SAT Question fe6f9678: Linear Functions answer and explanation with Desmos solution

Choice C 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. The slope of the graph of $y=f\left(x\right)$ can be found using the formula $m=\frac{y_2-y_1}{x_2-x_1}$, where $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are any two points that the graph passes through. If $f\left(0\right)=17$, it follows that the graph of $y=f\left(x\right)$ passes through the point $\left(0,17\right)$. If $f\left(1\right)=17$, it follows that the graph of $y=f\left(x\right)$ passes through the point $\left(1,17\right)$. Substituting $\left(0,17\right)$ and $\left(1,17\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the formula $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{17-17}{1-0}$, which is equivalent to $m=\frac{0}{1}$, or $m=0$. Since the graph of $y=f\left(x\right)$ passes through $\left(0,17\right)$, it follows that $b=17$. Substituting $0$ for $m$ and $17$ for $b$ in the equation $f\left(x\right)=mx+b$ yields $f\left(x\right)=0x+17$, or $f\left(x\right)=17$. Thus, the equation that defines $f$ is $f\left(x\right)=17$.