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Chapter 7: Problem 16

\(3-16\) Graph the function. $$ h(x)=|\sin x| $$

Short Answer

Expert verified
The graph of \( h(x) = |\sin x| \) resembles the sine function but is reflected above the x-axis wherever \( \sin x \) is negative.

Step by step solution

01

Understand the Function

The function we are dealing with is \( h(x) = |\sin x| \). The absolute value function ensures that the outputs are always non-negative. Thus, any negative value of \( \sin x \) is converted to a positive value.
02

Identify Characteristics of \( \sin x \)

The sine function \( \sin x \) is periodic with period \(2\pi\), oscillating between -1 and 1. It crosses the x-axis at multiples of \(\pi\).
03

Apply Absolute Value

When an absolute value is applied to \( \sin x \), the graph will have the same points as \(\sin x\) for values that are non-negative. When \( \sin x \) is negative, the graph will be reflected above the x-axis.
04

Graph from 0 to \(2\pi\)

For the interval \([0, 2\pi]\), the graph of \(|\sin x|\) will resemble the sine wave in the first half and a reflected sine wave in the second half. Specifically, it will start at 0, peak at 1 at \(\frac{\pi}{2}\), return to 0 at \(\pi\), then instead of going negative, it will peak to 1 at \(\frac{3\pi}{2}\) due to the reflection, and return to 0 at \(2\pi\).
05

Extend the Graph Beyond \(2\pi\)

The graph periodically repeats this pattern for every \(2\pi\) interval. Hence, for other intervals like \( [2\pi, 4\pi] \), the graph will look identical to the one between \( [0, 2\pi] \).
06

Draw the Graph

Plot the critical points determined in Steps 4 and extend periodically. Use symmetry property along the x-axis and the periodic nature to draw these repeated patterns over several periods effectively.

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Key Concepts

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

Periodic Functions
A periodic function is a function that repeats its values at regular intervals or periods. This repetition is what makes periodic functions so unique and applicable to many real-world situations:
  • They can describe harmonic motions, like sound waves or even the orbits of planets.
  • The "period" of a function is the smallest positive length for which the function's values repeat.
For example, in the function \(h(x) = |\sin x|\), the periodic nature of \(|\sin x|\) stems from the original sine function, \(\sin x\), which has a period of \(2\pi\). When the absolute value is applied, the negative parts of \(\sin x\) are reflected above the x-axis, but the overall period remains \(2\pi\). Thus, as you continually increase x, the pattern of the graph repeats every \(2\pi\) units.
Sine Function
The sine function, \(\sin x\), is a foundational trigonometric function. It oscillates in a smooth, wave-like pattern that is defined by several key characteristics:
  • It oscillates between -1 and 1, which means that the maximum value is 1 and the minimum value is -1.
  • The period of this wave is \(2\pi\), meaning the function completes one full cycle every \(2\pi\) units along the x-axis.
  • It crosses the x-axis at multiples of \(\pi\), specifically at \(x = 0\), \(\pi\), \(2\pi\), and so forth.
When we take the absolute value of the sine function, \(\left|\sin x\right|\), all negative values of the sine curve are transformed into positive ones. This retains the original peaks but reflects any troughs across the x-axis, creating a continuous series of arcs reaching up to +1.
Reflection of Graphs
Reflection of graphs is a transformation that "flips" the graph over a specific axis. In the context of the function \(h(x) = |\sin x|\):
  • The negative portions of the sine wave (which would normally dip below the x-axis) are reflected above the x-axis.
  • This means each negative crest becomes a positive peak.
  • The graph's symmetry about the x-axis is preserved due to this reflection.
Reflecting a function such as \(\sin x\) using absolute value ensures that all y-values are non-negative.This is why the graph of \(|\sin x|\) has a "V" shape where it previously had valleys below zero.
Intervals of Periodicity
Intervals of periodicity refer to the specific domain over which a periodic function completes one complete cycle before it repeats.
  • For the function \(h(x) = |\sin x|\), its periodicity is based on the original sine function, which has a cycle of \(2\pi\).
  • Therefore, each full pattern on the graph of \(|\sin x|\) spans from \(0\) to \(2\pi\), \(2\pi\) to \(4\pi\), and so on.
These intervals are crucial for graphing because they allow us to predictably replicate the function's behavior along the x-axis repeatedly.This predictability is utilized in many applications, such as sound engineering, where understanding the periodicity of waveforms is essential.

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Most popular questions from this chapter

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\). \(\left(\frac{3}{5}, \frac{4}{5}\right)\)

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Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\tan t, \cos t ; \quad\) Quadrant III

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