The given function is $f(t) = \left[ u(t-2) - u(t-5) \right](3 - 2t)$, where $u(t)$ is the unit step function. To draw the graph of this function, we can analyze it step by step.

Step 1: Understand the Unit Step Function

• $u(t-2)$ is 0 for $t < 2$ and 1 for $t \geq 2$.
• $u(t-5)$ is 0 for $t < 5$ and 1 for $t \geq 5$.

Step 2: Analyze the Difference $u(t-2) - u(t-5)$

• For $t < 2$: Both $u(t-2)$ and $u(t-5)$ are 0, so $u(t-2) - u(t-5) = 0$.
• For $2 \leq t < 5$: $u(t-2) = 1$ and $u(t-5) = 0$, so $u(t-2) - u(t-5) = 1$.
• For $t \geq 5$: Both $u(t-2)$ and $u(t-5)$ are 1, so $u(t-2) - u(t-5) = 0$.

Thus, the expression $u(t-2) - u(t-5)$ equals 1 only in the interval $2 \leq t < 5$ and is 0 elsewhere.

Step 3: Multiply by $3 - 2t$

• For $2 \leq t < 5$, the function becomes $f(t) = (3 - 2t)$.
• Outside this interval, $f(t) = 0$.

Step 4: Graph the Function

• Interval 1 ($t < 2$): $f(t) = 0$.
• Interval 2 ($2 \leq t < 5$): $f(t) = 3 - 2t$, which is a linear function starting at $t = 2$ with a value $f(2) = -1$ and ending at $t = 5$ with a value $f(5) = -7$.
• Interval 3 ($t \geq 5$): $f(t) = 0$.

Final Graph Description

The graph is zero before $t = 2$ and after $t = 5$. Between $t = 2$ and $t = 5$, it is a straight line decreasing from $-1$ to $-7$.

Would you like a detailed explanation on how to draw this graph, or do you have any questions?

Related Questions:

1. What is the significance of the unit step function in engineering applications?
2. How does shifting the argument of a unit step function affect the graph?
3. What other piecewise functions can be expressed using the unit step function?
4. How would the graph change if the slope of $3 - 2t$ were altered?
5. Can you extend this method to graph other functions involving unit step functions?

Tip:

When working with unit step functions, always break the problem into intervals based on the step points to simplify the analysis and graphing.