Finding the Laplace transform of a signal: How do you setup the step function $f(t)$ (equation of the graph on the image). Even though, I do know know how they setup the equations. I do know how to find the slope of line but I do not get it how they are getting the $u(t)$, $u(t-2)$ and $u(t-6)$ and multiplying each with their respective slopes $5$, $7.5$, and $2.5$ respectively. Thanks!
First, you need to parametrize your curve. You can check that $f(t)=5t$ for $t\in[0,2]$ and $f(t)=15-2.5t$ for $t\in(2,6]$. Then, the function can be written as $$f(t)=5t[u(t)-u(t-2)]+(15-2.5t)[u(t-2)-u(t-6)]$$ note that the function is recovering the value at $t=2$ if we take the convention $u(0)=1/2$. For the Laplace transform, you get two kind of terms: $u(t)\to\frac{1}{s}$ and $t \,u(t)\to\frac{1}{s^2}$. Note that you can use the time translation property of the Laplace transform to compute the transforms of the translated step functions