I am programming a small game (with other programming language) in which I have to use Mod function to express piecewise linear period functions.

In my case, two hard parts I think exist.

(1) The periodic function is linear in the piecewise sence, like the function below. In a full period it is NOT linear.

(2) There are NO powerful function like Piecewise function in Mathematica, and all I can use are some very basic functions including Mod, +,-, *, \, Abs, Floor, Ceiling,Exp,10^,Sin,Cos,Tan,ASin,ACos, ATan,Log and logic control commands If, If Then Else.

(3) After many times trying, I also found another challenging part is related to Mod of a negative number .My piecewise linear periodic function can thake negative integers whose Mod is so confusing to me. I can calculate the Mod of a negative number by hand, but I have no clear idea how it relates to a repeat pattern.

So is it possible to use Mathematica to find something like linear combinations of Mod functions for any periodic piecewise linear function?

For example, the periodic piecewise linear period function like this:

f[x_?IntegerQ] := 
 Piecewise[{{-10 x + 2, 1 <= x < 4}, {-5 x + 8, 4 <= x <= 10}}, True]
f[x_?IntegerQ] := f[x + 10] /; x < 1
f[x_?IntegerQ] := f[x - 10] /; x > 10
ListPlot[Table[{x, f[x]}, {x, -9, 20}]]

Any advices are highly appreciated! Thanks :)