Problem 93 - A Graphical Approach to Precalculus with Limits

Question

The monthly average temperatures in a Canadian city are shown in the table. $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline \text { Temperature ('F) } & 40 & 43 & 47 & 52 & 59 & 63\end{array}$$ $$\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Month } & 7 & 8 & 9 & 10 & 11 & 12 \\\\\hline \text { Temperature }\left(^{\circ} \mathrm{F}\right) & 68 & 67 & 61 & 54 & 47 & 43\end{array}$$ (a) Plot the average monthly temperature over a 24 -month period by letting \(x=1\) and \(x=13\) correspond to January. (b) Find the constants \(a, b, c,\) and \(d\) so that the function \(f(x)=a \sin [(b(x-c)]+d\) models the data. (c) Graph \(f\) together with the data.

Step-by-Step Solution

Answer

Plot data for two years, determine function constants, model with sinusoidal function, graph both the function and data.

Step 1: Plot the Temperatures for 24 Months

First, we will visualize the data by plotting the temperatures over a 24-month period. For the months 1 to 12, use the temperatures given directly. For months 13 to 24, reuse the temperatures for January through December (months 1 to 12). Plot each month as a point with the month as the x-coordinate and the temperature as the y-coordinate.

Step 2: Analyze Data to Determine Sinusoidal Behavior

Examine the plot to see if it exhibits sinusoidal behavior. Notice that the temperatures start low, rise to a maximum in the middle of the year, and then decrease again, which suggests a sinusoidal pattern. Confirm that the data repeats every 12 months, typical of a sinusoidal function.

Step 3: Identify Parameters for the Sinusoidal Model

Find the values of the constants in the sinusoidal model, \(f(x) = a \, \sin[b(x - c)] + d\). - **Amplitude (\(a\))**: Calculate the amplitude as half the difference between the maximum and minimum temperatures. Here, it's \((68 - 40) / 2 = 14\).- **Vertical Shift (\(d\))**: Find \(d\) as the average of the maximum and minimum temperatures, which is \((68 + 40) / 2 = 54\).- **Period (\(T\))**: Confirm the period is 12 months since the cycle repeats every year. Thus, \(b = \dfrac{2\pi}{T} = \dfrac{2\pi}{12} = \dfrac{\pi}{6}\).- **Horizontal Shift (\(c\))**: Determine \(c\) by finding the month where the temperature reaches its first maximum. From the data, the peak occurs in July (month 7), so adjust the sine function to reflect this shift: \(b(x - c) = \dfrac{\pi}{6}(x - 4)\).

Step 4: Write the Function

Substitute the calculated constants back into the sinusoidal function: \[f(x) = 14 \, \sin\left(\dfrac{\pi}{6} (x - 4) \right) + 54\] This function should model the periodic temperature fluctuations observed in the data.

Step 5: Graph the Function and Data

Plot the function \(f(x) = 14 \, \sin\left(\dfrac{\pi}{6} (x - 4) \right) + 54\) on the same graph as the data points for 24 months. Make sure the curve closely follows the plotted points, indicating the model accurately represents the temperature changes throughout the year.

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visualize the behavior of data points and identify patterns within them. In this exercise, the task is to plot monthly average temperatures over a 24-month period. By letting January correspond to both month 1 and month 13, we can create a seamless cycle of temperature data. This approach allows us to better observe the repetition and shape of the temperature changes over time.
Here's what to do:

• Start with plotting each month's temperature as a point. Use the month as the x-coordinate and the temperature as the y-coordinate.
• Repeat the temperature values for months 1 to 12 as months 13 to 24, continuing the pattern to cover two full years.

Once plotted, you should notice the wave-like pattern of the temperatures. This wave is characteristic of a sinusoidal function, revealing the periodic nature of temperature changes throughout the year. Graphing these functions helps us to identify when temperatures are highest and lowest, providing insights into seasonal trends.

Amplitude and Period

In sinusoidal modeling, amplitude and period are crucial parameters that shape the behavior of the function. These parameters help in closely fitting a sinusoidal model to data like temperature variations. Let's break down what they mean and how you determine them:
Amplitude tells us how far the temperature varies from its central value. To find it, calculate half the distance between the maximum (68°F) and minimum (40°F) temperatures. Hence, the amplitude in our exercise is \(a = \frac{68 - 40}{2} = 14\).

• The amplitude reflects the energy or the range of the temperature fluctuation.
• A large amplitude indicates a significant seasonal temperature change.

Period refers to how long it takes for the temperature cycle to repeat. The period in monthly temperature data is 12 months because temperatures repeat every year. This means the frequency component (\

Temperature Data Analysis

Analyzing temperature data to find patterns involves examining how it changes over time and identifying cycles. With sinusoidal modeling, like we've done here, you can see how well this type of function suits cyclical data such as temperatures.
Here's why temperature data analysis is valuable:

• It helps in understanding weather and climate trends, which are essential for agriculture, energy management, and daily life planning.
• Temperature analysis using sinusoidal models can predict future temperature patterns based on historical data, aiding preparation for extreme weather conditions.

By visualizing and fitting a sinusoidal model to the temperature data, we not only see current trends but also predict future patterns. This insight is invaluable in various fields such as environmental science and urban planning. The sinusoidal model provides a simplified yet effective way to interpret and predict temperature data, encapsulating the essence of how temperatures fluctuate with the seasons.