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

When dealing with functions, understanding transformations is crucial for analyzing and graphing functions. In this article, we will delve into the concept of horizontal stretching and how it affects the graph of a function. We will examine the given function $g(x) = f(4x)$ and determine which statement is true.

What is Horizontal Stretching?

Horizontal stretching is a type of transformation that occurs when a function is stretched horizontally by a scale factor. This means that the graph of the function is compressed or stretched in the horizontal direction, resulting in a change in the x-coordinates of the points on the graph.

The Effect of Horizontal Stretching on the Graph of a Function

When a function is stretched horizontally by a scale factor of $k$, the graph of the function is compressed in the horizontal direction by a factor of $k$. This means that the x-coordinates of the points on the graph are multiplied by $k$, resulting in a change in the shape of the graph.

Analyzing the Given Function

The given function is $g(x) = f(4x)$. To understand the effect of this function on the graph of $f$, we need to analyze the transformation that occurs.

Breaking Down the Transformation

The function $g(x) = f(4x)$ can be broken down into two parts:

• The function $f$ is the original function.
• The function $4x$ is a horizontal compression of the function $x$ by a factor of $4$.

Understanding the Effect of the Horizontal Compression

When the function $x$ is compressed horizontally by a factor of $4$, the x-coordinates of the points on the graph are multiplied by $4$. This means that the graph of the function $f$ is stretched horizontally by a factor of $4$.

Determining the Correct Statement

Based on our analysis, we can determine that the graph of function $f$ is stretched horizontally by a scale factor of $4$ to create the graph of function $g$. Therefore, the correct statement is:

A. The graph of function $f$ is stretched horizontally by a scale factor of 4 to create the graph of function $g$

Conclusion

In conclusion, understanding function transformations is crucial for analyzing and graphing functions. Horizontal stretching is a type of transformation that occurs when a function is stretched horizontally by a scale factor. By analyzing the given function $g(x) = f(4x)$, we can determine that the graph of function $f$ is stretched horizontally by a scale factor of $4$ to create the graph of function $g$.

Key Takeaways

• Horizontal stretching is a type of transformation that occurs when a function is stretched horizontally by a scale factor.
• The graph of a function is compressed or stretched in the horizontal direction by a scale factor of $k$ when the function is stretched horizontally by a scale factor of $k$.
• The x-coordinates of the points on the graph are multiplied by $k$ when the function is stretched horizontally by a scale factor of $k$.

Frequently Asked Questions

Q: What is horizontal stretching?

A: Horizontal stretching is a type of transformation that occurs when a function is stretched horizontally by a scale factor.

Q: How does horizontal stretching affect the graph of a function?

A: Horizontal stretching compresses or stretches the graph of a function in the horizontal direction by a scale factor of $k$.

Q: What is the effect of a horizontal compression on the x-coordinates of the points on the graph?

A: The x-coordinates of the points on the graph are multiplied by $k$ when the function is stretched horizontally by a scale factor of $k$.

Q: How can we determine the correct statement based on the given function?

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In our previous article, we explored the concept of horizontal stretching and its effect on the graph of a function. We analyzed the given function $g(x) = f(4x)$ and determined that the graph of function $f$ is stretched horizontally by a scale factor of $4$ to create the graph of function $g$. In this article, we will continue to answer frequently asked questions related to function transformations.

Q&A: Function Transformations

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

A: Horizontal stretching occurs when a function is stretched horizontally by a scale factor, resulting in a compression of the graph in the horizontal direction. Horizontal compression occurs when a function is compressed horizontally by a scale factor, resulting in an expansion of the graph in the horizontal direction.

Q: How do we determine the scale factor of a horizontal stretching or compression?

A: The scale factor of a horizontal stretching or compression can be determined by analyzing the transformation that occurs in the function. In the case of the given function $g(x) = f(4x)$, the scale factor is $4$.

Q: What is the effect of a horizontal stretching or compression on the x-coordinates of the points on the graph?

A: The x-coordinates of the points on the graph are multiplied by the scale factor when the function is stretched or compressed horizontally.

Q: Can a function be stretched or compressed horizontally by a scale factor of $0$?

A: No, a function cannot be stretched or compressed horizontally by a scale factor of $0$. A scale factor of $0$ would result in a function that is undefined, as the x-coordinates of the points on the graph would be multiplied by $0$.

Q: How do we graph a function that has been stretched or compressed horizontally?

A: To graph a function that has been stretched or compressed horizontally, we can use the following steps:

• Identify the scale factor of the horizontal stretching or compression.
• Multiply the x-coordinates of the points on the graph by the scale factor.
• Plot the resulting points on the graph.

Q: Can a function be stretched or compressed horizontally by a scale factor of $1$?

A: Yes, a function can be stretched or compressed horizontally by a scale factor of $1$. This would result in no change to the graph of the function.

Q: What is the effect of a horizontal stretching or compression on the y-coordinates of the points on the graph?

A: The y-coordinates of the points on the graph are not affected by a horizontal stretching or compression.

Q: Can a function be stretched or compressed horizontally by a negative scale factor?

A: Yes, a function can be stretched or compressed horizontally by a negative scale factor. This would result in a reflection of the graph across the y-axis.

Q: How do we determine the effect of a horizontal stretching or compression on the graph of a function?

A: We can determine the effect of a horizontal stretching or compression on the graph of a function by analyzing the transformation that occurs in the function.

Conclusion

In conclusion, understanding function transformations is crucial for analyzing and graphing functions. By answering frequently asked questions related to function transformations, we can gain a deeper understanding of the concepts and improve our ability to analyze and graph functions.

Key Takeaways

• Horizontal stretching and compression occur when a function is stretched or compressed horizontally by a scale factor.
• The scale factor of a horizontal stretching or compression can be determined by analyzing the transformation that occurs in the function.
• The x-coordinates of the points on the graph are multiplied by the scale factor when the function is stretched or compressed horizontally.
• A function cannot be stretched or compressed horizontally by a scale factor of $0$.
• A function can be stretched or compressed horizontally by a scale factor of $1$.
• The y-coordinates of the points on the graph are not affected by a horizontal stretching or compression.
• A function can be stretched or compressed horizontally by a negative scale factor.

Frequently Asked Questions: Additional Topics

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

A: A horizontal stretching occurs when a function is stretched horizontally by a scale factor, resulting in a compression of the graph in the horizontal direction. A vertical stretching occurs when a function is stretched vertically by a scale factor, resulting in an expansion of the graph in the vertical direction.

Q: How do we determine the scale factor of a vertical stretching or compression?

A: The scale factor of a vertical stretching or compression can be determined by analyzing the transformation that occurs in the function.

Q: What is the effect of a vertical stretching or compression on the x-coordinates of the points on the graph?

A: The x-coordinates of the points on the graph are not affected by a vertical stretching or compression.

Q: Can a function be stretched or compressed vertically by a scale factor of $0$?

A: No, a function cannot be stretched or compressed vertically by a scale factor of $0$. A scale factor of $0$ would result in a function that is undefined, as the y-coordinates of the points on the graph would be multiplied by $0$.

Q: How do we graph a function that has been stretched or compressed vertically?

A: To graph a function that has been stretched or compressed vertically, we can use the following steps:

• Identify the scale factor of the vertical stretching or compression.
• Multiply the y-coordinates of the points on the graph by the scale factor.
• Plot the resulting points on the graph.

Q: Can a function be stretched or compressed vertically by a scale factor of $1$?

A: Yes, a function can be stretched or compressed vertically by a scale factor of $1$. This would result in no change to the graph of the function.

Q: What is the effect of a vertical stretching or compression on the y-coordinates of the points on the graph?

A: The y-coordinates of the points on the graph are multiplied by the scale factor when the function is stretched or compressed vertically.

Q: Can a function be stretched or compressed vertically by a negative scale factor?

A: Yes, a function can be stretched or compressed vertically by a negative scale factor. This would result in a reflection of the graph across the x-axis.

Q: How do we determine the effect of a vertical stretching or compression on the graph of a function?

A: We can determine the effect of a vertical stretching or compression on the graph of a function by analyzing the transformation that occurs in the function.