To find the equation of a line that is perpendicular to \( y = -4x + 1 \) and passes through the point \( (4, -2) \), we first need to determine the slope of the given line.
The given line is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. From the equation \( y = -4x + 1 \), we see that the slope \( m \) is \( -4 \).
For two lines to be perpendicular, the product of their slopes must equal \( -1 \). Therefore, if the slope of the original line is \( -4 \), the slope of the line that is perpendicular to it will be:
\[ m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{-4} = \frac{1}{4} \]
Now, we have the slope of the line we want to find, which is \( \frac{1}{4} \). We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope. Plugging in our point \( (4, -2) \) and the slope \( \frac{1}{4} \):
\[ y - (-2) = \frac{1}{4}(x - 4) \]
Simplifying this equation:
\[ y + 2 = \frac{1}{4}(x - 4) \]
\[ y + 2 = \frac{1}{4}x - 1 \]
Now, subtract 2 from both sides to get \( y \) by itself:
\[ y = \frac{1}{4}x - 1 - 2 \]
\[ y = \frac{1}{4}x - 3 \]
Thus, the equation of the line that passes through the point \( (4, -2) \) and is perpendicular to \( y = -4x + 1 \) is:
\[ y = \frac{1}{4}x - 3 \]