why is it a straight line decreasing from -1 to -7

You're right to question that! My earlier explanation contained a mistake in evaluating the function. Let's correct it.

### Correct Analysis of the Function \( f(t) = \left[ u(t-2) - u(t-5) \right](3 - 2t) \)

#### Step 1: Understand the Function's Structure
1. **Unit Step Function** \( u(t-a) \):
   - \( u(t-2) \) activates at \( t = 2 \).
   - \( u(t-5) \) activates at \( t = 5 \).

2. **Expression \( u(t-2) - u(t-5) \)**:
   - This expression equals 1 when \( 2 \leq t < 5 \) and 0 elsewhere.

#### Step 2: Simplify the Function
- For \( t < 2 \), both \( u(t-2) \) and \( u(t-5) \) are 0, so \( f(t) = 0 \).
- For \( 2 \leq t < 5 \), \( u(t-2) = 1 \) and \( u(t-5) = 0 \), so \( f(t) = 3 - 2t \).
- For \( t \geq 5 \), both \( u(t-2) \) and \( u(t-5) = 1 \), so \( f(t) = 0 \).

#### Step 3: Evaluate \( f(t) \) within \( 2 \leq t < 5 \)
In the interval \( 2 \leq t < 5 \):
\[ f(t) = 3 - 2t \]

This is a linear equation:
- At \( t = 2 \): \( f(2) = 3 - 2(2) = 3 - 4 = -1 \)
- At \( t = 5 \): \( f(5) = 3 - 2(5) = 3 - 10 = -7 \)

Thus, the line decreases from \( -1 \) at \( t = 2 \) to \( -7 \) at \( t = 5 \).

### Final Graph:
- The graph is \( 0 \) for \( t < 2 \) and \( t \geq 5 \).
- For \( 2 \leq t < 5 \), the graph is a straight line with a negative slope, decreasing from \( -1 \) at \( t = 2 \) to \( -7 \) at \( t = 5 \).

So, the function \( 3 - 2t \) is linear with a negative slope, which explains why the graph decreases linearly from \( -1 \) to \( -7 \) as \( t \) increases from 2 to 5.

Let me know if you need any further clarification or details on this! Would you like me to provide additional examples or explanations?

### Related Questions:
1. How can the slope of a linear function affect its graph?
2. What would happen if the coefficient of \( t \) were positive in the linear term?
3. How can the piecewise definition be extended to model more complex behaviors?
4. What other functions can be modified by unit step functions to create piecewise linear graphs?
5. How can we analyze the discontinuity at points where the unit step function changes?

### Tip:
When analyzing functions that involve unit step functions, always carefully evaluate the function within each interval defined by the step points. This ensures accurate graphing and understanding of the function's behavior.

Explore how piecewise linear functions are defined using unit step functions. Learn about the function f(t) = [u(t-2) - u(t-5)](3 - 2t), its structure, and how it represents a straight line decreasing from -1 to -7 over the interval [2, 5]. Ideal for advanced high school students and learners interested in mathematical functions.

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