SAT Question 8ad64841: Linear Functions answer and explanation with Desmos solution

Choice D is correct. It's given that the relationship between $x$ and $f\left(x\right)$ is linear. A linear function can be written in the form $f\left(x\right)=mx+b$, where $m$ is the slope and $b$ is the y-coordinate of the y-intercept of the graph of $y=f\left(x\right)$ in the xy-plane. Given two points on a line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, the slope of the line can be found using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$. It's given that the company's profit is $$220$ when it sells $8$ items and the profit is $$320$ when it sells $10$ items. Since $f\left(x\right)$ represents the company's profit, in dollars, for selling $x$ items, the graph of $y=f\left(x\right)$ in the xy-plane passes through the points $\left(8,220\right)$ and $\left(10,320\right)$. Substituting $\left(8,220\right)$ and $\left(10,320\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{320-220}{10-8}$, which gives $m=\frac{100}{2}$, or $m=50$. Substituting $50$ for $m$, $8$ for $x$, and $220$ for $f\left(x\right)$ in $f\left(x\right)=mx+b$ yields $220=\left(50\right)\left(8\right)+b$, or $220=400+b$. Subtracting $400$ from each side of this equation yields $-180=b$. Substituting $50$ for $m$ and $-180$ for $b$ in $f\left(x\right)=mx+b$ yields $f\left(x\right)=50x-180$. Therefore, the equation that defines $f$ is $f\left(x\right)=50x-180$.