Problem 50 - College Algebra

Question

Graph each equation in a rectangular coordinate system. \(y=4\)

Step-by-Step Solution

Answer

The graph of the equation \(y=4\) is a horizontal line which passes through the point (0,4) on the y-axis.

Step 1: Understand the Equation

The equation \(y=4\) is a horizontal line passing through the point (0,4) on the y-axis. Regardless of the value of \(x\), \(y\) will always be 4.

Step 2: Draw the Coordinate System

Firstly, a rectangular coordinate system is needed, which must consist of a horizontal x-axis and a vertical y-axis. Their intersection point is called the origin.

Step 3: Plot the Line

The line \(y=4\) will be a horizontal straight line crossing the y-axis at point (0,4). So, mark point (0,4) on the y-axis then draw a horizontal line through it.

Key Concepts

Rectangular Coordinate System

A rectangular coordinate system is a fundamental concept in algebra and geometry, often used for graphing equations. It is composed of two number lines that intersect at a right angle:

• the horizontal line is known as the x-axis
• the vertical line is known as the y-axis
• their point of intersection is termed the origin, marked as (0,0)

The plane formed by these two axes is called the coordinate plane. Every point on this plane can be expressed as a coordinate pair, \(x, y\), where x is the horizontal position, and y is the vertical position. This system is crucial for graphing lines and curves, representing various mathematical relationships visually.

Horizontal Line

A horizontal line is a straight line that runs from left to right across the rectangular coordinate system without ever changing its vertical position. In the equation form, it is typically written as \(y = c\), where c is a constant. Here are some features of a horizontal line:

• For all points on the line, the y-coordinate remains constant, indicating a uniform height. In the case of the equation y=4, all points maintain a y-value of 4.
• This type of line is parallel to the x-axis.
• It does not intersect the x-axis unless the constant c equals zero, which would mean the line coincides with the x-axis.

Understanding the concept of a horizontal line helps in graphing equations where only the y-value is fixed.

Y-Axis

The y-axis is the vertical line in the rectangular coordinate system. It plays a pivotal role, as it represents the dependent variable when graphing equations. Here are some important characteristics:

• It is perpendicular to the x-axis.
• The y-intercept of a line is the point where the line crosses the y-axis. For y=4, this is at (0,4).
• The y-axis helps to determine how a variable behaves as it moves up and down across the coordinate system.
• It is typically labeled "y" and coordinates are marked vertically on it.

In any equation graph, knowing how the line interacts with the y-axis is crucial for accurate representation and understanding of the data or values involved.

Plotting Points

Plotting points is a method used to visualize mathematical equations and data on the rectangular coordinate system. Here's how to plot a point:

• Start at the origin point (0,0).
• Move horizontally to the x-coordinate value.
• From there, move vertically to the y-coordinate value.
• Mark this position as the point on the graph.

To graph a horizontal line like y=4, you specifically focus on placing points where y equals 4, such as (1,4), (2,4), or (-3,4). Once two or more points are plotted, the points can be connected to form the line. Plotting provides a visual aid, making comprehension of linear equations much simpler for learners.