To determine the equation of the function modeled by the graph of damped harmonic motion, we'll follow these steps:

Step 1: Identify the Damping Factor

The problem states that the function has a damping effect provided by a factor of $e^{-t}$. Therefore, the function will have a damping factor of $e^{-kt}$, where $k$ is a constant that will determine the rate of damping. Since the problem specifies $e^{-t}$ as the damping factor, we can conclude $k = 1$. Hence, the damping term is $e^{-t}$.

Step 2: Determine the Amplitude

The amplitude of the undamped function (i.e., the trigonometric part of the function) is given as 4, since the wave peaks at $y = 4$ and $y = -4$. However, due to damping, the amplitude decreases over time and is multiplied by $e^{-t}$. So, the amplitude at any time $t$ is $4e^{-t}$.

Step 3: Identify the Period of the Function

The problem specifies that the function completes a cycle every 12 units. The period $T$ of a trigonometric function is related to the angular frequency $\omega$ by the formula: $T = \frac{2\pi}{\omega}$ Given $T = 12$, we solve for $\omega$: $\omega = \frac{2\pi}{T} = \frac{2\pi}{12} = \frac{\pi}{6}$

Step 4: Combine the Components

The general form of the damped harmonic motion equation is: $y(t) = A \cdot e^{-kt} \cdot \cos(\omega t)$ Substituting the values we've determined: $y(t) = 4e^{-t} \cos\left(\frac{\pi}{6}t\right)$

Final Equation

The equation of the function in terms of $y$ and $t$ is: $\boxed{y(t) = 4e^{-t} \cos\left(\frac{\pi}{6}t\right)}$

This equation models the damped harmonic motion described in the problem.

Would you like further details or have any questions? Here are five related questions that could expand your understanding:

1. How does changing the value of $k$ in the damping factor $e^{-kt}$ affect the graph?
2. What would happen to the function if the amplitude were doubled?
3. How would the equation change if the period were different, say 6 units instead of 12?
4. Can you derive the equation for a function that has a vertical shift?
5. What is the physical interpretation of the damping factor in a real-world scenario?

Tip: The damping factor $e^{-t}$ causes the amplitude to decrease exponentially over time, which is a key feature in many real-world oscillatory systems, such as springs and electrical circuits.