Identify Amplitude, Period, And Phase Shift Of F(x)=4sin((1/2)(x-3)).

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In the realm of mathematics, sinusoidal functions stand as fundamental building blocks for modeling periodic phenomena. These functions, characterized by their smooth, oscillating waveforms, find applications in diverse fields, ranging from physics and engineering to economics and biology. Understanding the properties of sinusoidal functions, such as amplitude, period, and phase shift, is crucial for effectively interpreting and manipulating these mathematical tools.

Deciphering the Equation: A Step-by-Step Approach

To embark on our journey of understanding, let's consider the general form of a sinusoidal function:

f(x) = A sin(B(x - C)) + D

where:

  • A represents the amplitude, which dictates the vertical stretch of the function.
  • B governs the period, influencing the horizontal compression or expansion of the function.
  • C signifies the phase shift, determining the horizontal displacement of the function.
  • D indicates the vertical shift, dictating the vertical displacement of the function.

Now, let's apply this knowledge to our specific function:

f(x) = 4 sin((1/2)(x - 3))

By comparing this equation to the general form, we can readily identify the values of A, B, and C:

  • Amplitude (A): 4
  • B: 1/2
  • Phase Shift (C): 3

With these values in hand, we can proceed to calculate the period and provide a comprehensive interpretation of the function's characteristics.

Amplitude: Unveiling the Vertical Extent

Amplitude, denoted by A, quantifies the vertical distance between the function's midline and its maximum or minimum value. In simpler terms, it represents the "height" of the wave. For the function f(x) = 4 sin((1/2)(x - 3)), the amplitude is 4. This signifies that the function's graph oscillates between a maximum value of 4 and a minimum value of -4, with the midline situated at y = 0. Amplitude plays a crucial role in determining the intensity or strength of the periodic phenomenon being modeled. For instance, in sound waves, amplitude corresponds to loudness, while in electromagnetic waves, it relates to the wave's energy.

To further elaborate, consider a scenario where we are modeling the oscillation of a pendulum. The amplitude in this case would represent the maximum angular displacement of the pendulum from its resting position. A larger amplitude would indicate a wider swing, signifying a greater amount of energy stored in the system. Similarly, in the context of electrical circuits, the amplitude of an alternating current (AC) signal corresponds to the peak voltage or current. A higher amplitude implies a stronger electrical signal, capable of delivering more power.

The concept of amplitude extends beyond physical systems and finds applications in various fields. In economics, for example, the amplitude of a business cycle can represent the magnitude of fluctuations in economic activity, such as periods of expansion and contraction. In signal processing, amplitude modulation (AM) is a technique used to encode information onto a carrier wave by varying its amplitude. This technique is widely used in radio broadcasting, where the amplitude of the radio wave is modulated to transmit audio signals.

Understanding amplitude is not just about identifying a numerical value; it's about grasping the physical significance of that value in the context of the system being modeled. A thorough comprehension of amplitude allows us to interpret and analyze periodic phenomena with greater accuracy and insight.

Period: Measuring the Rhythmic Cycle

Period, denoted by P, measures the horizontal distance required for the function to complete one full cycle of oscillation. It essentially tells us how long it takes for the function to repeat itself. The period is inversely proportional to the value of B in the general equation f(x) = A sin(B(x - C)) + D, and is calculated using the formula:

P = (2π) / |B|

In our case, B = 1/2, so the period is:

P = (2π) / (1/2) = 4π

This indicates that the function f(x) = 4 sin((1/2)(x - 3)) completes one full cycle over an interval of 4π units along the x-axis. The period is a fundamental characteristic of any periodic function, as it dictates the frequency of oscillations. A shorter period implies a higher frequency, meaning the function oscillates more rapidly. Conversely, a longer period corresponds to a lower frequency, indicating slower oscillations.

To illustrate the significance of the period, let's consider the example of a simple harmonic oscillator, such as a mass attached to a spring. The period of oscillation represents the time it takes for the mass to complete one full cycle of movement, returning to its initial position and velocity. A shorter period would mean the mass oscillates more quickly, while a longer period would indicate slower oscillations. The period is influenced by factors such as the mass and the spring constant, with heavier masses and weaker springs resulting in longer periods.

In the context of sound waves, the period is inversely related to the frequency, which determines the pitch of the sound. A shorter period corresponds to a higher frequency and a higher pitch, while a longer period corresponds to a lower frequency and a lower pitch. For example, a high-pitched note on a musical instrument has a shorter period than a low-pitched note.

The concept of period extends beyond physical systems and finds applications in diverse fields. In astronomy, the period of a planet's orbit around the sun represents the time it takes for the planet to complete one full revolution. In financial markets, the period of a market cycle can represent the time it takes for the market to complete a full cycle of expansion and contraction. Understanding the period of a periodic phenomenon is crucial for predicting its future behavior and making informed decisions.

Phase Shift: Unveiling the Horizontal Displacement

Phase shift, denoted by C, represents the horizontal displacement of the function's graph from its standard position. In the general equation f(x) = A sin(B(x - C)) + D, the phase shift is the value of C. For our function f(x) = 4 sin((1/2)(x - 3)), the phase shift is 3. This indicates that the graph of the function is shifted 3 units to the right compared to the standard sine function f(x) = sin(x). Phase shift is a crucial parameter for aligning sinusoidal functions with real-world data.

The phase shift determines the starting point of the sinusoidal cycle. A positive phase shift shifts the graph to the right, while a negative phase shift shifts the graph to the left. The magnitude of the phase shift indicates the amount of horizontal displacement. In the context of our example, the phase shift of 3 means that the graph of f(x) = 4 sin((1/2)(x - 3)) starts its cycle 3 units to the right of the origin.

To illustrate the significance of phase shift, consider the example of modeling the tides. The tides are caused by the gravitational forces of the moon and the sun, which create periodic variations in sea level. The phase shift in this context can represent the time delay between the lunar cycle and the occurrence of high tide at a particular location. This delay is influenced by factors such as the shape of the coastline and the depth of the ocean.

In the context of alternating current (AC) circuits, the phase shift can represent the time difference between the voltage and current waveforms. This phase difference is crucial for understanding the power flow in the circuit and designing efficient electrical systems. A large phase shift between voltage and current can indicate a reactive load, which consumes energy without performing useful work.

The concept of phase shift extends beyond physical systems and finds applications in various fields. In signal processing, phase shift is used to manipulate the timing of signals, which is crucial for applications such as audio and video processing. In image processing, phase shift is used to enhance the features of an image, such as edges and textures. Understanding phase shift is essential for accurately modeling and analyzing periodic phenomena in diverse contexts.

Putting It All Together: A Comprehensive Analysis

In summary, for the function f(x) = 4 sin((1/2)(x - 3)), we have identified the following characteristics:

  • Amplitude: 4
  • Period: 4π
  • Phase Shift: 3 units to the right

These parameters provide a complete description of the sinusoidal function's behavior. The amplitude of 4 indicates the vertical extent of the oscillations, the period of 4π determines the length of each cycle, and the phase shift of 3 units to the right specifies the horizontal displacement of the graph. By understanding these parameters, we can effectively interpret and manipulate sinusoidal functions in various mathematical and real-world applications.

Choosing the Correct Answer

Based on our analysis, we can conclude that the correct answer is:

  • Amplitude is 4; period is 4π; phase shift is 3 to the right.

This comprehensive analysis demonstrates the importance of understanding amplitude, period, and phase shift in characterizing sinusoidal functions. These parameters provide valuable insights into the function's behavior and allow us to apply sinusoidal functions effectively in various contexts.

Mastering Sinusoidal Functions: A Path to Mathematical Proficiency

Understanding the amplitude, period, and phase shift of sinusoidal functions is a cornerstone of mathematical proficiency. These concepts not only unlock the secrets of trigonometric functions but also provide a powerful framework for modeling and analyzing periodic phenomena across diverse disciplines. By mastering these fundamental properties, you'll gain a deeper appreciation for the elegance and versatility of mathematics in describing the world around us.

In conclusion, identifying the amplitude, period, and phase shift of a sinusoidal function involves a systematic approach of comparing the given equation to the general form and applying the appropriate formulas. A thorough understanding of these parameters empowers us to interpret and manipulate sinusoidal functions effectively, paving the way for advanced mathematical explorations and real-world applications.