far above or below a point is relative to the origin.

In equation \( x = -2 \), even though the focus is on the x-value, the vertical alignment echoes the utility of the y-axis in showing the line's straight, up and down continuity.