SAT Question 4a0d0399: Nonlinear Functions answer and explanation with Desmos solution

It's given that the function $f$ is defined by $f\left(x\right)=a^x+b$, where $a$ and $b$ are constants. It's also given that the graph of $y=f\left(x\right)$ has a y-intercept at $\left(0,-323\right)$. It follows that $f\left(0\right)=-323$. Substituting $0$ for $x$ and $-323$ for $f\left(x\right)$ in $f\left(x\right)=a^x+b$ yields $-323=a^0+b$, or $-323=1+b$. Subtracting $1$ from each side of this equation yields $-324=b$. Therefore, the value of $b$ is $-324$.