SAT Question 4b0c156b: Linear Functions answer and explanation with Desmos solution

Choice D is correct. An equation defining $h$ can be written in the form $y=mx+b$, where $y=h\left(x\right)$, $m$ represents the slope of the graph of $y=h\left(x\right)$ in the xy-plane, and $b$ represents the y-coordinate of the y-intercept of the graph. It's given that $h\left(28\right)=15$ and $h\left(26\right)=22$. It follows that the points $\left(28,15\right)$ and $\left(26,22\right)$ are on the graph of $y=h\left(x\right)$ in the xy-plane. The slope can be found by using any two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, and the formula $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting $\left(28,15\right)$ and $\left(26,22\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{22-15}{26-28}$, or $m=-\frac{7}{2}$. Substituting $-\frac{7}{2}$ for $m$ and $\left(28,15\right)$ for $\left(x,y\right)$ in the equation $y=mx+b$ yields $15=\left(-\frac{7}{2}\right)\left(28\right)+b$, or $15=-98+b$. Adding $98$ to both sides of this equation yields $113=b$. Substituting $-\frac{7}{2}$ for $m$ and $113$ for $b$ in the equation $y=mx+b$ yields $y=-\frac{7}{2}x+113$. Since $y=h\left(x\right)$, it follows that the equation that defines $h$ is $h\left(x\right)=-\frac{7}{2}x+113$.