What could be the mathematical formula of this function?

Question

I need to change with one parameter (i) the way how y changes over the x like in the image below. What would be the approximated formula?

Answer 1

Try graphing $$ y = x^r $$ for $0 \le x \le 1$, varying $r$ from, say, $0.2$ to $5$. That might be what you're looking for. It at least looks a little like what you drew. Hint for graphing: desmos.com.

Answer 2

There are too many possible answers, so it helped if you were more specific about what exactly you try to accomplish.

Generally speaking, assume that the line segment in the drawing spans the segment $(x_0,y_0),(x_1,y_1)$ and $f(x)$ is the line passing through the two points.

Let $g(x)$ be any function such that:

• $g(x_0) = g(x_1) = 0$

• $g(x) \gt 0$ for $x \in (x_0,x_1)$

Then (using $\lambda$ instead of $i$ for the parameter to avoid possible confusions) the function:

$$ f(x) + \lambda g(x) $$

will match $f(x)$ at the endpoints $\{x_0,x_1\}$, and run either above ($\lambda \gt 0$) or below ($\lambda \lt 0$) $f(x)$ on the interior interval $(x_0,x_1)$.

Examples of such $g(x)$ functions are:

• $g(x) = -(x-x_0)(x-x_1)$

• $g(x) = \sin(\pi \cdot \frac{x-x_0}{x_1-x_0})$

• ...countless others.