SAT Question 7e1bff94: Linear Functions answer and explanation with Desmos solution

Choice C is correct. It's given that there is a linear relationship between a plumber's hours of labor, $h$, and the plumber's total charge $f\left(h\right)$, in dollars. It follows that the relationship can be represented by an equation of the form $f\left(h\right)=mh+b$, where $m$ is the rate of change of the function $f$ and $b$ is a constant. The rate of change of $f$ can be calculated by dividing the difference in two values of $f\left(h\right)$ by the difference in the corresponding values of $h$. Based on the values given in the table, the rate of change of $f$ is $\frac{285-155}{3-1}$, or $65$. Substituting $65$ for $m$ in the equation $f\left(h\right)=mh+b$ yields $f\left(h\right)=65h+b$. The value of $b$ can be found by substituting any value of $h$ and its corresponding value of $f\left(h\right)$ for $h$ and $f\left(h\right)$, respectively, in this equation. Substituting $1$ for $h$ and $155$ for $f\left(h\right)$ yields $155=65\left(1\right)+b$, or $155=65+b$. Subtracting $65$ from both sides of this equation yields $90=b$. Substituting $90$ for $b$ in the equation $f\left(h\right)=65h+b$ yields $f(h)=65h+90$.