How to sketch/plot this discrete time signal?
Begin with defining the base signal as $x[n] = n ~u[n]$ and recognize the output as a linear compositions (superpositions) of the base signal as:
$$y[n] = x[n+5] + 2 x[n] + x[n-10]$$

Now what you have to answer is the result of translations and scalings applied on the basic signal $x[n]$. This ability is a simple yet an indispensibly necessary skill for a clear understanding and successful applications of all subsequent signal processing theory. Therefore you must train yourself enough for this task.

Incidentally, the signal $x[n+5]$ is an advanced (shift left) version of $x[n]$ whereas $x[n-10]$ is a delayed (shift right) version.

Graphically the result is obtained by summing the three components drawn on the on the same plot. A more detailed approach would use the piecewise definitions of the components to define the sum. It would proceed as follows:

Let $x[n]$be defined as:
$$ x[n] = n ~\text{u}[n] = 
\begin{cases}
n , \text{ for } n \geq 0 \\
0 , \text{ for } n &lt; 0 \\
\end{cases}
$$

Then you can define the three superposed signals as:

$$
\begin{align}
x_1[n] = x[n+5] = (n+5) ~u[n+5] &amp;= \begin{cases} n+5, &amp;\text{ for } &amp;n+5\geq 0 \\ 0, &amp;\text{ for } &amp;n+5&lt; 0 \\ \end{cases} \\
x_2[n] = 2x[n] = 2n ~u[n] &amp;= \begin{cases} 2n,~~~~ &amp;\text{ for } &amp;2n \geq 0 \\ 0,~~~ &amp;\text{ for } &amp;2n &lt; 0 \\ \end{cases} \\
x_3[n] = x[n-10] = (n-10) ~u[n-10] &amp;= \begin{cases} n-10, &amp;\text{for} &amp;n-10 \geq 0 \\0, &amp;\text{for} &amp;n-10&lt;0 \\ \end{cases} \\
\end{align}
$$

Modifiy the intervals:
$$
\begin{align}
x_1[n] = x[n+5] = (n+5) ~u[n+5] &amp;= \begin{cases} n+5, &amp;\text{ for } &amp;n \geq -5 \\ 0, &amp;\text{ for } &amp;n&lt;-5 \\ \end{cases} \\
x_2[n] = 2x[n] = 2n ~u[n] &amp;= \begin{cases} 2n,~~~~ &amp;\text{ for } &amp;n \geq 0 \\ 0,~~~ &amp;\text{ for } &amp;n&lt;0 \\ \end{cases} \\
x_3[n] = x[n-10] = (n-10) ~u[n-10] &amp;= \begin{cases} n-10, &amp;\text{for} &amp;n \geq 10 \\0, &amp;\text{for} &amp;n&lt;10 \\ \end{cases} \\
\end{align}
$$

Therefore $y[n] = x_1[n] + x_2[n] + x_3[n]$ is:
$$
y[n] = 
\begin{cases}
0 , &amp;\text{ for }     &amp;-\infty&lt;n&lt;-5 \\
n+5 , &amp;\text{ for } &amp;-5 \leq n &lt;0 \\
3n+5 , &amp;\text{ for } &amp;0 \leq n &lt;10 \\
4n-5 , &amp;\text{ for } &amp;10 \leq n &lt;\infty \\
\end{cases} 
$$