Problem 188 - 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. What are the amplitude, period, and phase shift for the function?

Step-by-Step Solution

Answer

Amplitude: 8,000; Period: 10 years; Phase Shift: 0.

Step 1: Identifying the Amplitude

The amplitude of a sine or cosine function is the coefficient in front of the sine or cosine term. In the given function, the coefficient of the sine term is 8,000. Hence, the amplitude of the function is 8,000.

Step 2: Calculating the Period

The period of a sine function is given by the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\). In this function, \(b = 0.628\). Thus, the period is \(\frac{2\pi}{0.628} \approx 10\). This means the function will repeat every 10 years.

Step 3: Determining the Phase Shift

The phase shift of a sine function is determined by the horizontal shift, typically represented by \(\frac{-c}{b}\), where \(c\) is any horizontal translation (in this case, 0). Since there is no horizontal shift specified in the function, and \(c = 0 \), the phase shift is 0.

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function determines the height of its peaks. It measures the maximum distance from the middle of the wave to its farthest point. Simply put, it's how "tall" the wave is. In our example, the function is modeled as \(y=12,000+8,000\sin(0.628x)\). Here, the amplitude is given by the coefficient of the sine term, which is 8,000.

Some things to remember about amplitude:

• It always represents a positive value even though the coefficient may be negative.
• The larger the amplitude, the more extreme the highs and lows of the wave will be.
• The amplitudes don't shift the graph horizontally or vertically but affect the wave's height only.

Now you know that the sine wave varies 8,000 above and 8,000 below the middle value of 12,000.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. For sine functions, the formula to calculate the period is \(\frac{2\pi}{b}\), where \(b\) is the coefficient of the variable \(x\). In our given equation, \(b = 0.628\).

By using the formula, we can find the period:

• Calculate \(\frac{2\pi}{0.628} \approx 10\).

This tells us that the cycle of the population model repeats every 10 years. It means that every 10 years, the sinusoidal pattern of population change will reset and start over from the beginning. Knowing the period helps us anticipate when the same values in the sequence will occur again.

Phase Shift

Phase shift measures how much a wave is shifted horizontally from its usual position. In the typical sine function \(y = a\sin(bx + c) + d\), the phase shift formula is \(\frac{-c}{b}\). To apply this formula, we take \(c\) as the horizontal translation component.

In our scenario, since there is no horizontal shift (\(c = 0\)), the phase shift is zero. This means the wave starts exactly at the origin point without any shift to the left or right.

• A positive phase shift moves the wave to the left.
• A negative phase shift moves it to the right.

Since the phase shift in our model is zero, the population's sinusoidal fluctuations start right from 1980 without any adjustments.

Sine Function

The sine function, which is a fundamental trigonometric function, creates a smooth, periodic wave. It's essential for modeling natural phenomena swaying back and forth or events repeating over time, like our city's population dynamics. The standard form of a sine function is \(y = a\sin(bx+c)+d\).

Some key components of the sine function include:

• \(a\) is the amplitude, affecting the height of the wave.
• \(b\) controls the period, or how quickly the sine wave cycles.
• \(c\) represents the phase shift, indicating horizontal displacement.
• \(d\) is the vertical shift, which adjusts the entire wave up or down.

In the example equation, the sine function effectively models how a city's population rises and falls in a cyclic pattern. Understanding each of these components helps in visualizing how variables interact to reflect real-world changes.