In the xy -plane, the graph of the linear function f contains the points 0 ,

Choice C is correct. In the xy-plane, the graph of a linear function can be written in the form $f\left(x\right)=mx+b$, where $m$ represents the slope and $\left(0,b\right)$ represents the y-intercept of the graph of $y=f\left(x\right)$. It’s given that the graph of the linear function $f$, where $y=f\left(x\right)$, in the xy-plane contains the point $\left(0,2\right)$. Thus, $b=2$. The slope of the graph of a line containing any two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be found using the slope formula, $m=\frac{y_2-y_1}{x_2-x_1}$. Since it’s given that the graph of the linear function $f$ contains the points $\left(0,2\right)$ and $\left(8,34\right)$, it follows that the slope of the graph of the line containing these points is $m=\frac{34-2}{8-0}$, or  $m=4$. Substituting $4$ for $m$ and $2$ for $b$ in $f\left(x\right)=mx+b$ yields $f\left(x\right)=4x+2$.

Choice A is incorrect. This function represents a graph with a slope of $2$ and a y-intercept of $\left(0,42\right)$.

Choice B is incorrect. This function represents a graph with a slope of $32$ and a y-intercept of $\left(0,36\right)$.

Choice D is incorrect. This function represents a graph with a slope of $8$ and a y-intercept of $\left(0,2\right)$.