To find the rate of change of \( y \) with respect to \( x \), we can use the formula for the slope between two points on a Cartesian plane. The slope \( m \) between the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Given the points \( (-4, 7) \) and \( (4, 1) \):
- Let \( (x_1, y_1) = (-4, 7) \)
- Let \( (x_2, y_2) = (4, 1) \)
Substituting in the values:
\[ m = \frac{1 - 7}{4 - (-4)} \] \[ m = \frac{-6}{4 + 4} \] \[ m = \frac{-6}{8} \] \[ m = -\frac{3}{4} \]
Thus, the rate of change of \( y \) with respect to \( x \) is \( -\frac{3}{4} \).
The correct answer is \( \boxed{d. -\frac{3}{4}} \).