Problem 54 - Contemporary Precalculus

Question

Initial push is downward from the equilibrium point. [Hint: What does the graph of \(A\) sin \(b t\) look like when \(A<0 ?\)

Step-by-Step Solution

Answer

Answer: The graph of a negative sinusoidal function with initial push downward from equilibrium, represented by the function \(-A \sin(bt)\), will begin at its minimum amplitude (downward from equilibrium position). From there, it will follow the same pattern as a regular sine wave, but flipped across the horizontal axis due to the negative amplitude. This means that the wave will proceed to the equilibrium point, reach its maximum value above the equilibrium position, return to the equilibrium point, and end with its minimum value again.

Step 1: Analyze the given function

First, we need to recognize the general form of a sinusoidal function: \(A \sin(bt)\). In this function, \(A\) represents the amplitude (the maximum value or the distance from the maximum displacement to the equilibrium point) and \(b\) represents the frequency (how many cycles occur in a given time).

Step 2: Identify the amplitude and frequency

Given that the initial push is downward from the equilibrium point and we have a negative amplitude, we can rewrite the function as \(-A \sin(bt)\) with \(A > 0\) (since amplitude is always positive) and \(b\) as the frequency.

Step 3: Graph the negative sinusoidal function

To graph the function \(-A \sin(bt)\), start by drawing the regular sine wave \(A \sin(bt)\) with a positive amplitude. The sine wave starts from the equilibrium position (horizontal axis), goes up to its maximum value, comes back to the equilibrium position, goes down to its minimum value, and ends back at the equilibrium position. Now, take the reflections of all the points on the sine wave about the horizontal axis (this changes the signs of all the points). This gives us the graph of the negative sinusoidal function \(-A \sin(bt)\). The wave now starts with its minimum value (downward from equilibrium), goes up to the equilibrium point, reaches its maximum value above the equilibrium position, comes back to the equilibrium point, and ends with its minimum value again.

Step 4: Interpret the graph

When the amplitude is negative, the graph of \(-A \sin(bt)\) will start with its minimum amplitude (downward from equilibrium position), reflecting the initial push is downward from the equilibrium point. The rest of the wave will follow the same pattern as a regular sine wave, but flipped across the horizontal axis due to the negative amplitude.

Key Concepts

Amplitude

In a sinusoidal function like \(A \sin(bt)\), the amplitude is a key feature. It determines how far the wave moves away from its central axis or equilibrium point. The amplitude \(A\) measures the maximum distance each point on the wave reaches from the center.
In simple terms, it tells you how tall or deep the waves are.

A positive amplitude makes the sine wave start upwards, reaching a high point first, while a negative amplitude causes the wave to start downwards, reflecting it across the horizontal axis.
This concept shows why the graph of \(-A \sin(bt)\) starts below the axis when \(A < 0\).

• The amplitude determines the wave's height.
• A negative amplitude flips the wave direction.

Knowing how amplitude affects a wave helps in predicting and understanding the behavior of physical systems, like sound waves or tides.

Frequency

The frequency in the function \(A \sin(bt)\) indicates how often the wave cycles repeat within a given time.
It is represented by the variable \(b\), and it affects the wave's speed or how compressed the wave looks.

Frequency is commonly thought of in terms of cycles per unit of time, such as per second, often identified as Hertz (Hz). The larger the value of \(b\), the more oscillations you'll see in the same time period.

• Higher frequency means more cycles in less time.
• Influences the wave's rate of oscillation.

Understanding frequency is crucial for decoding various natural phenomena or technologies, such as radio waves and signals. It essentially tells you how fast the wave is vibrating.

Graph of Sine Function

Visualizing the graph of a sine function helps grasp its dynamics. The standard sine wave \(A \sin(bt)\) forms a repeating pattern, symmetrically moving above and below the horizontal axis, known as the equilibrium line.
It starts at the equilibrium, rises to a peak, returns to the center, drops to a trough, and comes back up.

When you graph \(-A \sin(bt)\), you flip this motion across the axis:

• Begins below the center point.
• Rises through the equilibrium.
• Reaches a peak opposite the starting point of a positive sine graph.

This reflection helps explain how changes in signs or values impact the wave's orientation. Creating or analyzing these waves can be vital for tasks like sound synthesis, where negative or positive influences reshape the produced sounds.