Examine the following graph of a function modeling damped harmonic motion. Find the equation for the function pictured in terms of y and t. Assume that a factor of e−t provides the desired damping effect and that the graph has no vertical or horizontal shifts.

t
1
2
3
4
y
−4
−2
2
4
Cartesian Coordinate PlaneThe horizontal axis is labeled with t. The horizontal axis contains 9 grid lines, starts at 0, and ends at 4; and the axis is labeled with the values: 1, 2, 3, 4.The vertical axis is labeled with y. The vertical axis contains 11 grid lines, starts at -5, and ends at 5; and the axis is labeled with the values: -4, -2, 2, 4.Upper Dampening FunctionLower Dampening FunctionFunctionA trigonometric curve is drawn which is dampened by the exponential functions which cross the vertical axis -4 units above and below the origin. At t=0 the function has a value of 0, the wave is below the horizontal axis for the first half of each cycle, the wave is above the horizontal axis for the second half of each cycle, and the function completes a cycle every 12 units.

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.

Learn how to find the equation for damped harmonic motion from a given graph, analyzing a trigonometric curve dampened by exponential functions. Understand the concept of exponential decay and trigonometric functions in modeling oscillatory systems. Suitable for advanced high school students and learners interested in mathematical modeling.

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