We Want to find $a,b,c,d$ of $\frac{x^2 +ax + b}{x^2 +cx +d}$ from its plot. If they can be found accurately, found the exact number. If they can be an interval, find that interval. (note that the extremum of plot is not at an accurately known point, we just know that its $x$ is between $(0,1)$ and $y$ is negative)
From the plot, I can find that the numerator must be $(x+1)(x-2) = x^2 - x -2$ so I get $a=-1 ,b=-2$. I can Also found that the denominator must not have real roots so $c^2 - 4d <0$ but I cannot find $c,d$ Note that it is known that $c , d$ has accurate values or intervals like $c=-4,d>10$ (just for example). So I don't want parametric answers like $-2\sqrt d < c < 2 \sqrt d$.
Disclaimer: This answer is subject to possible errors due to inaccurate readings of the graph.
The $y$-intercept of the graph seems to be at $y=-\frac23$, whence $\frac{b}{d}=-\frac23$ should be the case. As $b=-2$, we conclude $d=3$.
The graph occurs to be symmetric about the line $x=\frac12$. Therefore, the function is symmetric about the same axis. Since the numerator $x^2+ax+b=x^2-x-2=\left(x-\frac12\right)^2-\frac{9}{4}$ is symmetric about this axis, the denominator must also be symmetric about $x=\frac12$ too. That is, $x^2+cx+d=\left(x-\frac12\right)^2+k$ for some constant $k$. Hence, $c=-1$.
The plot below shows the plot for $y=\frac{x^2-x-2}{x^2-x+3}$. This looks almost identical to the OP's graph.