Final answer:
The equation of the line that is perpendicular to the given line and passes through (4, -1) is y = -4x + 15. This is found by using the negative reciprocal of the original line's slope and the point-slope formula.
Step-by-step explanation:
To find the equation of the line that is perpendicular to the given line y = \( \frac{x}{4} \) - 2 and intersects the point (4,-1), we need to determine the slope of the perpendicular line. Since the slope of the given line is \( \frac{1}{4} \), the slope of the perpendicular line will be the negative reciprocal, which is -4. Now, we use the point-slope form to write the equation of the perpendicular line; let's call it y2:
y - y1 = m(x - x1)
Where (x1, y1) is the point (4, -1) and m is the slope -4:
y - (-1) = -4(x - 4)
Simplify to get the standard form of the equation:
y + 1 = -4x + 16
Now, rewrite it to match the desired form y = -4x + b:
y = -4x + 15
The equation of the perpendicular line is therefore y = -4x + 15.