Please provide more context, like what are you trying to do, what part of your course is this from.
One of the most common reasons for such a process is related to the numericali ntegration of ODEs, but there is no unique way of doing this.
The simplest method is to use that approximation:
$$
f(x+h)=f(x)+hf'(x)
$$
So defining $\Delta_h f(x)=f(x+h)-f(x)=hf'(x)$.
But these are approximations of a continuous process by discrete.
As I said before we need more context.
.
We want to convert the ODE $y''=-y$ into a difference equation. I will use repeated approximation by a difference quotient:
$$
y''(x)=\frac{y'(x+h)-y'(x)}{h}=\frac{\frac{y(x+2h)-y(x+h)}{h}-\frac{y(x+h)-y(x)}{h}}{h}=\frac{y(x+2h)-2y(x+h)+y(x)}{h^2}=-y(x)
$$
so:
$$
y(x+2h)-2y(x+h)+y(x)=-h^2y(x)
$$
with initial conditions $y(0)=0,\ y(h)=h+y(0)=h$
and the rest is algebra