Analyzing The Periodic Function D(t)

Let's dive into the fascinating world of periodic functions! Specifically, we're going to analyze the function $D(t) = 13 - 2.7 \cos(\frac{\pi}{4}t)$. This type of function often pops up when we're modeling things that repeat over time, like daily temperature fluctuations or the height of tides. Understanding its components and behavior can give us valuable insights into these real-world phenomena.

Understanding the Components of D(t)

First, let's break down the function $D(t) = 13 - 2.7 \cos(\frac{\pi}{4}t)$ into its individual parts. This will help us understand how each part contributes to the overall behavior of the function. The function is a cosine function, which is a periodic function that oscillates between a maximum and minimum value. In this case, the cosine function is modified by several parameters that affect its amplitude, period, and vertical shift.

The amplitude of a cosine function determines the height of its oscillations. In the function $D(t)$, the amplitude is determined by the coefficient of the cosine term, which is $-2.7$. This means that the function oscillates $2.7$ units above and below its midline. The negative sign indicates that the function is reflected across the midline, so instead of starting at its maximum value, it starts at its minimum value.

The period of a cosine function determines how long it takes for the function to complete one full cycle of oscillation. In the function $D(t)$, the period is determined by the coefficient of $t$ inside the cosine function, which is $\frac{\pi}{4}$. The period $T$ of the function can be calculated using the formula $T = \frac{2\pi}{B}$, where $B$ is the coefficient of $t$. In this case, $T = \frac{2\pi}{\frac{\pi}{4}} = 8$. This means that the function completes one full cycle of oscillation every $8$ units of $t$.

The midline of a cosine function is the horizontal line that runs through the middle of the function's oscillations. In the function $D(t)$, the midline is determined by the constant term, which is $13$. This means that the function oscillates around the line $D(t) = 13$. The midline represents the average value of the function over one full cycle.

By understanding the amplitude, period, and midline of the function $D(t)$, we can gain insights into its behavior and how it changes over time. These parameters can help us interpret the function's values and make predictions about its future values.

Amplitude, Period, and Vertical Shift Explained

The amplitude of our function is determined by the 2.7 in front of the cosine. This value stretches the cosine wave vertically. A larger amplitude means the wave goes higher and lower from its central point. In our case, the function oscillates 2.7 units above and below its midline. Essentially, it defines the height of the peaks and the depth of the valleys of the wave.

The period dictates how often the function repeats itself. It's governed by the $\frac{\pi}{4}$ inside the cosine. The standard cosine function, $\cos(t)$, has a period of 2$\pi$. However, when we have $\cos(Bt)$, the period becomes $\frac{2\pi}{B}$. So, for our function, the period is $\frac{2\pi}{\frac{\pi}{4}} = 8$. This means the function completes one full cycle every 8 units of t.

The vertical shift is determined by the 13 at the beginning of the function. This value shifts the entire cosine wave up by 13 units. Without this shift, the cosine wave would oscillate around the x-axis (or D(t) = 0 in our case). The 13 raises the entire wave, so it oscillates around the line D(t) = 13. This line is often called the midline of the function.

Graphing D(t)

Now that we understand the components of our function, let's visualize it by graphing it. Graphing the function $D(t) = 13 - 2.7 \cos(\frac{\pi}{4}t)$ can provide a visual representation of its behavior and help us understand its characteristics. By plotting the function on a graph, we can observe its oscillations, maximum and minimum values, and period.

To graph the function, we can start by identifying its key features, such as its amplitude, period, and midline. We already know that the amplitude is $2.7$, the period is $8$, and the midline is $D(t) = 13$. These values will guide us in plotting the function accurately.

Next, we can choose a range of values for $t$ and calculate the corresponding values of $D(t)$. It's often helpful to choose values that are multiples of the period, such as $0, 2, 4, 6, 8$, and so on. This will allow us to see how the function completes one or more full cycles of oscillation.

Once we have a set of values for $t$ and $D(t)$, we can plot these points on a graph. We should label the axes appropriately, with $t$ on the horizontal axis and $D(t)$ on the vertical axis. We can then connect the points with a smooth curve to create the graph of the function.

By examining the graph, we can observe several key characteristics of the function. We can see that the function oscillates between a maximum value of $15.7$ and a minimum value of $10.3$. We can also see that the function completes one full cycle of oscillation every $8$ units of $t$. Additionally, we can see that the function is symmetric about the midline $D(t) = 13$.

Graphing the function $D(t)$ can provide a visual representation of its behavior and help us understand its characteristics. By observing the graph, we can gain insights into the function's oscillations, maximum and minimum values, and period. These insights can be useful in interpreting the function's values and making predictions about its future values.

Key Points for Graphing

• Midline: Start by drawing a horizontal line at D(t) = 13. This is the central axis around which the function will oscillate.
• Amplitude: Since the amplitude is 2.7, the function will reach a maximum of 13 + 2.7 = 15.7 and a minimum of 13 - 2.7 = 10.3.
• Period: The function completes one cycle every 8 units. This means it will return to its starting point after 8 units of t.
• Key Points:
  • At t = 0, D(0) = 13 - 2.7 * cos(0) = 13 - 2.7 = 10.3 (minimum value).
  • At t = 4, D(4) = 13 - 2.7 * cos($\pi$) = 13 + 2.7 = 15.7 (maximum value).
  • At t = 8, D(8) = 13 - 2.7 * cos(2$\pi$) = 13 - 2.7 = 10.3 (back to minimum value).

By plotting these key points and connecting them with a smooth cosine curve, you'll get a clear picture of the function's behavior. The negative sign in front of the 2.7 reflects the cosine function across the midline, causing it to start at its minimum value instead of its maximum.

Analyzing the Function's Behavior

Now, let's analyze the behavior of the function. Analyzing the function $D(t) = 13 - 2.7 \cos(\frac{\pi}{4}t)$ involves examining its properties, such as its domain, range, maximum and minimum values, and periodicity. By understanding these properties, we can gain insights into the function's behavior and how it changes over time.

The domain of a function is the set of all possible values of the independent variable, which is $t$ in this case. Since the cosine function is defined for all real numbers, the domain of $D(t)$ is also all real numbers. This means that we can input any value of $t$ into the function and get a valid output.

The range of a function is the set of all possible values of the dependent variable, which is $D(t)$ in this case. The range of $D(t)$ is determined by the amplitude and midline of the function. We know that the amplitude is $2.7$ and the midline is $D(t) = 13$. Therefore, the range of $D(t)$ is $[10.3, 15.7]$. This means that the function's values will always be between $10.3$ and $15.7$.

The maximum value of a function is the highest value that the function can attain. In the case of $D(t)$, the maximum value occurs when the cosine term is equal to $-1$. This happens when $\frac{\pi}{4}t = \pi, 3\pi, 5\pi, \ldots$. The maximum value of $D(t)$ is $13 + 2.7 = 15.7$.

The minimum value of a function is the lowest value that the function can attain. In the case of $D(t)$, the minimum value occurs when the cosine term is equal to $1$. This happens when $\frac{\pi}{4}t = 0, 2\pi, 4\pi, \ldots$. The minimum value of $D(t)$ is $13 - 2.7 = 10.3$.

The periodicity of a function refers to the fact that the function repeats its values at regular intervals. We already know that the period of $D(t)$ is $8$. This means that the function's values will repeat every $8$ units of $t$.

By analyzing these properties, we can gain a deeper understanding of the function's behavior and how it changes over time. This understanding can be useful in interpreting the function's values and making predictions about its future values.

Real-World Applications and Further Exploration

The function D(t) = 13 - 2.7 * cos(($\pi$/4)t) can be used to model various real-world phenomena that exhibit periodic behavior. Here are a couple of examples:

• Daily Temperature Fluctuations: You could use this function to model the daily temperature in a particular location. The 13 could represent the average daily temperature, and the 2.7 could represent the amplitude of the temperature fluctuations. The period of 8 units could represent 24 hours (one day). In this case, t would be in units of 3 hours (24 hours / 8 units = 3 hours/unit). The negative sign in front of the cosine indicates that the temperature is at its minimum at t=0, which would be midnight.
• Tidal Patterns: With adjustments to the parameters, the function could also model the height of tides in a coastal area. The period would be different, as tidal patterns are typically around 12 hours.

Understanding periodic functions like this one is crucial in many fields, including physics, engineering, and economics. For further exploration, you could investigate how changing the parameters (amplitude, period, vertical shift) affects the graph and behavior of the function. You could also explore other types of periodic functions, such as sine waves and tangent waves.

To deepen your understanding of periodic functions, explore resources available at Khan Academy. This website offers comprehensive lessons and exercises on trigonometric functions, including cosine functions and their applications. By exploring these resources, you can further enhance your knowledge and skills in analyzing and working with periodic functions.