SAT Question 768b60d2: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. Each of the given choices is an equation of the form $f\left(x\right)=a{\left(b\right)}^{x-k}$, where $a$, $b$, and $k$ are constants. For an equation of this form, the coefficient, $a$, is equal to the value of the function when the exponent is equal to $0$, or when $x=k$. It follows that in the equation $f\left(x\right)=33{\left(1.5\right)}^x$, the coefficient, $33$, is equal to the value of $f\left(0\right)$. Substituting $0$ for $x$ in this equation yields $f\left(0\right)=33{\left(1.5\right)}^0$, which is equivalent to $f\left(0\right)=33\left(1\right)$, or $f\left(0\right)=33$. Thus, the value of $c$ is $33$ and the equation $f\left(x\right)=33{\left(1.5\right)}^x$ shows the value of $c$ as the coefficient.