How do you find an analytical solution from a graph?

Question

Given is a graph with three functions, g(x), G(x) and g'(x).

I don't need simply the answer. I want to know how to do it, or at least how to begin.

Answer 1

Starting with the red graph:

This is obviously a sine wave. A sine wave is $$A\sin(\omega x+\varphi),$$ where $A$ is the amplitude, $f$ the angular frequency and $\varphi$ the phase. We can easily see that $A=1$ and $\varphi=0$. The angular frequency is defined as $\frac{2\pi}{T}$, where $T$ is the lenght of a period, which in this case is $\pi$. So the red wave is the graph of $$\sin(2x).$$

Continuing with the blue graph

First calculate the antiderivative of $\sin(2x)$. It is $-\cos(2x)/2+C_1$. At $x=0$ the function,with $C_1=0$, is $-1/2$. To bring it up to $y=0$ set $C=1/2$.

Finishing with the green graph

Calculate the antiderivative again. This yealds $x/2-\sin(2x)/4+C_2$. At $x=0$ this function is equal to the green graph if $C_2=0$.

Conclusion

We have $$g'(x)=\sin(2x), \quad g(x) = -\frac{\cos(2x)}{2}+\frac{1}{2},\quad G(x) = \frac{x}{2}-\frac{\sin(2x)}{4}.$$