On a precalculus test I got recently, the following question stumped the entire class, and even weeks later, none of us could figure it out! The problem is to
Write the equation of the following rational function, given this information:
Domain: $x\neq -0.5, \ 5,6$
$f(0) = 8$
$f(3) = 0$
$f(-8) = 0$
$\lim_{x\to -\infty} f(x) = 4$
$\lim_{x\to 2} f(x) = 0.5$
$f'(x) < 0$ when $x<-0.5$
$f''(x) > 0$ when $-0.5<x<5 \cap 5<x<6 \cap x>6$
I broke it down to this graph, but my problem is that if I change the scaling to fix the points that don't line up (the y-intercept and the ($2$, $0.5$)), the horizontal asymptote moves as well. How can you scale the short-term behavior without affecting the long-term behavior?
I also tried adding factors to the top and bottom that take the form $x^2 + a$ that way no roots are added but you can add factors, but it hasn't worked out nicely for me yet.
Update: I had a minor breakthrough with multiplying the original function I had in Desmos by $\frac{x^2 +a}{x^2 +2a}$, and then I increased a to approach $+\infty$, which seemed to work. However, I'm not sure I'm allowed to write $\lim_{a \to +\infty}$ on this question for a rational function, so how would I go about rewriting this without having to introduce a limit?