The following function fits exactly, but has the wrong asymptotic behaviour. $$y= 0.75x^4-2.5 x^3+0.75x^2+3x$$
The following cubic nearly fits. $$2 y = x^2(2x-9)+11x$$
This following quintic exactly fits. $$4y = 4x^5-21x^4+42x^3-45x^2+28x$$
The differential set to zero gives $$ 20x^4-84x^3+126x^2-90x+28 = (x-1)(x-2)(20x^2-24x+14)=0\;.$$
This is by choosing the coefficient of $x^5$ to be $1$.
This curve only has 2 real turning points.
You can see these graphs on the attached