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 \(-1\). This is a key property of perpendicular lines. 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 line perpendicular to it, we use the formula for perpendicular gradients:
\[ m_{\text{perpendicular}} = -\frac{1}{m} \]

Substituting \(-4\) for \( m \):
\[ m_{\text{perpendicular}} = -\frac{1}{-4} = \frac{1}{4} \]

Therefore, the gradient of the line perpendicular to \( y = -4x + 1 \) is \(\frac{1}{4}\). This means that for every 4 units you move horizontally, the line will move 1 unit vertically. This is a much gentler slope compared to the steepness of the original line with a gradient of \(-4\).