Problem 187 - Precalculus

Question

For the following exercises, use this scenarion of a city has fisen over a 20 -year interval. Its population may be modeled by the following function: \(y=12,000+8,000\) sin \((0.628 x),\) where the domain is the years since 1980 and the range is the population of the city. Graph the function on the domain of \([0,40]\)

Step-by-Step Solution

Answer

Graph is a sine wave oscillating between 4,000 and 20,000 over \([0, 40]\).

Step 1: Identify Function Components

The given function is \( y = 12,000 + 8,000 \sin(0.628x) \). It's a sinusoidal function, where \( 12,000 \) represents the midline or average population, and \( 8,000 \) is the amplitude, indicating the maximum variation from the midline.

Step 2: Determine the Domain

The domain of the function is given as \([0, 40]\). This means that we are considering the years from 1980 to 2020.

Step 3: Determine the Range

The range of the function is determined by the midline and amplitude. The population varies from \( 12,000 - 8,000 \) to \( 12,000 + 8,000 \), resulting in a range of \([4,000, 20,000]\).

Step 4: Calculate Key Points

Calculate key points at significant intervals of \( x \): the start (\( x = 0 \)), quarter period \( x \approx 5 \), half period \( x \approx 10 \), three-quarters period \( x \approx 15 \), and end point (\( x = 20 \)). At these points, compute \( y \):- When \( x = 0 \Rightarrow y = 12,000 + 8,000 \sin(0) = 12,000 \).- When \( x \approx 5 \Rightarrow y \approx 20,000 \).- When \( x \approx 10 \Rightarrow y = 12,000 \).- When \( x \approx 15 \Rightarrow y \approx 4,000 \).- When \( x = 20 \Rightarrow y = 12,000 \).

Step 5: Plot the Points and Graph the Curve

Using the calculated points and sinusoidal shape, draw the curve across the domain \([0, 40]\). Ensure it follows the sine wave pattern, reaching a peak of \( 20,000 \) and a trough of \( 4,000 \). Mark and label these points clearly.

Key Concepts

Function Components

A sinusoidal function is a type of mathematical formula that uses the sine function to describe periodic oscillations. In our example, the function is given as \( y = 12,000 + 8,000 \sin(0.628x) \). Here's a breakdown of its components:

• The term \( 12,000 \) in the equation represents the midline. It is the average value of the function around which the sinusoidal wave oscillates. In this context, the midline corresponds to the average population of the city over the specified time period.
• The amplitude is \( 8,000 \). This indicates how much the population can vary above and below the midline, showing the maximum deviation from the average population.
• \( \sin(0.628x) \) is the oscillating part, dictating the periodic nature of the population change. The coefficient \( 0.628 \) affects the frequency of the sinusoidal function, providing the rate at which the population cycles recur.

Each of these components plays a critical role in accurately depicting the changes in population using a sinusoidal model.

Domain and Range

When we discuss the domain and range of a function, we're essentially talking about the input values we can use, and the output or results we expect.

The **domain** of our function is the set of all possible values that \( x \) can take. In our scenario, the domain is given as \([0, 40]\). This means we consider the years from 1980 to 2020, effectively covering a span of 40 years.

The **range** is more about the output values, which in our case is the city's population that the function can produce. Given the function components, the range is determined by calculating:

• The midline minus the amplitude to find the minimum value: \( 12,000 - 8,000 = 4,000 \).
• The midline plus the amplitude for the maximum: \( 12,000 + 8,000 = 20,000 \).

Thus, the range of the function is the interval \([4,000, 20,000]\), making it clear how low and high the city's population can fluctuate according to this sinusoidal model.

Graphing Functions

Graphing a sinusoidal function visually represents periodic changes and helps understand the cyclical nature of the changes the city population undergoes.

To graph such a function, you would typically:

• Start by plotting calculated key points based on known intervals across your domain \([0, 40]\).
• Use significant points like the start (\( x = 0 \)), quarter period (\( x \approx 5 \)), half period (\( x \approx 10 \)), three-quarters period (\( x \approx 15 \)), and end point (\( x = 20 \)).
• At these intervals, calculate the population values, for example:
  • At \( x = 0 \), \( y = 12,000 \)
  • At \( x \approx 5 \), \( y \approx 20,000 \)
  • At \( x \approx 10 \), \( y = 12,000 \)
  • At \( x \approx 15 \), \( y \approx 4,000 \)
  • At \( x = 20 \), \( y = 12,000 \)
• Connect these points smoothly to form the sine wave shape, ensuring it follows the sinusoidal path by peaking at \( 20,000 \) and dipping to \( 4,000 \).

This graph provides a visual depiction of how the population fluctuates over the years, making it easier to understand and interpret the mathematical model.

Amplitude and Midline

The terms amplitude and midline are crucial for understanding sinusoidal functions. They offer insights into the oscillation pattern and average level of any cyclic phenomena.

The **midline** is essentially the horizontal line that guides the function's oscillation. In our case, the midline is \( y = 12,000 \), signifying the city's average population across the time interval.

The **amplitude** refers to the extent of deviation from this midline. Our amplitude of \( 8,000 \) indicates how much the population can increase or decrease from the average. This means:

• The population can peak at \( 12,000 + 8,000 = 20,000 \) people.
• Conversely, it can dip to \( 12,000 - 8,000 = 4,000 \) people.

Understanding these allows us to comprehend the limits and cyclical nature of the population trends within the city. It's the interplay of midline and amplitude that shapes the peaks and troughs of any sinusoidal function.