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Chapter 0: Problem 25

Identify the amplitude, period and frequency. $$f(x)=3 \sin 2 x$$

Short Answer

Expert verified
The amplitude is 3, the period is \( \pi \), and the frequency is \( \frac{1}{\pi} \).

Step by step solution

01

Identify the Amplitude

The amplitude is the absolute value of the coefficient of the sine function. In this case, it is the absolute value of 3 which gives us 3 as the amplitude.
02

Identify the Period

The period of a sine function is given by \( \frac{2\pi}{|B|} \). In this case, B is 2. So the period is \( \frac{2\pi}{2} = \pi \).
03

Identify the Frequency

The frequency is the reciprocal of the period. So, it is \( \frac{1}{\pi} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
The amplitude of a trigonometric function, like the sine function, represents the height from the middle of the wave to its peak. It shows the extent of the vibration or oscillation. For a function in the form of \( f(x) = a \sin(bx) \), the amplitude is the absolute value of \( a \). This means the amplitude is a positive number, indicating how far up and down the graph will go from its central axis.
  • In the equation \( f(x) = 3 \sin 2x \), the coefficient of the sine term is 3.
  • Thus, the amplitude is \( |3| = 3 \).
The amplitude shapes the vertical stretch of the wave and determines how tall the peaks and low the troughs will be. It is crucial in applications like sound waves, where precise amplitude can alter loudness.
Sine Function
The sine function is one of the fundamental trigonometric functions. It is periodic, meaning it repeats its values in regular intervals. The standard form of a sine function is \( f(x) = a \sin(bx + c) + d \), where:
  • \( a \) represents the amplitude.
  • \( b \) influences the period and frequency.
  • \( c \) shifts the graph horizontally.
  • \( d \) shifts the graph vertically.

The sine wave is smooth, continuous, and symmetric around the origin. It starts from zero, moves to its peak, falls back through zero to its trough, and finally returns to zero. Understanding the properties of the sine function is crucial for mastering topics involving waves and oscillations.
Period
The period of a trigonometric function, like the sine function, is the length of one complete cycle of the wave. It indicates the horizontal length before a pattern repeats itself. The formula to calculate the period for \( f(x) = a \sin(bx) \) is \( \frac{2\pi}{|b|} \).
  • In our example, \( f(x) = 3 \sin 2x \), \( b = 2 \).
  • Thus, the period is \( \frac{2\pi}{2} = \pi \).

A shorter period results in more cycles over the same interval, leading to a higher frequency. Periodic functions, like this, help describe repeating patterns found in real-world phenomena, such as tides and electrical currents.
Frequency
Frequency describes how often a cycle repeats in a unit interval. It is essential in understanding the behavior of waves and oscillations. Mathematically, frequency is the reciprocal of the period. If the period of a function is \( T \), then the frequency \( f \) is \( \frac{1}{T} \).
  • From our function \( f(x) = 3 \sin 2x \), the period is \( \pi \).
  • Thus, the frequency is \( \frac{1}{\pi} \).

Frequency is typically measured in cycles per second (Hertz). By analyzing the frequency of a sine wave, one can determine how fast the wave oscillates. In practical terms, high frequencies can affect the pitch of a sound or the color of light.

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Most popular questions from this chapter

Use a graphing calculator or computer graphing utility to estimate all zeros. $$f(x)=x^{4}-7 x^{3}-15 x^{2}-10 x-1410$$

Discuss whether the function described has an inverse. Suppose that your boss informs you that you have been awarded a \(10 \%\) raise. The next week, your boss announces that due to circumstances beyond her control, all employees will have their salaries cut by \(10 \% .\) Are you as well off now as you were two weeks ago? Show that increasing by \(10 \%\) and decreasing by \(10 \%\) are not inverse processes. Find the inverse for adding \(10 \%\) (Hint: To add \(10 \%\) to a quantity you can multiply the quantity by \(1.10 .)\)

A standard graphing window will not reveal all of the important details of the graph. Adjust the graphing window to find the missing details. $$f(x)=x^{4}-11 x^{3}+5 x-2$$

Adjust the graphing window to identify all vertical asympotes. $$f(x)=\frac{2 x}{x+4}$$

Rewrite the expression as a single logarithm. $$\ln 3-\ln 4$$
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