SAT Question 55c5d3c2: Nonlinear Functions answer and explanation with Desmos solution

It's given that function $f$ is defined by $f\left(x\right)=a^x+b$, where $a$ and $b$ are constants and $a>0$. It's also given that the graph of $y=f\left(x\right)$ in the xy-plane has a y-intercept at $\left(0,-25\right)$ and passes through the point $\left(2,23\right)$. Since the graph has a y-intercept at $\left(0,-25\right)$, $f\left(0\right)=-25$. Substituting $0$ for $x$ in the given equation yields $f\left(0\right)=a^0+b$, or $f\left(0\right)=1+b$, and substituting $-25$ for $f\left(0\right)$ in this equation yields $-25=1+b$. Subtracting $1$ from each side of this equation yields $-26=b$. Substituting $-26$ for $b$ in the equation $f\left(x\right)=a^x+b$ yields $f\left(x\right)=a^x-26$. Since the graph also passes through the point $\left(2,23\right)$, $f\left(2\right)=23$. Substituting $2$ for $x$ in the equation $f\left(x\right)=a^x-26$ yields $f\left(2\right)=a^2-26$, and substituting $23$ for $f\left(2\right)$ yields $23=a^2-26$. Adding $26$ to each side of this equation yields $49=a^2$. Taking the square root of both sides of this equation yields $\pm 7=a$. Since it's given that $a>0$, the value of $a$ is $7$. It follows that the value of $a+b$ is $7-26$, or $-19$.