Select The Correct Answer.If $g(x) = F(4x$\], Which Statement Is True?A. The Graph Of Function $f$ Is Stretched Horizontally By A Scale Factor Of 4 To Create The Graph Of Function $g$.B. The Graph Of Function $f$ Is

Introduction

In mathematics, function transformations play a crucial role in understanding the behavior of functions and their graphs. When we encounter a function transformation, it's essential to identify the type of transformation and its effect on the original function. In this article, we will delve into the concept of horizontal stretching and explore how it affects the graph of a function.

What is Horizontal Stretching?

Horizontal stretching is a type of function transformation that involves stretching the graph of a function horizontally. This occurs when the input variable of the function is multiplied by a constant factor, resulting in a change in the x-coordinates of the graph. In other words, the graph of the function is stretched out horizontally, making it wider.

The Effect of Horizontal Stretching on the Graph of a Function

When a function is horizontally stretched, its graph is affected in several ways. The most notable effect is the change in the x-coordinates of the graph. As the input variable is multiplied by a constant factor, the x-coordinates of the graph are also multiplied by the same factor. This results in a horizontal stretching of the graph, making it wider.

The Relationship Between $f(x)$ and $g(x)$

In the given problem, we have $g(x) = f(4x)$. To understand the relationship between $f(x)$ and $g(x)$, let's analyze the transformation that occurs when we substitute $4x$ for $x$ in the function $f(x)$.

Horizontal Stretching vs. Horizontal Compression

It's essential to note that horizontal stretching is the opposite of horizontal compression. While horizontal compression involves shrinking the graph of a function horizontally, horizontal stretching involves stretching the graph out horizontally.

The Correct Answer

Now that we have a better understanding of horizontal stretching and its effect on the graph of a function, let's revisit the original problem. We are given the statement: "The graph of function $f$ is stretched horizontally by a scale factor of 4 to create the graph of function $g$."

Is the Statement True?

To determine whether the statement is true, let's analyze the transformation that occurs when we substitute $4x$ for $x$ in the function $f(x)$. As we discussed earlier, this substitution results in a horizontal stretching of the graph of $f(x)$ by a scale factor of 4.

Conclusion

In conclusion, the statement "The graph of function $f$ is stretched horizontally by a scale factor of 4 to create the graph of function $g$" is true. The graph of function $f$ is indeed stretched horizontally by a scale factor of 4 to create the graph of function $g$.

Key Takeaways

• Horizontal stretching is a type of function transformation that involves stretching the graph of a function horizontally.
• The graph of a function is affected by horizontal stretching in several ways, including a change in the x-coordinates of the graph.
• The relationship between $f(x)$ and $g(x)$ is that $g(x)$ is obtained by substituting $4x$ for $x$ in the function $f(x)$.
• Horizontal stretching is the opposite of horizontal compression.

Additional Examples

To further illustrate the concept of horizontal stretching, let's consider a few additional examples.

Example 1

Suppose we have the function $f(x) = x^2$. To create a new function $g(x)$ that is obtained by horizontally stretching $f(x)$ by a scale factor of 2, we can substitute $2x$ for $x$ in the function $f(x)$.

Example 2

Suppose we have the function $f(x) = \sin x$. To create a new function $g(x)$ that is obtained by horizontally stretching $f(x)$ by a scale factor of 3, we can substitute $3x$ for $x$ in the function $f(x)$.

Example 3

Suppose we have the function $f(x) = \cos x$. To create a new function $g(x)$ that is obtained by horizontally stretching $f(x)$ by a scale factor of 5, we can substitute $5x$ for $x$ in the function $f(x)$.

Final Thoughts

Introduction

In our previous article, we explored the concept of horizontal stretching and its effect on the graph of a function. We also discussed the relationship between $f(x)$ and $g(x)$, where $g(x) = f(4x)$. In this article, we will continue to delve into the world of function transformations and answer some frequently asked questions.

Q&A Guide

Q1: What is the difference between horizontal stretching and horizontal compression?

A1: Horizontal stretching involves stretching the graph of a function horizontally, making it wider. On the other hand, horizontal compression involves shrinking the graph of a function horizontally, making it narrower.

Q2: How do I determine whether a function is stretched or compressed horizontally?

A2: To determine whether a function is stretched or compressed horizontally, you need to analyze the transformation that occurs when you substitute a constant factor for the input variable of the function. If the constant factor is greater than 1, the function is stretched horizontally. If the constant factor is less than 1, the function is compressed horizontally.

Q3: What is the effect of horizontal stretching on the graph of a function?

A3: The effect of horizontal stretching on the graph of a function is that the x-coordinates of the graph are multiplied by the constant factor. This results in a horizontal stretching of the graph, making it wider.

Q4: How do I create a new function that is obtained by horizontally stretching a given function?

A4: To create a new function that is obtained by horizontally stretching a given function, you need to substitute a constant factor for the input variable of the function. For example, if you have the function $f(x) = x^2$ and you want to create a new function $g(x)$ that is obtained by horizontally stretching $f(x)$ by a scale factor of 2, you can substitute $2x$ for $x$ in the function $f(x)$.

Q5: What is the relationship between $f(x)$ and $g(x)$, where $g(x) = f(4x)$?

A5: The relationship between $f(x)$ and $g(x)$ is that $g(x)$ is obtained by substituting $4x$ for $x$ in the function $f(x)$. This results in a horizontal stretching of the graph of $f(x)$ by a scale factor of 4.

Q6: How do I determine whether a function is stretched or compressed horizontally using the graph of the function?

A6: To determine whether a function is stretched or compressed horizontally using the graph of the function, you need to analyze the shape of the graph. If the graph is wider than the original graph, the function is stretched horizontally. If the graph is narrower than the original graph, the function is compressed horizontally.

Q7: What is the effect of horizontal stretching on the amplitude of a function?

A7: The effect of horizontal stretching on the amplitude of a function is that the amplitude remains the same. However, the period of the function is affected by the horizontal stretching.

Q8: How do I create a new function that is obtained by horizontally compressing a given function?

A8: To create a new function that is obtained by horizontally compressing a given function, you need to substitute a constant factor for the input variable of the function. For example, if you have the function $f(x) = x^2$ and you want to create a new function $g(x)$ that is obtained by horizontally compressing $f(x)$ by a scale factor of 1/2, you can substitute $2x$ for $x$ in the function $f(x)$.

Conclusion

In conclusion, function transformations are an essential concept in mathematics that helps us understand the behavior of functions and their graphs. By analyzing the transformation that occurs when we substitute a constant factor for the input variable of a function, we can determine whether the graph of the function is stretched or compressed horizontally. We hope that this Q&A guide has helped you understand function transformations better.

Key Takeaways

• Horizontal stretching involves stretching the graph of a function horizontally, making it wider.
• Horizontal compression involves shrinking the graph of a function horizontally, making it narrower.
• The effect of horizontal stretching on the graph of a function is that the x-coordinates of the graph are multiplied by the constant factor.
• To create a new function that is obtained by horizontally stretching a given function, you need to substitute a constant factor for the input variable of the function.
• The relationship between $f(x)$ and $g(x)$, where $g(x) = f(4x)$, is that $g(x)$ is obtained by substituting $4x$ for $x$ in the function $f(x)$.

Additional Resources

For more information on function transformations, we recommend the following resources:

• Khan Academy: Function Transformations
• Mathway: Function Transformations
• Wolfram Alpha: Function Transformations

Final Thoughts

In conclusion, function transformations are an essential concept in mathematics that helps us understand the behavior of functions and their graphs. By analyzing the transformation that occurs when we substitute a constant factor for the input variable of a function, we can determine whether the graph of the function is stretched or compressed horizontally. We hope that this Q&A guide has helped you understand function transformations better.