Choice A is correct. An equation that defines a linear function $f$ can be written in the form $f (x) = m x + b$ , where $m$ is the slope of the graph of $y = f (x)$ in the xy-plane and $left parenthesis 0 comma b right parenthesis$ is the y-intercept of the graph. It’s given that for the linear function $f$ , the graph of $y = f (x)$ in the xy-plane has a slope of $\frac{1}{4}$ . Therefore, $m = \frac{1}{4}$ . It’s also given that the graph of $y = f (x)$ passes through the point $left parenthesis 0 comma 5 right parenthesis$ . Therefore, the y-intercept of the graph is $left parenthesis 0 comma 5 right parenthesis$ , and it follows that $b equals 5$ . Substituting $\frac{1}{4}$ for $m$ and $5$ for $b$ in the equation $f (x) = m x + b$ yields $f (x) = \frac{1}{4} x + 5$ .

Choice B is incorrect. This equation defines a function whose graph has a y-intercept of $left parenthesis 0 comma one fifth right parenthesis$ , not $left parenthesis 0 comma 5 right parenthesis$ .

Choice C is incorrect. This equation defines a function whose graph has a y-intercept of $left parenthesis 0 comma negative five fourths right parenthesis$ , not $left parenthesis 0 comma 5 right parenthesis$ .

Choice D is incorrect. This equation defines a function whose graph has a y-intercept of $left parenthesis 0 comma negative 5 right parenthesis$ , not $left parenthesis 0 comma 5 right parenthesis$ .