Type The Correct Answer In The Box. Use Numerals Instead Of Words. If Necessary, Use / For The Fraction Bar.Function \[$ G \$\] Is A Transformation Of The Parent Sine Function, \[$ F(x) = \sin(x) \$\].\[$ G(x) = \frac{1}{3} \sin(2x

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Introduction

In the realm of trigonometry, function transformations play a crucial role in understanding various mathematical concepts. One such transformation is the function { g(x) = \frac{1}{3} \sin(2x) $}$, which is a transformation of the parent sine function, { f(x) = \sin(x) $}$. In this article, we will delve into the world of function transformations and explore the properties of the given function.

What are Function Transformations?

Function transformations refer to the process of modifying a parent function to create a new function. These transformations can be vertical, horizontal, or a combination of both. In the case of the given function, { g(x) = \frac{1}{3} \sin(2x) $}$, we can observe two transformations: vertical and horizontal.

Vertical Transformation

The vertical transformation in the given function is represented by the coefficient { \frac{1}{3} $}$. This coefficient compresses or stretches the parent function vertically. In this case, the coefficient is { \frac{1}{3} $}$, which means the function is compressed vertically by a factor of { \frac{1}{3} $}$.

Horizontal Transformation

The horizontal transformation in the given function is represented by the term { 2x $}$ inside the sine function. This term stretches or compresses the parent function horizontally. In this case, the term { 2x $}$ compresses the function horizontally by a factor of { \frac{1}{2} $}$.

Properties of the Given Function

Now that we have understood the vertical and horizontal transformations in the given function, let's explore some of its properties.

Period

The period of a function is the distance between two consecutive points on the graph that have the same y-coordinate. In the case of the given function, the period can be calculated using the formula:

{ \text{Period} = \frac{2\pi}{\text{coefficient of x}} $}$

Substituting the value of the coefficient of x, we get:

{ \text{Period} = \frac{2\pi}{2} = \pi $}$

Therefore, the period of the given function is { \pi $}$.

Amplitude

The amplitude of a function is the maximum value that the function can attain. In the case of the given function, the amplitude can be calculated using the formula:

{ \text{Amplitude} = \frac{1}{\text{coefficient of the function}} $}$

Substituting the value of the coefficient of the function, we get:

{ \text{Amplitude} = \frac{1}{\frac{1}{3}} = 3 $}$

Therefore, the amplitude of the given function is { 3 $}$.

Phase Shift

The phase shift of a function is the horizontal distance between the graph of the function and the graph of the parent function. In the case of the given function, the phase shift can be calculated using the formula:

{ \text{Phase Shift} = \frac{-\text{constant term}}{\text{coefficient of x}} $}$

Substituting the values of the constant term and the coefficient of x, we get:

{ \text{Phase Shift} = \frac{-0}{2} = 0 $}$

Therefore, the phase shift of the given function is { 0 $}$.

Conclusion

In conclusion, the function { g(x) = \frac{1}{3} \sin(2x) $}$ is a transformation of the parent sine function, { f(x) = \sin(x) $}$. The function undergoes a vertical transformation represented by the coefficient { \frac{1}{3} $}$ and a horizontal transformation represented by the term { 2x $}$. The properties of the given function, including its period, amplitude, and phase shift, have been explored in this article.

Key Takeaways

  • Function transformations refer to the process of modifying a parent function to create a new function.
  • The given function { g(x) = \frac{1}{3} \sin(2x) $}$ undergoes a vertical transformation represented by the coefficient { \frac{1}{3} $}$ and a horizontal transformation represented by the term { 2x $}$.
  • The period of the given function is { \pi $}$.
  • The amplitude of the given function is { 3 $}$.
  • The phase shift of the given function is { 0 $}$.

Practice Problems

  1. What is the period of the function { g(x) = \frac{1}{2} \sin(3x) $}$?
  2. What is the amplitude of the function { g(x) = 2 \sin(x) $}$?
  3. What is the phase shift of the function { g(x) = \sin(2x) $}$?

Answer Key

  1. { \frac{2\pi}{3} $}$
  2. { 2 $}$
  3. { 0 $}$

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Functions" by Khan Academy

Introduction

In our previous article, we explored the concept of function transformations in trigonometry, specifically the function { g(x) = \frac{1}{3} \sin(2x) $}$. In this article, we will answer some frequently asked questions related to function transformations in trigonometry.

Q: What is the difference between a vertical transformation and a horizontal transformation?

A: A vertical transformation is a change in the amplitude of the function, while a horizontal transformation is a change in the period of the function.

Q: How do I determine the period of a function?

A: To determine the period of a function, you can use the formula:

{ \text{Period} = \frac{2\pi}{\text{coefficient of x}} $}$

Q: What is the amplitude of a function?

A: The amplitude of a function is the maximum value that the function can attain. It is determined by the coefficient of the function.

Q: How do I determine the phase shift of a function?

A: To determine the phase shift of a function, you can use the formula:

{ \text{Phase Shift} = \frac{-\text{constant term}}{\text{coefficient of x}} $}$

Q: What is the difference between a sine function and a cosine function?

A: A sine function is a function of the form { f(x) = \sin(x) $}$, while a cosine function is a function of the form { f(x) = \cos(x) $}$. The main difference between the two is the phase shift.

Q: How do I graph a function with a vertical transformation?

A: To graph a function with a vertical transformation, you can multiply the parent function by a constant. For example, if you want to graph the function { g(x) = 2 \sin(x) $}$, you can multiply the parent function { f(x) = \sin(x) $}$ by 2.

Q: How do I graph a function with a horizontal transformation?

A: To graph a function with a horizontal transformation, you can replace the x variable with a new variable that is a multiple of x. For example, if you want to graph the function { g(x) = \sin(2x) $}$, you can replace the x variable with 2x.

Q: What are some common function transformations?

A: Some common function transformations include:

  • Vertical transformations: { f(x) = af(x) $}$
  • Horizontal transformations: { f(x) = f(bx) $}$
  • Phase shifts: { f(x) = f(x - c) $}$

Conclusion

In conclusion, function transformations are an essential concept in trigonometry. By understanding how to apply vertical and horizontal transformations, you can graph and analyze various functions. We hope this Q&A article has provided you with a better understanding of function transformations in trigonometry.

Key Takeaways

  • Function transformations refer to the process of modifying a parent function to create a new function.
  • Vertical transformations change the amplitude of the function, while horizontal transformations change the period of the function.
  • The period of a function can be determined using the formula { \text{Period} = \frac{2\pi}{\text{coefficient of x}} $}$.
  • The amplitude of a function is determined by the coefficient of the function.
  • The phase shift of a function can be determined using the formula { \text{Phase Shift} = \frac{-\text{constant term}}{\text{coefficient of x}} $}$.

Practice Problems

  1. What is the period of the function { g(x) = \frac{1}{2} \sin(3x) $}$?
  2. What is the amplitude of the function { g(x) = 2 \sin(x) $}$?
  3. What is the phase shift of the function { g(x) = \sin(2x) $}$?

Answer Key

  1. { \frac{2\pi}{3} $}$
  2. { 2 $}$
  3. { 0 $}$

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Functions" by Khan Academy

Note: The references provided are for informational purposes only and are not a substitute for the original sources.