Problem 48 - Algebra A Combined Function

Question

Graph each equation. See Sections 3.2 and 3.3. $$ x=-2 $$

Step-by-Step Solution

Answer

It's a vertical line at \( x = -2 \).

Step 1: Identify the Type of Equation

The equation given is \( x = -2 \). This is a vertical line equation because it is formatted as \( x = a \), where \( a \) is a constant.

Step 2: Locate the X-Coordinate

Since the equation is \( x = -2 \), every point on this line will have an \( x \) coordinate of \(-2\). This means the line is located at \( x = -2 \) on the coordinate plane.

Step 3: Draw the Vertical Line

Draw a straight line parallel to the y-axis passing through the point \( (-2, 0) \). Extend this line up and down across the grid.

Step 4: Verify Consistency

Ensure that the line covers all points \( (-2, y) \) for various values of \( y \). This confirms that the line is indeed a vertical line where \( x = -2 \) holds true for all y-values.

Key Concepts

Vertical lines

Vertical lines are a unique type of line on the coordinate plane. Unlike horizontal lines, which run parallel to the x-axis, vertical lines run parallel to the y-axis. This means they never tilt in any direction and always maintain a perfectly up-and-down orientation.
When identifying vertical lines in equations, you'll notice they have a distinct form: \( x = a \). Here, \( a \) represents a constant value. This constant specifies the exact position on the x-axis where the line will run through.
Vertical lines have some notable characteristics:

• Every point on a vertical line has the same x-coordinate, the value of \( a \).
• They do not have a defined slope, often described as "undefined."
• A vertical line could be any length, always spanning infinitely in both directions along the y-axis.

Understanding these features can help you quickly identify and graph vertical lines on a coordinate plane.

Coordinate plane

The coordinate plane is a two-dimensional surface defined by two axes: the x-axis and the y-axis. The intersection of these two axes is known as the origin, represented by the point \( (0, 0) \).
This plane is crucial for graphing various types of equations, including vertical lines. It allows us to visually represent relationships between two variables. Each point on this plane is marked by a pair of numerical values \( (x, y) \), known as coordinates. The x-coordinate indicates a point's horizontal position, while the y-coordinate shows its vertical position.
When dealing with vertical lines, you focus on fixing the x-coordinate to a particular value. For instance, with \( x = -2 \), you draw a straight line through all points where x remains constant at -2, regardless of the y-value. This ability to visually map out equations on the coordinate plane aids in comprehending algebraic relationships.

Equations of the form x=a

Equations of the form \( x = a \) are simplistic but powerful. These equations specify a vertical line on the coordinate plane, where every point along the line has the same x-coordinate that equals \( a \).
For example, if we consider \( x = -2 \), every point along this line will have an x-value of -2. The y-value can be any number, allowing the line to extend infinitely in the vertical direction. This type of equation emphasizes a single, fixed x-coordinate while leaving the y-coordinate unconstrained.
A few key insights about equations \( x = a \) include:

• They create a vertical line which is perpendicular to the x-axis.
• Such equations often arise when looking at boundaries or constraints in mathematical problems.

By understanding these principles, you can quickly graph and analyze vertical lines on the coordinate plane, illustrating their simplicity and versatility.