Select The Correct Answer.Consider Function { G $} . . . { G(x) = 3 \sin (\pi X) \} Function { G $}$ Is Horizontally Stretched By A Factor Of 2 And Then Translated 2 Units Down To Obtain Function { F $}$. Which

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Introduction

In mathematics, functions are used to describe relationships between variables. Understanding how functions can be transformed is crucial in various mathematical and real-world applications. In this article, we will explore the transformation of a given function, g(x)=3sin(πx)g(x) = 3 \sin (\pi x), to obtain a new function, f(x)f(x). We will analyze the transformations applied to g(x)g(x) to determine the correct answer.

Understanding the Original Function

The original function is given by g(x)=3sin(πx)g(x) = 3 \sin (\pi x). This function represents a sine wave with an amplitude of 3 and a period of 2π\frac{2}{\pi}. The sine function oscillates between -1 and 1, but in this case, the amplitude is multiplied by 3, resulting in a wave that oscillates between -3 and 3.

Horizontal Stretching

The function g(x)g(x) is horizontally stretched by a factor of 2. This means that the x-values are multiplied by 2, resulting in a new function g1(x)=3sin(πx2)g_1(x) = 3 \sin (\pi \frac{x}{2}). The period of the function is doubled, resulting in a wave that oscillates more slowly.

Translation Down

The function g1(x)g_1(x) is then translated 2 units down to obtain the function f(x)f(x). This means that the y-values are shifted down by 2 units, resulting in a new function f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2. The amplitude of the function remains the same, but the wave is shifted down by 2 units.

Determining the Correct Answer

To determine the correct answer, we need to analyze the transformations applied to the original function g(x)g(x). The function is horizontally stretched by a factor of 2 and then translated 2 units down. This results in a new function f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2.

Conclusion

In conclusion, the correct answer is the function f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2. This function represents a sine wave that has been horizontally stretched by a factor of 2 and then translated 2 units down.

Key Takeaways

  • The original function g(x)=3sin(πx)g(x) = 3 \sin (\pi x) represents a sine wave with an amplitude of 3 and a period of 2π\frac{2}{\pi}.
  • The function is horizontally stretched by a factor of 2, resulting in a new function g1(x)=3sin(πx2)g_1(x) = 3 \sin (\pi \frac{x}{2}).
  • The function is then translated 2 units down, resulting in a new function f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2.
  • The correct answer is the function f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2.

Mathematical Representation

The mathematical representation of the transformations applied to the original function g(x)g(x) is as follows:

  • Horizontal stretching: g1(x)=3sin(πx2)g_1(x) = 3 \sin (\pi \frac{x}{2})
  • Translation down: f(x)=3sin(πx2)2f(x) = 3 \sin (\pi \frac{x}{2}) - 2

Graphical Representation

The graphical representation of the original function g(x)g(x) and the transformed function f(x)f(x) is as follows:

  • Original function g(x)g(x): a sine wave with an amplitude of 3 and a period of 2π\frac{2}{\pi}
  • Transformed function f(x)f(x): a sine wave with an amplitude of 3 and a period of 2π\frac{2}{\pi}, shifted down by 2 units

Real-World Applications

Understanding function transformations has numerous real-world applications, including:

  • Signal processing: understanding how functions can be transformed is crucial in signal processing, where signals are often modified to extract useful information.
  • Image processing: understanding function transformations is also important in image processing, where images are often modified to enhance their quality.
  • Physics: understanding function transformations is essential in physics, where functions are used to describe the behavior of physical systems.

Conclusion

Introduction

In our previous article, we explored the transformation of a given function, g(x)=3sin(πx)g(x) = 3 \sin (\pi x), to obtain a new function, f(x)f(x). We analyzed the transformations applied to g(x)g(x) to determine the correct answer. In this article, we will answer some frequently asked questions about function transformations.

Q: What is function transformation?

A: Function transformation refers to the process of modifying a function to obtain a new function. This can be done by applying various transformations, such as horizontal stretching, vertical stretching, translation, and reflection.

Q: What are the different types of function transformations?

A: There are several types of function transformations, including:

  • Horizontal stretching: this involves multiplying the x-values by a constant factor, resulting in a new function with a wider or narrower graph.
  • Vertical stretching: this involves multiplying the y-values by a constant factor, resulting in a new function with a taller or shorter graph.
  • Translation: this involves shifting the graph of the function up or down, left or right, or a combination of these.
  • Reflection: this involves flipping the graph of the function over a horizontal or vertical line.

Q: How do I determine the correct answer when applying function transformations?

A: To determine the correct answer when applying function transformations, you need to analyze the transformations applied to the original function. This involves identifying the type of transformation, the direction of the transformation, and the magnitude of the transformation.

Q: What are some common mistakes to avoid when applying function transformations?

A: Some common mistakes to avoid when applying function transformations include:

  • Misidentifying the type of transformation: make sure to identify the correct type of transformation, such as horizontal stretching or vertical stretching.
  • Incorrectly applying the transformation: make sure to apply the transformation correctly, taking into account the direction and magnitude of the transformation.
  • Failing to consider the original function: make sure to consider the original function and how it will be affected by the transformation.

Q: How do I graph a function after applying transformations?

A: To graph a function after applying transformations, you need to follow these steps:

  • Identify the original function: identify the original function and its graph.
  • Apply the transformation: apply the transformation to the original function, taking into account the type, direction, and magnitude of the transformation.
  • Graph the new function: graph the new function, taking into account the changes made by the transformation.

Q: What are some real-world applications of function transformations?

A: Function transformations have numerous real-world applications, including:

  • Signal processing: understanding how functions can be transformed is crucial in signal processing, where signals are often modified to extract useful information.
  • Image processing: understanding function transformations is also important in image processing, where images are often modified to enhance their quality.
  • Physics: understanding function transformations is essential in physics, where functions are used to describe the behavior of physical systems.

Conclusion

In conclusion, understanding function transformations is crucial in various mathematical and real-world applications. By analyzing the transformations applied to the original function, we can determine the correct answer and apply function transformations to solve problems.