Problem 49 - Precalculus Essentials

Question

Graph equation in a rectangular coordinate system. $$y=-2$$

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

Draw a rectangular or Cartesian coordinate system. The x-axis (horizontal) and y-axis (vertical) intersect at a point called the origin. The positive direction for x is to the right and for y is upwards.

Step 2: Identify the Line Equation

The given equation is y = -2. This is a horizontal line, parallel to x-axis, crossing the y-axis at -2.

Step 3: Plot the Line

Using a ruler, draw a horizontal line across the graph that intersects the y-axis at -2. All the points along this line have y-coordinate equal to -2. This line represents the graph of the equation.

Key Concepts

Coordinate System

A coordinate system is a grid that helps us locate points on a plane. In a 2D space, the most commonly used coordinate system is the Cartesian plane, which consists of two main components:

• The x-axis, which is a horizontal line.
• The y-axis, which is a vertical line.

These two lines intersect at a point called the origin, noted as (0, 0). The plane is divided into four quadrants by these axes. Each point in this system is defined by a pair of numbers, \(x, y\), called coordinates. The x value shows the point's distance from the y-axis, while the y value shows its distance from the x-axis. Together, they allow us to pinpoint exact locations on the graph. Understanding the coordinate system is crucial for plotting and graphing equations effectively.

Horizontal Line

A horizontal line on a graph runs left to right, parallel to the x-axis. In the equation form, a horizontal line can be represented as \(y = c\), where \(c\) is a constant. This equation means that no matter what value x takes, y remains the same.
For instance, in the equation \(y = -2\), every point on this line has a y-coordinate of -2.

• The line cuts across the graph, intersecting the y-axis at -2.
• It does not intersect the x-axis because its y-value does not change with x.

Horizontal lines are unique in that their slope is zero, indicating that there is no vertical change when moving along the line. This is important to know when dealing with linear equations and graph interpretations.

Plotting Points

Plotting points on a graph involves locating each point on the coordinate plane using their respective coordinates \(x, y\). Here’s how you can plot points effectively:

• Start at the origin (0,0).
• Move horizontally to the x-coordinate value.
• Then, move vertically to reach the y-coordinate value.

Once you reach the exact spot, you mark it with a dot. To graph a line like \(y = -2\), you need several points that satisfy the equation. However, since \(y = -2\) is a horizontal line, any points have the same y-coordinate value (-2).
For example, some points \( (1, -2), (0, -2), (-1, -2) \) can be plotted anywhere along this line. Thus, once a few of these points are plotted, they can be connected to show the full extent of the line across the graph.