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Chapter 4: Problem 21

Identify the horizontal translation for each equation. Do not sketch the graph. $$ y=\sin \left(x-\frac{2 \pi}{3}\right) $$

Short Answer

Expert verified
The function is translated right by \( \frac{2\pi}{3} \) units.

Step by step solution

01

Understand the Standard Form of the Transformation

To identify the horizontal translation, start with the standard form for a horizontal translation of a sine function: \( y = \sin(x-h) \). In this expression, \( h \) represents the horizontal shift. A positive \( h \) translates the function to the right, while a negative \( h \) translates it to the left.
02

Identify the Horizontal Shift Value

Compare the given equation \( y = \sin(x-\frac{2\pi}{3}) \) to the standard form. Here, \( h = \frac{2\pi}{3} \). Since \( h \) is positive in the expression \( x-h \, \) the translation will be to the right.
03

Conclude the Direction and Magnitude of the Shift

The function \( y = \sin(x-\frac{2\pi}{3}) \) represents a sine function that is horizontally translated to the right by \( \frac{2\pi}{3} \) units. This means the graph of the original sine function moves right but amplitude and period remain unchanged.

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Key Concepts

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

Sine Function Translation
In trigonometry, the sine function is known for its smooth, wave-like pattern that repeats every 2蟺 radians. Understanding the translation of the sine function can be quite useful. Let鈥檚 break down how a sine wave is shifted horizontally.
  • The basic sine function is represented by the equation: \( y = \sin(x) \).
  • When a horizontal translation is applied, you can express it as: \( y = \sin(x - h) \), where \( h \) represents the horizontal shift.
The value of \( h \) determines how the function is translated:
  • If \( h \) is positive, the graph shifts to the right by \( h \) units.
  • If \( h \) is negative, the graph shifts to the left by \( |h| \) units.
The formula remains quite intuitive as it indicates movement in direct relation to \( h \). This translation does not change the function鈥檚 amplitude or its period, allowing the sine wave to maintain its fundamental shape and properties.
Trigonometric Transformations
Aside from horizontal translations, there are several key ways trigonometric functions can be transformed. Each transformation alters the function's graph in specific ways without changing its inherent nature.
  • **Vertical Shift:** When a constant is added to a function, such as \( y = \sin(x) + k \), it results in a vertical shift. The sine wave moves up by \( k \) units if \( k \) is positive, or down by \( |k| \) units if negative.
  • **Amplitude Change:** By multiplying the function by a constant, \( y = a\sin(x) \), the amplitude changes. The function鈥檚 peaks and troughs stretch taller or shrink smaller, depending on the value of \( a \).
  • **Period Modification:** A change in period is expressed as \( y = \sin(bx) \). This alters the frequency of the waves - the greater the \( b \), the more waves appear over the same interval.
  • **Reflection:** Reflecting a trigonometric function can occur across the x-axis with \( y = -\sin(x) \), reversing the graph鈥檚 orientation vertically.
Recognizing these transformations and how they affect a function provides a deeper understanding of trigonometric equations and their versatile applications.
Horizontal Shift in Graphs
Understanding horizontal shifts in trigonometric graphs can make analyzing these functions more straightforward. When a curve is horizontally translated, its position moves laterally on the x-axis.For example, in the equation \( y = \sin(x - \frac{2\pi}{3}) \), there is a horizontal shift occurring.
  • This specific shift is seen in the form \( y = \sin(x - h) \).
  • With \( h = \frac{2\pi}{3} \), this means the graph shifts \( \frac{2\pi}{3} \) units to the right.
Here鈥檚 what doesn鈥檛 change:
  • The **shape** of the wave remains the same.
  • The **amplitude**, or height, doesn't change.
  • The **period**, or horizontal length of one complete cycle, stays constant.
Horizontal shifts are a versatile tool in interpreting and adjusting trigonometric functions. Such shifts can align functions to fit different phases or conditions, making them key in mathematical modeling and analysis.

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Most popular questions from this chapter

Use your graphing calculator to graph each family of functions for \(-2 \pi \leq x \leq 2 \pi\) together on a single coordinate system. (Make sure your calculator is set to radian mode.) What effect does the value of \(h\) have on the graph? $$ y=\sin (x-h) \quad \text { for } h=0,-\frac{\pi}{4},-\frac{\pi}{2} $$

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. $$ y=-2 \sin \frac{\pi}{2} x $$

The oscillations in air pressure representing the sound wave for a particular musical tone can be modeled by the equation \(y=0.3 \sin (500 \pi t)\), where \(y\) is the sound pressure in pascals after \(t\) seconds. What is the frequency of the tone?

Graph each of the following over the given interval. Label the axes so that the amplitude and period are easy to read. $$ y=-3 \cos \frac{1}{2} x,-2 \pi \leq x \leq 6 \pi $$

Convert each of the following to radians without using a calculator. $$ 70^{\circ} $$
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