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Draw this graph using common engineering function

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.

Learn how to draw the graph of the function f(t) = [u(t-2) - u(t-5)](3 - 2t) step-by-step. Understand the unit step function and its application in piecewise linear functions. Suitable for advanced high school students or anyone interested in understanding piecewise functions with unit steps.

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