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Chapter 6: Problem 15

What is the amplitude of the function \(4 \sin x ?\)

Short Answer

Expert verified
The amplitude of the function \(4 \sin x\) is 4.

Step by step solution

01

Identify the sine function and its coefficient

The given function is \(4 \sin x\). Here, the sine function is \(\sin x\) and its coefficient is 4.
02

Find the absolute value of the coefficient

The amplitude is the absolute value of the coefficient of the sine function, which is \(|4|\).
03

Determine the amplitude

Since the absolute value of 4 is 4 itself, the amplitude of the function \(4 \sin x\) is 4.

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Key Concepts

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

Sine Function
The sine function is a fundamental building block in trigonometry and is usually represented as \( \sin x \).
The sine function produces a wave-like graph, often termed a sinusoidal wave. This wave has some specific properties, such as amplitude, period, and frequency. In mathematics, the sine function is used to model periodic phenomena like sound waves, light waves, and other cyclical data.
  • The standard sine function, \( \sin x \), oscillates between -1 and 1.
  • It has a period of \( 2\pi \), meaning the function repeats its pattern every \( 2\pi \) units.
  • The sine function is important in studying angles and triangles, especially in Cartesian coordinates.
Understanding the sine function's behavior aids in tackling problems involving trigonometric functions and periodic patterns.
Coefficient
A coefficient in mathematics is a constant multiplier of a certain term, and it provides important information on how terms are scaled in expressions or functions.
In the function \( 4 \sin x \), the number 4 is a coefficient.
  • The coefficient can be any real number and dictates the "steepness" or "stretch" of the function graph along the y-axis.
  • In trigonometric functions like the sine function, the coefficient represents the amplitude when it is multiplied by the function.
  • This means that the graph of the function will stretch or compress vertically based on the coefficient鈥檚 absolute value.
The role of the coefficient in a sine function specifically determines how far from the center line (usually the x-axis) the peaks and valleys of the graph reach.
Absolute Value
Absolute value is a concept that plays a significant role in understanding various mathematical functions, including trigonometric functions.
It is denoted by vertical bars, like this: \(|x|\).
  • The absolute value of a number is always non-negative. For positive numbers and zero, the absolute value is the number itself; for negative numbers, it is the positive equivalent.
  • In the context of our function \( 4 \sin x \), the absolute value \(|4| = 4\) gives the amplitude.
  • This means the waveform of \( 4 \sin x \) will reach up to 4 units above and below the central axis (typically the x-axis).
Understanding the absolute value helps in numerous mathematical calculations and is essential for determining characteristics like the amplitude in trigonometric functions.

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