Problem 25 - Elementary Algebra

Question

Graph the equations. $$ y=0 $$

Step-by-Step Solution

Answer

Answer: The equation $$y=0$$ represents a horizontal line passing through the origin (0,0). To graph it, locate the point (0,0) on the coordinate plane, draw a horizontal line passing through the point (0,0) and extending indefinitely to the left and right, and label the line with the equation $$y=0$$.

Step 1: Identify the line type

The given equation is $$y=0$$, which represents a horizontal line passing through the point (0,0).

Step 2: Plot the horizontal line

To graph this equation, follow these steps: 1. Locate the point (0,0) on the coordinate plane. 2. Since y = 0 for all x-values, draw a horizontal line passing through the point (0,0), extending to the left and right as far as desired. 3. Label the line with the equation $$y=0$$ to denote that the line represents all points where y has a value of zero. The graph of the equation $$y=0$$ is a horizontal line passing through the origin (0,0) and extending indefinitely to the left and right.

Key Concepts

Understanding the Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It is used to graph mathematical relationships and visualize concepts. Let's break it down into its core components:

• Axes: There are two axes intersecting at a right angle. The horizontal axis is called the x-axis and the vertical axis is the y-axis.
• Quadrants: These axes divide the plane into four sections, known as quadrants, which are numbered I to IV, usually in counterclockwise order starting from the upper right quadrant.
• Coordinates: A point on the plane is identified by a pair of numbers \(x, y\), known as coordinates, where x represents a position on the x-axis, and y a position on the y-axis.

The coordinate plane is essential for graphing linear equations, like the one given in the exercise, as it provides a framework to visually represent the relationships expressed by the equations.

What is a Horizontal Line?

A horizontal line is one where all the points on the line have the same y-value and extend indefinitely to the left and right on the coordinate plane. Here are some key characteristics of horizontal lines:

• Slope: A horizontal line has a slope of zero. This means it does not rise or fall as you move along the line; rather, it stays at a constant vertical position. In mathematical terms, this is because the change in y (rise) over the change in x (run) is zero.
• Equation Form: The equation of a horizontal line can be represented by \(y = k\), where \(k\) is a constant, meaning every point on the line has a y-coordinate of \(k\).
• Graphical Representation: On the coordinate plane, a horizontal line gives a visual representation of a situation where the dependent variable y remains the same, irrespective of any changes in x.

In the exercise, the equation \(y=0\) is such a line and it tells us to graph where the y-value is consistently zero across all points.

The Importance of the Origin

The origin is the point where the x-axis and y-axis intersect, denoted by the coordinates \(0, 0\). It serves as a central reference point on the coordinate plane. Why is the origin significant?

• Common Reference Point: It's used to define other points on the plane. For any point \(x, y\), the origin provides a baseline to determine the distance and direction from the center.
• Symmetry and Geometry: Many geometric properties and symmetry operations are centered at the origin. For example, reflections, rotations, and translations often use the origin as a focal point.
• Simplifies Calculations: Being at the center, calculations such as slope, distances, and midpoints start from or involve the origin.

In the exercise, the origin is pivotal since the horizontal line \(y=0\) passes through it, emphasizing that \(y\) is zero when \(x\) is any number, making it essential for plotting our graph correctly.