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Chapter 4: Problem 13

In Exercises 5-18, find the period and amplitude. \(y\ =\ 3\ sin\ 10x\)

Short Answer

Expert verified
The amplitude of the function \(y = 3 \sin 10x\) is 3 and the period is \( \pi/5\).

Step by step solution

01

Determine the amplitude

The amplitude of the sine function is the absolute value of the coefficient of the sine function. Looking at the equation \(y = 3 \sin 10x\), it can be noted that the coefficient of the sine function is 3. Therefore, the amplitude of the function is |3| = 3.
02

Determine the period

The period of a sine function is given by \(2\pi / B\) where \(B\) is the coefficient of \(x\). In the given equation \(y = 3 \sin 10x\), \(B = 10\). Therefore, the period of the function can be calculated by dividing \(2\pi\) by 10, which gives \( \pi/5\).

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Key Concepts

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

Amplitude of a Sine Function
The amplitude of a sine function represents the height of the wave from the midpoint to its highest or lowest point. In simple terms, it determines how "tall" or "short" the sine wave is.

In the standard sine function equation, written as \( y = a \sin(bx) \), the amplitude is represented by \(|a|\). Here, \(|a|\) means the absolute value of \(a\), which ensures that the amplitude is always a positive number.

  • For the equation \(y = 3\sin(10x)\), the coefficient of \(\sin(10x)\) is 3.
  • The amplitude, therefore, is \(|3| = 3\).
So in this case, the sine wave reaches up to 3 units above and 3 units below its central axis. Understanding amplitude helps in visualizing the vertical stretch or compression of the sine wave.
Period of a Sine Function
The period of a sine function tells us how long it takes for the wave to complete one full cycle. It's like finding out the length of a loop before the motion repeats.

For the general function \( y = a \sin(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \). Here, \(b\) is the value that affects the frequency of the wave.

  • In \(y = 3\sin(10x)\), \(b\) is 10.
  • The period is thus \( \frac{2\pi}{10} = \frac{\pi}{5} \).
This tells us that, before the sine wave of the equation completes and starts anew, it spans a horizontal length of \( \frac{\pi}{5} \) units on the graph. A shorter period results in more waves within a given interval, so understanding the period is crucial for graphing accurately.
Sine Function Properties
The sine function is a periodic function commonly represented by the formula \( y = a \sin(bx + c) + d \). Each component of this equation changes the shape and position of its wave.

Here are key properties of the sine function:

  • Amplitude: \( |a| \) defines the wave's height.
  • Period: Calculated as \( \frac{2\pi}{b} \), it shows the horizontal stretch.
  • Horizontal Shift (Phase Shift): Determined by \(c\), it shifts the wave left or right. The shift formula is \(-\frac{c}{b}\).
  • Vertical Shift: Represented by \(d\), it moves the wave up or down.
For our specific example \( y = 3 \sin(10x) \), there's no shift because both \(c\) and \(d\) are zero. Thus, the sine wave is centered on the origin and only affected vertically by amplitude and horizontally by period. This foundational knowledge of sine function properties allows one to adjust function parameters to reflect real-world sinusoidal data.

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