What is the gradient of a line perpendicular to y = 4x - 1?

The gradient of a line perpendicular to \( y = 4x - 1 \) is \( -\frac{1}{4} \).

To understand why, let's start by looking at the gradient of the given line. The equation \( y = 4x - 1 \) is in the form \( y = mx + c \), where \( m \) represents the gradient. Here, the gradient \( m \) is 4.

When two lines are perpendicular, the product of their gradients is always -1. This is a key property in coordinate geometry. So, if the gradient of one line is \( m \), the gradient of the line perpendicular to it will be \( -\frac{1}{m} \).

In this case, the gradient of the given line is 4. To find the gradient of the perpendicular line, we take the negative reciprocal of 4. The reciprocal of 4 is \( \frac{1}{4} \), and the negative reciprocal is \( -\frac{1}{4} \).

Therefore, the gradient of a line perpendicular to \( y = 4x - 1 \) is \( -\frac{1}{4} \). This means that for every 4 units the perpendicular line moves horizontally, it moves 1 unit downwards. Understanding this relationship helps in solving many problems involving perpendicular lines in coordinate geometry.