What is the equation for this graph?

Question

Here is what y(x) = -x+100 graph looks like:

I am trying to change this equation, so that the graph becomes a curve which still crosses the axis Y at (0,100) and axis X at (100,0):

Can anybody give me a hint on this?

Answer 1

Probably the easiest way is to choose a third point $(a,b),\ \ b\in(0,100)$, what your curve also contains, and then construct a quadratic function to fit these $3$ points, using $$ \begin{aligned} p_0(x) &:=\frac{(x-a)(x-100)}{(-a)(-100)} \\ p_a(x) &:=\frac{x(x-100)}{a(a-100)} \\ p_{100}(x) &:=\frac{x(x-a)}{100(100-a)} \end{aligned}$$ These satisfy $p_i(j)=1$ if $i=j$ and $p_i(j)=0$ if $i\ne j$ for $i,j\in\{0,a,100\}$. So, a quadratic curve can be obtained as $$f(x):=100\cdot p_0(x)+b\cdot p_a(x)+0\cdot p_{100}(x)\,.$$