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

Choice B is correct. An equation defining a linear function can be written in the form $f\left(x\right)=mx+b$, where $m$ and $b$ are constants, $m$ is the slope of the graph of $y=f\left(x\right)$ in the xy-plane, and $\left(0,b\right)$ is the y-intercept of the graph. It's given that for the function $f$, the graph of $y=f\left(x\right)$ in the xy-plane has a slope of $3$. Therefore, $m=3$. It's also given that this graph passes through the point $\left(0,-8\right)$. Therefore, the y-intercept of the graph is $\left(0,-8\right)$, and it follows that $b=-8$. Substituting $3$ for $m$ and $-8$ for $b$ in the equation $f\left(x\right)=mx+b$ yields $f\left(x\right)=3x-8$. Thus, the equation that defines $f$ is $f\left(x\right)=3x-8$.

Choice A is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane passes through the point $\left(0,0\right)$, not $\left(0,-8\right)$.

Choice C is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane passes through the point $\left(0,5\right)$, not $\left(0,-8\right)$.

Choice D is incorrect. For this function, the graph of $y=f\left(x\right)$ in the xy-plane passes through the point $\left(0,11\right)$, not $\left(0,-8\right)$.