Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each equation. $$y-2.5=0$$

Step-by-Step Solution

Answer

The graph of \(y - 2.5 = 0\) is a horizontal line plotted at y = 2.5 for all x-values.

Step 1: Write the equation in general form

Rewrite the equation in the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Since there are no terms with x in the equation, it can be rewritten as \(y = 2.5\). This implies that the slope, m, is 0 and the y-intercept, b, is 2.5.

Step 2: Interpret the result

Since the slope is 0 and the y-intercept is 2.5, the line to be plotted is a horizontal line running through the y-coordinates at 2.5, for all x-values.

Step 3: Graph the line

To graph the line, identify the y-intercept at 2.5 and draw a straight, horizontal line across the graph at y = 2.5. This line will not slope upwards or downwards but lie flat, indicating that for any x-value, y will always be 2.5.

Key Concepts

Slope-Intercept Form

In mathematics, the slope-intercept form is a powerful tool for understanding and graphing linear equations. This form is expressed as \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept.
The slope \(m\) indicates the steepness of the line and its direction. However, in our given equation \(y = 2.5\), there is no \(x\) term, which means that \(m = 0\).
Hence, the line is horizontal, and you need not worry about rise or fall along the x-axis. The equation showcases how simple a line can be in its representation: a constant y-value that doesn't change.
Recognizing this form helps quickly identify how a line behaves, including its horizontal, vertical, or slanting nature. It also makes graphing lines fast and straightforward.

Horizontal Line Graph

A horizontal line graph is one of the simplest types of graphs you will encounter. It is a straight line drawn parallel to the x-axis without any incline or decline. In the equation \(y = 2.5\), the absence of an \(x\) term confirms this.
Horizontal lines can be easily identified by looking at their slope. When the slope \(m\) is 0, it indicates a constant y-value across all x-coordinates. This characteristic means that the line will remain flat and unchanging, no matter where you measure along the x-axis.

• Slope \(m = 0\): Indicates no change in height with change in x.
• Y-value is the same everywhere on the line.
• Run parallelly to the x-axis, appearing flat and steady.

Understanding horizontal lines helps not only in graphing but also in recognizing scenarios where change happens along just one axis.

Y-Intercept

The y-intercept is a concept intrinsic to understanding and plotting graphs. It represents the point where a line crosses the y-axis. In the slope-intercept form \(y = mx + b\), \(b\) denotes the y-intercept.
For our equation, \(y = 2.5\), the y-intercept is 2.5. This means that the line will touch the y-axis at the point (0, 2.5). It provides a starting spot on the graph when you begin plotting.

• The value of \(b\) directly shows where the line meets the y-axis.
• A higher \(b\) signifies a higher starting point on the y-axis.
• Allows visualization of lines, even without x-value calculations.

Knowing how to identify and use the y-intercept is a primary step in sketching any linear graph correctly and also explains the initial position of a line on a graph grid.