Function $f$ is defined by $f(x)=-a^x+b$, where $a$ and $b$ are constants. In the $xy$-plane, the graph of $y=f(x)-10$ has a $y$-intercept at $(0,-\frac{43}{4})$. The product of $a$ and $b$ is $\frac{15}{4}$. What is the value of $a$?