Problem 49 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The graph of the equation \( y = -2 \) is a horizontal line parallel to the x-axis crossing the y-axis at -2.

Step 1: Set up the coordinate system

First, set up a Cartesian coordinate system. Draw two perpendicular lines intersecting at a point called the origin. Label the horizontal line as \( x \) and the vertical line as \( y \). Ensure to mark the point -2 on the \( y \) line.

Step 2: Identify the equation

Observe the equation to be graphed. Here, the equation \( y = -2 \) means that no matter what the value of \( x \) is, \( y \) will always be -2.

Step 3: Graph the line

To graph the line, draw a straight horizontal line passing through the point where \( y \) equals -2. As a result, you should have a straight horizontal line parallel to the x-axis, passing through y=-2.

Key Concepts

Understanding the Coordinate System

The coordinate system is a framework we use to locate points in a plane. It consists of two number lines, the x-axis (horizontal) and the y-axis (vertical), that intersect at a point called the origin, denoted as (0, 0). Every point in this system can be described by an ordered pair (x, y). This means for any point, you have a specific x-value that tells you how far along it is from the origin horizontally, and a y-value that tells you how far up or down it is vertically.

To set up a coordinate system yourself:

• Draw two perpendicular lines on paper.
• Label the horizontal line as the x-axis.
• Label the vertical line as the y-axis.
• Mark the intersection as the origin (0,0).

This system helps us easily graph equations and visualize the relationships between variables.

Graphing Horizontal Lines

A horizontal line in a graph has the same y-value for all x-values. This means it runs parallel to the x-axis. For instance, in the equation \( y = -2 \), no matter what x-values you choose, the y-value always remains at -2.

To graph a horizontal line:

• Identify the constant y-value, which in this example is -2.
• Draw a line that runs parallel to the x-axis, passing through the point where y is -2.

Horizontal lines are unique because their slope is 0. This signifies there is no vertical change as we move along the line.

Exploring the Cartesian Plane

The Cartesian plane, also known simply as the coordinate plane, is the area created by the intersection of the x-axis and y-axis. This plane is divided into four quadrants, each containing ordered pairs that describe specific locations on the plane.

The quadrants are typically labeled as:

• Quadrant I: Here, both x and y values are positive.
• Quadrant II: x-values are negative, and y-values are positive.
• Quadrant III: Both x and y values are negative.
• Quadrant IV: x-values are positive, and y-values are negative.

Understanding this layout helps us not only in graphing functions but also in comprehending their behaviors and relationships in different parts of the plane.