Question

True or False The slope of a vertical line is zero. Justify your answer.

Short answer

The statement is False. The slope of a vertical line is not zero but undefined because the change in x (run) for a vertical line is zero, and dividing by zero causes the slope to be undefined.

Step-by-step solution

Understand the definition of slope

The slope of a line is the rise (change in y) over the run (change in x) between any two points on a line. It is usually represented by the letter 'm'.

Recognize the characteristics of a vertical line

A vertical line goes straight up and down, which means there's a lot of change in the y direction but no change in the x direction.

Apply the definition to a vertical line

Since a vertical line has no change in the x-direction (run), and the denominator of the slope cannot be zero, the slope of a vertical line is undefined, not zero.

Key concepts

Slope Definition

When we talk about the slope of a line in math, we're referring to a measurement that represents how steep a line is. This is crucial in understanding various concepts in algebra and geometry.

The slope is calculated by finding the ratio of the vertical change to the horizontal change between two points on a line. This ratio is typically represented by the letter 'm.' It's important to note that the slope can be positive, negative, zero, or undefined depending on the direction and steepness of the line.

For instance, a positive slope means the line is going uphill from left to right, while a negative slope indicates a downhill trend. When the slope is zero, the line is flat—it has no incline at all. However, if the slope is undefined, that tells us that the line is vertical, which brings us to our next key concept.

Rise Over Run

The expression 'rise over run' is essentially a straightforward way to remember how to calculate the slope of a line. The 'rise' represents the vertical change, and the 'run' is the horizontal change between any two points on a line.

Let's break it down: suppose you're on a hike. The 'rise' would be the altitude you gain (or lose) as you walk, and the 'run' would be the distance you travel forward. Translated into a graph, 'rise' corresponds to the change in the y-coordinate (up or down), and 'run' correlates to the change in the x-coordinate (left or right).

By dividing the rise by the run, you get the slope of the line, giving you a sense of how steep your 'mathematical hike' is. It’s a simple yet robust concept enabling us to graphically represent and analyze linear relationships.

Undefined Slope

In mathematics, an undefined slope might sound like an oxymoron. After all, don't we always seek a definitive answer? Yet, that's precisely what happens when we try to find the slope of a vertical line. As we have learned, the slope is the ratio of the 'rise' to the 'run,' but what if our 'run,' the horizontal change, is zero?

This gives us a division by zero situation, which in mathematics is undefined. Simply put, you cannot divide anything by zero; it's a fundamental rule. Therefore, when a line is vertical, the x-value for each point on the line doesn't change, resulting in a zero run and consequently, an undefined slope.

This concept is critical when dealing with equations of vertical lines, where we cannot describe the line in terms of 'y = mx + b,' because 'm' cannot be determined. Therefore, vertical lines stand in their unique category when it comes to slope.

Vertical Line Characteristics

Vertical lines are one of the fundamental shapes in geometry, and they have unique properties that set them apart from other types of lines.

First and foremost, vertical lines are defined as lines that go straight up and down; they run parallel to the y-axis on a Cartesian plane. This means that for every point on a vertical line, the x-coordinate is the same while the y-coordinate can change.

Because of this consistent x-value, we can define vertical lines by an equation such as 'x = a constant.' This equation tells us that no matter what value 'y' takes, 'x' will always be the same.

Vertical lines are essential in understanding cartesian coordinate systems and are particularly noteworthy because of their undefined slope, distinguishing them from all lines with a defined slope.