How to generate a function from a given graph?

Question

Please provide any formula or step-by-step guide on how to generate a function for the following graph.

Graph logic:

Part 1 $\longrightarrow$ where $x \leq 3$ $\longrightarrow$ linear relation.

Part 2 $\longrightarrow$ when $3 \leq x \leq 6$ $\longrightarrow$ $y$ remains unchanged.

Then, again part 1 with $3 < x < 9$ $\longrightarrow$ linear relation.

Then part 2 when $y$ remains unchanged.

Thanks!

EDIT: Values for $x$ and $y$

x	y
0	0
1	1
2	2
3	3
4	3
5	3
6	3
7	3
8	3
9	3
10	4
11	5
12	6
13	6
14	6
15	6
16	6
17	6

and so on.

I am trying to create an interpolator for the following case:

Interpolation time 3500 ms
Value start = 0, end = 1

Firstly, the 500 ms value increases by the formula (end/3500)*interpolated time.

Then 1000 ms value remains unchanged.

Then 500 ms again value increases.

Then 1000 ms value remains unchanged.

Then 500 ms again value increases.

Answer 1

$y = x - 2a/3$ for $a \leq x < a + 3$

$y = a/3 +3$ for $a+3 \leq x < a + 9$

Where $a$ is defined as all numbers for which $a \bmod 9 = 0$.

Basically, $a$ is divisible by 9.

Tried this on Desmos. It works but you’ll have to manually type in every value of $a$.

Answer 2

If you don’t want to use discrete functions, you can use sigmoid approximation.

$$f_1(x) = \lim_{k\rightarrow \infty}\frac{1}{1 + e^{-k(x)}} \cdot \frac{1}{1 + e^{k(x-3)}}$$

The above function will return approximately $0$ if $x$ is not in the interval $(0,3)$, otherwise $1$. Likewise you can create your own functions for other intervals.

\begin{align} f_2(x) &= \lim_{k\rightarrow \infty}\frac{1}{1 + e^{-k(x-3)}} \cdot \frac{1}{1 + e^{k(x-9)}} \\ f_3(x) &= \lim_{k\rightarrow \infty}\frac{1}{1 + e^{-k(x-9)}} \cdot \frac{1}{1 + e^{k(x-12)}} \\ f_4(x) &= \lim_{k\rightarrow \infty}\frac{1}{1 + e^{-k(x-12)}} \cdot \frac{1}{1 + e^{k(x-17)}} \end{align}

Now, multiply all the functions with their lines.

$$f(x) = f_1 \cdot x + f_2 \cdot 3 + f_3 \cdot (x-6) + f_4 \cdot 6$$

Here is an example MATLAB code and the result graph.