Question

Determine if a line with the following description has a positive slope, a negative slope, or an undefined slope. Vertical line

Short answer

The slope of a vertical line is undefined.

Step-by-step solution

Understanding The Concept

The slope of a line is calculated using the difference in the y-coordinates divided by difference in the x-coordinates.In the case of a vertical line, the x-coordinates are the same, leading us to an undefined slope as any number divided by zero is undefined.

Key concepts

vertical line

A vertical line is a line that goes straight up and down on a graph. It does not tilt to the left or right. This makes vertical lines unique because they run parallel to the y-axis.

On a coordinate plane, any point on a vertical line will have the same x-coordinate. For example, if you have a vertical line at x = 3, every point on this line will have an x-coordinate of 3, but the y-coordinates could be anything, like 2, 1, or -5.

Vertical lines are significant in geometry and algebra because their distinct direction and behavior impact how we calculate their slope.

undefined slope

When we talk about the slope of a line, we usually refer to how steep it is. Slope is calculated by taking the difference between the y-coordinates of two points on the line, and dividing by the difference between the x-coordinates of those points.

Mathematically, this is described by the formula: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]However, with a vertical line, all the x-coordinates are the same. So, when we try to calculate the slope, we end up dividing by zero because there is no difference between the x-coordinates. Dividing by zero is not defined in mathematics, leading to the term 'undefined slope'.

This means that when you have a vertical line, you don't label it as having a positive or negative slope because it's simply not defined that way. Instead, we just call it 'undefined.' This serves as a key property of vertical lines in math.

x-coordinates and y-coordinates

In the context of graphing a line, x-coordinates and y-coordinates are crucial to define the position of a point in a two-dimensional plane.

• x-coordinates are the horizontal values that describe how far left or right a point is from the origin (0,0).
• y-coordinates are the vertical values that specify how far up or down a point is from the origin.

When plotting any line, you select points with both x and y values, also called coordinates. Each point is plotted as \((x, y)\). For instance, the point (3, 2) is located 3 units to the right and 2 units up from the origin.

Understanding these coordinates helps calculate the slope of a line. In the case of a vertical line, since all x-coordinates are the same, this creates unique situations like the undefined slope we talked about earlier.