design the equation of a decreasing parabola (between 2 points)

Question

I am trying to come up with equations that can create a smooth downward decrease along $y$ so it falls from $0.997$ down to $0.990$. I am interested at the interval [N/5, N] along the $x$-axis.

I almost got it to work (0.997 is in the correct place at x=6.4), however, the value $0.99$ occurs at $x=37.4$ not at $x=32$ like I would desire.

How to tweak my formulas to get it?

... and N can change, it's a positive integer above zero)

Answer 1

You seem to want

• the maximum of the parabola to be at $(\frac{N}5,0.997)$
  • so the curve will be something like $y=a(x-\frac N5)^2+0.997$
• and another point to be $(N,0.99)$
  • so $0.99=a(N-\frac N5)^2+0.997$
  • making $a= (0.99-0.997)/(\frac 45N)^2=-0.0109375/ N^2$
  • and the parabola $$y=-0.0109375\frac{x^2}{N^2}+0.004375\frac xN +0.9965625$$