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Chapter 4: Problem 46

Graph one complete cycle for each of the following. In each case, label the axes so that the amplitude and period are easy to read. $$ y=2 \sin 4 x $$

Short Answer

Expert verified
Amplitude is 2, period is \( \frac{\pi}{2} \). Label axes accordingly and plot critical points.

Step by step solution

01

Identify Amplitude and Period

The function given is \( y = 2 \sin 4x \). The amplitude is the coefficient of the sine function, which is 2. The period of a sine function is given by \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \). In this case, \( b = 4 \), so the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
02

Set Up the Axes

Label the vertical axis with multiples of the amplitude. Since the amplitude is 2, the vertical axis should be labeled from -2 to 2. For the horizontal axis, since the period is \( \frac{\pi}{2} \), label it such that one complete cycle fits within this interval. This means marking the horizontal axis with divisions at \( 0, \frac{\pi}{8}, \frac{\pi}{4}, \frac{3\pi}{8}, \) and \( \frac{\pi}{2} \).
03

Plot Critical Points and Sketch the Graph

At \( x = 0 \), \( \sin(0) = 0 \), so \( y = 0 \). At \( x = \frac{\pi}{8} \), \( \sin\left(4 \times \frac{\pi}{8}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \), so \( y = 2 \). At \( x = \frac{\pi}{4} \), \( \sin\left(4 \times \frac{\pi}{4}\right) = \sin(\pi) = 0 \), so \( y = 0 \). At \( x = \frac{3\pi}{8} \), \( \sin\left(4 \times \frac{3\pi}{8}\right) = \sin\left(\frac{3\pi}{2}\right) = -1 \), so \( y = -2 \). At \( x = \frac{\pi}{2} \), \( \sin(2\pi) = 0 \), so \( y = 0 \). Plot these points and sketch the sine curve connecting them smoothly.
04

Finalize the Graph

Ensure the graph shows one complete cycle from \( x = 0 \) to \( x = \frac{\pi}{2} \). The vertical axis should range from -2 to 2, showing the maximum and minimum values of the sine function. The horizontal axis confirms that the period is \( \frac{\pi}{2} \), illustrating one full cycle of the function.

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Key Concepts

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

Graphing Trigonometric Functions
Graphing trigonometric functions like the sine or cosine can provide a visual representation of their cyclical nature.
These graphs are periodic and repeat after a specific interval, known as the period. When graphing, it's important to capture one complete cycle to show this pattern clearly.

When graphing the function \( y = 2 \sin 4x \):
  • The first step is to identify key parameters like amplitude and period.
  • Next, set up your axes based on these parameters for better visibility.
  • Mark and plot critical points within the determined interval.
  • Draw a smooth curve through these points to complete the graph.
Graphing helps us visualize complex mathematical functions, turning abstract numbers into something tangible and easy to understand.
Amplitude of Sine Function
Amplitude is a critical feature of trigonometric functions that dictates their vertical stretch. It tells us how far the graph stretches from its midline.
For a sine function \( y = a \sin bx \), the amplitude is the absolute value of \( a \).

In our exercise with \( y = 2 \sin 4x \):
  • The amplitude is 2, meaning the wave reaches a maximum and minimum of 2 and -2, respectively.
  • The vertical axis should be marked from -2 to 2 to reflect these values.
  • This clearly shows the function oscillating between its maximum and minimum values.
Understanding amplitude helps in determining how intense or mellow the swings of the sine wave are. It's crucial for applications in physics, engineering, and sound waves.
Period of Trigonometric Functions
The period of a trigonometric function indicates the interval after which the function repeats its pattern.
For functions of the form \( y = a \sin bx \), the period is calculated using \( \frac{2\pi}{|b|} \).

In our equation \( y = 2 \sin 4x \):
  • The coefficient \( b = 4 \), so the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).
  • This tells us that a complete cycle of the sine function occurs between \( x = 0 \) and \( x = \frac{\pi}{2} \).
  • Mark and divide the horizontal axis accordingly to fit one cycle within this range.
Analyzing the period is vital for predicting future values and understanding the rhythm of oscillations in various scientific and technological applications.

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Most popular questions from this chapter

Make a table using multiples of \(\pi / 4\) for \(x\) to sketch the graph of \(y=\sin 2 x\) from \(x=0\) to \(x=2 \pi\). After you have obtained the graph, state the number of complete cycles your graph goes through between 0 and \(2 \pi\).

Graph one complete cycle for each of the following. In each case, label the axes so that the amplitude and period are easy to read. $$ y=4 \sin 2 x $$

Identify the amplitude for each of the following. Do not sketch the graph. $$ y=-\frac{1}{4} \cos x $$

Graph one complete cycle for each of the following. In each case, label the axes accurately and state the period, vertical translation, and horizontal translation for each graph. \(y=\frac{3}{2}-\frac{1}{2} \cot \left(\frac{\pi}{2} x-\frac{3 \pi}{2}\right)\)

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find the amplitude of \(y=-4 \cos (2 x)\). a. \(-2\) b. 2 c. \(-4\) d. 4
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