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Plot x(t) = t(u(t+3) − u(t−3))

Short answer

The factor u(t+3)-u(t-3) yields a rectangular window from -3 to 3. Multiplying by t gives x(t)=t for -3≤t<3 and zero elsewhere. Key points: x(-3)=-3, x(3-)=3, with a line through the origin.

Plot the signal x(t) = t(u(t+3) − u(t−3)).

The term u(t+3)−u(t−3)u(t+3) - u(t-3) is a rectangular window equal to 1 for −3≤t<3-3 \le t < 3 and 0 elsewhere. Multiplying by tt keeps the ramp tt only inside the window:

x(t)={t,−3≤t<30,otherwisex(t) = \begin{cases} t, & -3 \le t < 3 \\ 0, & \text{otherwise} \end{cases}

Key points: x(−3)=−3x(-3) = -3, x(0)=0x(0) = 0, x(3−)=3x(3^-) = 3; the signal drops to 0 for t≥3t \ge 3 and is 0 for t<−3t < -3. It is a straight line of slope 1 through the origin, cut off at t=±3t = \pm 3 (an odd signal).

          x(t)
            |      /| 3
            |    /  |
            |  /    |
  --+-------+/------+------> t
   -3      /0       3
    |    /  |
    |  /    |
 -3 |/      |

(The vertical jumps are at t=−3t = -3, from 0 down to −3-3, and at t=3t = 3, from 3 down to 0.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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