Problem 21 - Algebra A Combined Function

Question

Write an equation of each line. See Examples 3 and \(4 .\) Horizontal; through (-3,1)

Step-by-Step Solution

Answer

The equation of the horizontal line is \( y = 1 \).

Step 1: Understand the properties of a horizontal line

A horizontal line has a constant y-value for all x-values, meaning the slope of a horizontal line is 0.

Step 2: Identify the y-coordinate of the given point

The line passes through the point (-3, 1). In a horizontal line, the y-coordinate stays the same for all points on the line. Thus, the y-value is 1.

Step 3: Write the equation for the horizontal line

Since a horizontal line maintains the same y-value for all x-values, the equation of the line is simply given by setting y equal to that constant value. Therefore, the equation is: \( y = 1 \).

Key Concepts

Horizontal Line

A horizontal line is one of the simplest types of lines you can find on a graph. It is completely flat and runs parallel to the x-axis on the coordinate plane. Because it is flat, every point on a horizontal line shares the same y-coordinate.
To illustrate this, let’s consider the horizontal line that passes through the point r(-3, 1). Regardless of what x-value you choose along this line, the y-value will always remain 1. This characteristic is crucial in understanding and writing the equation of a horizontal line.
Unlike other types of lines that might rise or fall, horizontal lines do not change in y-position as you move from left to right or right to left across the graph. This results in their unique property: the slope of a horizontal line is always zero.

Slope

The slope of a line describes its steepness and direction. It is typically calculated by dividing the change in the y-coordinates by the change in the x-coordinates between two points on the line.
For a horizontal line, however, this calculation behaves differently.

• The change in y is always zero, because a horizontal line doesn’t rise or fall.
• The change in x can be any non-zero value, but since it is divided by zero in the change of y, this makes the slope of the line 0.

For any horizontal line, the lack of vertical change ensures that the slope will always be zero. This concept is perfectly illustrated with our previous example of the line through (-3, 1), where calculating the slope confirms it as zero. This contributes to understanding why the equation of such a line is always in the form of \(y = k\), where \(k\) is the constant y-value for that line.

Coordinate System

The coordinate system is a fundamental mathematical concept used to locate points on a plane. Comprised of two axes, the horizontal axis is known as the x-axis, and the vertical one is the y-axis. Where the axes intersect is called the origin (0, 0).
When we talk about plotting a line, like our horizontal line through (-3, 1), we use these axes to determine the line’s location.
Each point on the plane is expressed as \((x, y)\), with x indicating the horizontal position and y indicating the vertical position. For a horizontal line, such as \(y = 1\), the y-value is constant, cutting across different x-values at the same height.
This consistent height is a visual representation of the coordinate system's ability to show different paths across the plane, illustrating both straight lines and different curves. The coordinate system is essential for understanding how horizontal lines behave and how to write their equations.