https://file.aiquickdraw.com/mathbot/file/1725407480882b2twbljq.jpg

Let's solve the problem step by step.

### Part (a): Determine constants \(a\), \(b\), and \(c\)

Given:
- \( y(t) = -\frac{1}{3}(t - 3) + 2 \) for \( t \) between 0 and 6.
- The signal \( z(t) = ax(bt + c) \).

We need to determine the constants \(a\), \(b\), and \(c\) so that the signal \(z(t)\) matches the form in the figure.

The figure for \(z(t)\) is a piecewise function composed of linear segments:

1. For \(t\) between \(-4\) and \(0\), the line has a positive slope.
2. For \(t\) between \(0\) and \(6\), the line is flat at \(3\).

Let’s assume \(z(t) = ax(bt + c)\) can be written as:

\[ z(t) = at + b \]

To match the given \(z(t)\) with the shape in the figure:

- At \(t = 0\), \( z(t) = 3 \), hence:
  \[ 3 = a(0) + b \Rightarrow b = 3 \]

Now, consider the line segment from \(t = -4\) to \(t = 0\):
- For \(t = -4\), the value of \(z(t)\) is \(0\), so:
  \[ 0 = a(-4) + 3 \Rightarrow -4a = -3 \Rightarrow a = \frac{3}{4} \]

Therefore, \(z(t)\) can be expressed as:

\[ z(t) = \frac{3}{4}t + 3 \]

### Part (b): Determine and sketch a signal \(v(t)\) such that \(z(t) = \int_{-\infty}^{t} v(\tau) d\tau\)

Since \(z(t)\) is the integral of \(v(t)\), we need to find the derivative of \(z(t)\) to obtain \(v(t)\):

1. For \(t \) in the interval \([-4, 0]\):
   \[ z(t) = \frac{3}{4}t + 3 \]
   Therefore, \(v(t) = \frac{dz(t)}{dt} = \frac{3}{4}\)

2. For \(t \) in the interval \([0, 6]\):
   \[ z(t) = 3 \]
   Therefore, \(v(t) = \frac{dz(t)}{dt} = 0\)

Thus, the signal \(v(t)\) can be represented as:
\[
v(t) =
\begin{cases}
\frac{3}{4}, & \text{for } -4 \leq t < 0 \\
0, & \text{for } 0 \leq t \leq 6
\end{cases}
\]

Would you like me to provide more details or have any questions? Here are 5 related questions you might consider:

1. How would the shape of \(v(t)\) change if \(z(t)\) had different slopes in its linear segments?
2. Can you express \(z(t)\) using a different method, such as convolution?
3. What would the integral of \(v(t)\) look like if \(v(t)\) had a quadratic shape instead of a piecewise constant function?
4. How does the presence of a constant term \(c\) in \(ax(bt + c)\) influence the shape of \(z(t)\)?
5. How would the analysis change if \(y(t)\) was defined differently, say with a quadratic form instead of linear?

**Tip:** When sketching signals based on given conditions, always carefully analyze the slopes and intersections to determine the correct constants.

Define signals y(t) and z(t) as in Fig. P1.2-7. (a) Determine constants a, b, and c to produce z(t) = ax(bt + c). (b) Determine and sketch a signal v(t) such that z(t) = ∫v(τ)dτ.

Learn how to determine the constants a, b, and c to produce a signal z(t) as shown in a given figure. Also, find and sketch a signal v(t) such that z(t) is the integral of v(t). This problem involves concepts from signals and systems, including piecewise functions and integration. Suitable for undergraduate students.

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