SAT Question 43926bd9: Nonlinear Functions answer and explanation with Desmos solution

The values of $f(x)$ for the exponential function $f$ shown in the table increase by a factor of $a^4$ for each increase of $1$ in $x$. This relationship can be represented by the equation $f(x)=a^{4x+b}$, where $b$ is a constant. It's given that when $x=2$, $f(x)=a^5$. Substituting $2$ for $x$ and $a^5$ for $f(x)$ into $f(x)=a^{4x+b}$ yields $a^5=a^{4\times 2+b}$. Since $4\times 2+b=5$, it follows that $b=-3$. Thus, an equation that defines the function $f$ is $f(x)=a^{4x-3}$. It follows that the value of $k$ such that $f(k)=a^{29}$ can be found by solving the equation $4k-3=29$, which yields $k=8$.