To determine which equation has a constant of proportionality equal to [tex]\(\frac{1}{4}\)[/tex], let's go through each option and see where the constant appears:
A. [tex]\(y = x + \frac{1}{4}\)[/tex]
In this equation, [tex]\(y\)[/tex] is dependent on [tex]\(x\)[/tex], but the [tex]\(\frac{1}{4}\)[/tex] is an added constant, not a multiplicative factor. The equation does not represent a direct proportional relationship because the form is [tex]\(y = mx + b\)[/tex], where [tex]\(b\)[/tex] is not zero. Therefore, the constant of proportionality here is not [tex]\(\frac{1}{4}\)[/tex].
B. [tex]\(y = \frac{1}{4} - x\)[/tex]
This equation shows [tex]\(y\)[/tex] being equal to the difference between [tex]\(\frac{1}{4}\)[/tex] and [tex]\(x\)[/tex]. This setup does not represent a direct proportionality, since the constant [tex]\(\frac{1}{4}\)[/tex] is being subtracted by [tex]\(x\)[/tex]. Hence, this equation does not have a constant of proportionality equal to [tex]\(\frac{1}{4}\)[/tex].
C. [tex]\(y = \frac{1}{4} x\)[/tex]
In this equation, [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex] with the factor [tex]\(\frac{1}{4}\)[/tex]. The equation is in the form [tex]\(y = mx\)[/tex], where [tex]\(m\)[/tex] is the constant of proportionality. Here, [tex]\(m = \frac{1}{4}\)[/tex]. Therefore, this equation has a constant of proportionality equal to [tex]\(\frac{1}{4}\)[/tex].
D. [tex]\(y = 4x\)[/tex]
In this equation, [tex]\(y\)[/tex] is directly proportional to [tex]\(x\)[/tex] with the factor 4. The equation is in the form [tex]\(y = mx\)[/tex], where [tex]\(m\)[/tex] is the constant of proportionality. Here, [tex]\(m = 4\)[/tex]. Therefore, this equation has a constant of proportionality equal to 4, not [tex]\(\frac{1}{4}\)[/tex].
Given the options, the correct answer is:
C. [tex]\(y = \frac{1}{4} x\)[/tex]
