SAT Question 295a41f0: Linear Functions answer and explanation with Desmos solution

Choice B is correct. The graph of a line in the xy-plane can be represented by the equation $y=mx+b$, where $m$ is the slope of the line and $\left(0,b\right)$ is the y-intercept. The slope of a line that passes through the points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ can be calculated using the formula $m=\frac{y_2-y_1}{x_2-x_1}$. The line shown passes through the points $\left(-1,-2\right)$ and $\left(0,-4\right)$. Substituting $\left(-1,-2\right)$ and $\left(0,4\right)$ 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 $m=\frac{-4-\left(-2\right)}{0-\left(-1\right)}$, which is equivalent to $m=\frac{-2}{1}$, or $m=-2$. Since the line shown passes through the point $\left(0,-4\right)$, it follows that $b=-4$. Substituting $-2$ for $m$ and $-4$ for $b$ in the equation $y=mx+b$ yields $y=-2x-4$. It's given that the graph shown is the graph of $y=f\left(x\right)-11$. Substituting $-2x-4$ for $y$ in the equation $y=f\left(x\right)-11$ yields $-2x-4=f\left(x\right)-11$. Adding $11$ to both sides of this equation yields $-2x+7=f\left(x\right)$. Therefore, the equation $f\left(x\right)=-2x+7$ defines the linear function $f$.