For the linear function f , the graph of y = f ( x ) in

Choice D is correct. An equation defining the linear function $f$ can be written in the form $f\left(x\right)=mx+b$, where $m$ is the slope and $\left(0,b\right)$ is the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane. It’s given that the graph of $y=f\left(x\right)$ has a slope of $2$. Therefore, $m=2$. It’s also given that the graph of $y=f\left(x\right)$ has a y-intercept at $\left(0,-5\right)$. Therefore, $b=-5$. Substituting $2$ for $m$ and $-5$ for $b$ in the equation $f\left(x\right)=mx+b$ yields $f\left(x\right)=2x-5$. Thus, the equation that defines $f$ is $f\left(x\right)=2x-5$.

Choice A is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane has a slope of $\frac{1}{2}$, not $2$.

Choice B is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane has a slope of $-\frac{1}{2}$, not $2$.

Choice C is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane has a slope of $-2$, not $2$.