To determine the gradient (or slope) of a line perpendicular to the given line [tex]\( y = -4x + 9 \)[/tex], we need to use the concept of perpendicular slopes.
Here's a step-by-step solution:
1. Identify the gradient of the given line:
The equation of the line is [tex]\( y = -4x + 9 \)[/tex]. This equation is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the gradient. From the equation, we see that the gradient [tex]\( m \)[/tex] is [tex]\(-4\)[/tex].
2. Determine the gradient of the perpendicular line:
The gradient of a line perpendicular to another is the negative reciprocal of the original gradient. The reciprocal of a number [tex]\( a \)[/tex] is [tex]\( \frac{1}{a} \)[/tex]. Therefore, the negative reciprocal of [tex]\(-4\)[/tex] is [tex]\(-\frac{1}{-4}\)[/tex].
3. Simplify the negative reciprocal:
Simplifying [tex]\(-\frac{1}{-4}\)[/tex] gives us [tex]\(\frac{1}{4}\)[/tex]. Since the reciprocal must also be multiplied by -1 (to ensure it is the "negative reciprocal"), we end up with [tex]\(\frac{1}{4}\)[/tex].
4. Simplify the negative character:
Thus, the final gradient of the line that is perpendicular to the given line is [tex]\(-\frac{1}{4}\)[/tex].
After these steps, we conclude that the gradient of a line perpendicular to [tex]\( y = -4x + 9 \)[/tex] is [tex]\(-\frac{1}{4}\)[/tex].
Therefore, the correct answer is:
B) [tex]\(-\frac{1}{4}\)[/tex]
