\( 2 \begin{array}{c}\text { Write the equation of } \\ \text { the line perpendicular to } \\ y=1 / 4 x+8 \text { through }(-4,-2)\end{array} \)

To find the equation of the line perpendicular to \( y = \frac{1}{4}x + 8 \), we first need to determine the slope of the original line, which is \( \frac{1}{4} \). The slope of a line perpendicular to it is the negative reciprocal, so the perpendicular slope is \( -4 \). Using the point-slope form of the line's equation \( y - y_1 = m(x - x_1) \) with point \( (-4, -2) \) gives us: \[ y - (-2) = -4(x - (-4)) \] \[ y + 2 = -4(x + 4) \] \[ y + 2 = -4x - 16 \] \[ y = -4x - 18 \] So, the equation of the line is \( y = -4x - 18 \). To double-check our work, we can ensure that our line is indeed perpendicular to the original by confirming the slopes. Given that the product of the slopes of both lines is \(-1\) (i.e., \( \frac{1}{4} \cdot (-4) = -1\)), our solution checks out perfectly!