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

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

Transformation of the Parent Sine Function

The parent sine function, denoted as { f(x) = \sin(x) $}$, is a fundamental trigonometric function that has numerous applications in mathematics, physics, and engineering. However, in many real-world scenarios, the sine function needs to be transformed to better represent the given situation. One such transformation is the function { g(x) = \frac{1}{3} \sin(2x) $}$, which is a transformation of the parent sine function.

Understanding the Transformation

To understand the transformation, let's break down the given function { g(x) = \frac1}{3} \sin(2x) $}$. The function { f(x) = \sin(x) $}$ is the parent sine function, and the transformation involves two main components the coefficient { \frac{1{3} $}$ and the argument { 2x $}$.

The coefficient { \frac{1}{3} $}$ is a vertical stretch factor that compresses the graph of the parent sine function vertically by a factor of { \frac{1}{3} $}$. This means that the amplitude of the transformed function is { \frac{1}{3} $}$ times the amplitude of the parent sine function.

The argument { 2x $}$ is a horizontal compression factor that compresses the graph of the parent sine function horizontally by a factor of { \frac{1}{2} $}$. This means that the period of the transformed function is { \frac{1}{2} $}$ times the period of the parent sine function.

Graphical Representation

To visualize the transformation, let's graph the parent sine function and the transformed function on the same coordinate plane.

### Parent Sine Function

Graph: { y = \sin(x) $}$
Period: { 2\pi $}$
Amplitude: { 1 $}$

Transformed Function

Graph: { y = \frac{1}{3} \sin(2x) $}$
Period: { \pi $}$
Amplitude: { \frac{1}{3} $}$

As we can see from the graphical representation, the transformed function has a period of { \pi $}$ and an amplitude of { \frac{1}{3} $}$, which are both { \frac{1}{2} $}$ times the period and amplitude of the parent sine function, respectively.

Key Takeaways

In conclusion, the transformation { g(x) = \frac{1}{3} \sin(2x) $}$ is a vertical compression and horizontal compression of the parent sine function. The coefficient { \frac{1}{3} $}$ compresses the graph vertically by a factor of { \frac{1}{3} $}$, and the argument { 2x $}$ compresses the graph horizontally by a factor of { \frac{1}{2} $}$.

Answer

The correct answer is { \frac{1}{3} \sin(2x) $}$.

Additional Resources

For more information on transformations of trigonometric functions, please refer to the following resources:

• Wikipedia: Trigonometric Functions
• Khan Academy: Trigonometric Functions
• Mathway: Trigonometric Functions
  Q&A: Transformation of the Parent Sine Function

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to the transformation of the parent sine function.

Q1: What is the parent sine function?

A1: The parent sine function is the basic trigonometric function { f(x) = \sin(x) $}$, which is a fundamental building block for many other trigonometric functions.

Q2: What is the transformation { g(x) = \frac{1}{3} \sin(2x) $}$ of the parent sine function?

A2: The transformation { g(x) = \frac{1}{3} \sin(2x) $}$ is a vertical compression and horizontal compression of the parent sine function. The coefficient { \frac{1}{3} $}$ compresses the graph vertically by a factor of { \frac{1}{3} $}$, and the argument { 2x $}$ compresses the graph horizontally by a factor of { \frac{1}{2} $}$.

Q3: What is the period of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A3: The period of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ is { \pi $}$, which is { \frac{1}{2} $}$ times the period of the parent sine function.

Q4: What is the amplitude of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A4: The amplitude of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ is { \frac{1}{3} $}$, which is { \frac{1}{3} $}$ times the amplitude of the parent sine function.

Q5: How can I graph the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A5: To graph the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$, you can use a graphing calculator or a computer algebra system. Alternatively, you can use a table of values to plot the function.

Q6: What are some real-world applications of the transformation { g(x) = \frac{1}{3} \sin(2x) $}$?

A6: The transformation { g(x) = \frac{1}{3} \sin(2x) $}$ has numerous real-world applications in fields such as physics, engineering, and economics. For example, it can be used to model periodic phenomena such as sound waves, light waves, and population growth.

Q7: How can I find the inverse of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A7: To find the inverse of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$, you can use the following steps:

1. Replace { x $}$ with { y $}$ and { y $}$ with { x $}$.
2. Solve for { x $}$ in terms of { y $}$.
3. Replace { y $}$ with { x $}$ and { x $}$ with { y $}$.

Q8: What is the domain and range of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A8: The domain of the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ is all real numbers, and the range is { [-\frac{1}{3}, \frac{1}{3}] $}$.

Q9: How can I use the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ to model a real-world phenomenon?

A9: To use the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ to model a real-world phenomenon, you can follow these steps:

1. Identify the periodic phenomenon you want to model.
2. Determine the period and amplitude of the phenomenon.
3. Use the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ to model the phenomenon.
4. Use the model to make predictions and analyze the data.

Q10: What are some common mistakes to avoid when working with the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$?

A10: Some common mistakes to avoid when working with the transformed function { g(x) = \frac{1}{3} \sin(2x) $}$ include:

• Confusing the period and amplitude of the transformed function with those of the parent sine function.
• Failing to account for the vertical and horizontal compression of the graph.
• Not using the correct notation and terminology when working with the transformed function.

Conclusion

In conclusion, the transformation { g(x) = \frac{1}{3} \sin(2x) $}$ is a vertical compression and horizontal compression of the parent sine function. The coefficient { \frac{1}{3} $}$ compresses the graph vertically by a factor of { \frac{1}{3} $}$, and the argument { 2x $}$ compresses the graph horizontally by a factor of { \frac{1}{2} $}$. By understanding the transformation and its properties, you can use it to model real-world phenomena and make predictions.