What is the gradient of a vertical line?

The gradient of a vertical line is undefined because it does not have a horizontal change.

In more detail, the gradient (or slope) of a line measures how steep the line is. It is calculated as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). For most lines, you can find the gradient using the formula:

\[ \text{Gradient} = \frac{\text{change in } y}{\text{change in } x} \]

However, for a vertical line, all the points on the line have the same x-coordinate. This means that the change in x (the run) is zero. When you try to divide by zero, the result is undefined in mathematics. Therefore, the gradient of a vertical line is considered undefined.

To visualise this, imagine a line that goes straight up and down, like the y-axis on a graph. No matter how far up or down you go, the x-coordinate remains constant. Since you cannot divide by zero, the concept of gradient does not apply in the usual sense, making it undefined.

Understanding this is important because it helps you recognise different types of lines and their properties. Horizontal lines, for example, have a gradient of zero because there is no vertical change. Knowing these differences can help you analyse and graph equations more effectively.