find the Laplace transform of a piecewise function using the graph

Question

enter image description here

what is the Laplace transform of the function shown in the graph above. note that I tried to write the function in terms of the unit step function $u(t)=1$ if $t\ge 0$ and $u(t)=0$ if $t<0$. But I get I different answer from the doctor. Also I didn't understand how to write the function of the piece that is parallel to the y-axis. Here is the solution of the doctor $$f(t)= a.u(t)-t.u(t)+(t-a).u(t-a)-a.u(t-2a)+(t-2a).u(t-2a)-(t-3a).u(t-3a)$$

Answer 1

Might be helpful if you write the graph as sum of characteristic functions $$(a-t)\chi_{[0,a]}(t)\ +(t-a)\chi_{[a,2a]}(t)\ +(t-2a)\chi_{[2a,3a]}(t)\ +a\chi_{[3a,\infty]}(t)$$ with $\chi_{[a,b]}(t)=u_a(t)-u_b(t)$.