The Function $f(x) = 2^x$ Is Dilated To Become $g(x) = 2^{\frac{1}{3}x}$. What Is The Effect On $ F ( X ) F(x) F ( X ) [/tex]?A. F ( X F(x F ( X ] Is Stretched Vertically By A Factor Of 3.B. F ( X F(x F ( X ] Is Stretched Horizontally

The Function Transformation: Understanding the Effect of Dilation on $f(x) = 2^x$

In mathematics, a function transformation is a process of changing the graph of a function by applying certain operations. One of the most common transformations is dilation, which involves scaling the graph of a function by a certain factor. In this article, we will explore the effect of dilation on the function $f(x) = 2^x$ when it is transformed into $g(x) = 2^{\frac{1}{3}x}$. We will analyze the changes in the graph and understand the implications of this transformation.

Dilation is a transformation that involves scaling a figure by a certain factor. In the context of functions, dilation can be applied to the graph of a function by multiplying the input or output values by a certain factor. When a function is dilated, its graph is stretched or compressed in a specific direction.

The Original Function: $f(x) = 2^x$

The original function $f(x) = 2^x$ is an exponential function that represents a curve that increases rapidly as the input value $x$ increases. The graph of this function is a classic example of an exponential growth curve.

The Dilated Function: $g(x) = 2^{\frac{1}{3}x}$

The dilated function $g(x) = 2^{\frac{1}{3}x}$ is obtained by applying a dilation transformation to the original function $f(x) = 2^x$. The dilation factor is $\frac{1}{3}$, which means that the input value $x$ is multiplied by $\frac{1}{3}$.

Effect of Dilation on $f(x)$

When the function $f(x) = 2^x$ is dilated to become $g(x) = 2^{\frac{1}{3}x}$, the effect on the graph of $f(x)$ is a horizontal stretch. This means that the graph of $f(x)$ is stretched out horizontally, resulting in a wider graph.

To understand why this is the case, let's analyze the transformation. When the input value $x$ is multiplied by $\frac{1}{3}$, the output value $2^x$ is also affected. Since the exponent is $\frac{1}{3}x$, the output value is $2^{\frac{1}{3}x}$, which is a smaller value than $2^x$.

As a result, the graph of $g(x) = 2^{\frac{1}{3}x}$ is stretched out horizontally, resulting in a wider graph. This is because the input values are being multiplied by $\frac{1}{3}$, which means that the graph is being stretched out in the horizontal direction.

In conclusion, when the function $f(x) = 2^x$ is dilated to become $g(x) = 2^{\frac{1}{3}x}$, the effect on the graph of $f(x)$ is a horizontal stretch. This means that the graph of $f(x)$ is stretched out horizontally, resulting in a wider graph.

• Dilation is a transformation that involves scaling a figure by a certain factor.
• The original function $f(x) = 2^x$ is an exponential function that represents a curve that increases rapidly as the input value $x$ increases.
• The dilated function $g(x) = 2^{\frac{1}{3}x}$ is obtained by applying a dilation transformation to the original function $f(x) = 2^x$.
• The effect of dilation on $f(x)$ is a horizontal stretch, resulting in a wider graph.

For further reading on function transformations and dilation, we recommend the following resources:

• Khan Academy: Function Transformations
• Mathway: Dilation of Functions
• Wolfram Alpha: Function Dilation

• [1] Khan Academy. (n.d.). Function Transformations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7f/x2f6b7f/x2f6b7f
• [2] Mathway. (n.d.). Dilation of Functions. Retrieved from https://www.mathway.com/subjects/Algebra/Transformations/Dilation
• [3] Wolfram Alpha. (n.d.). Function Dilation. Retrieved from https://www.wolframalpha.com/input/?i=function+dilation
  Q&A: Understanding the Effect of Dilation on $f(x) = 2^x$

In our previous article, we explored the effect of dilation on the function $f(x) = 2^x$ when it is transformed into $g(x) = 2^{\frac{1}{3}x}$. We analyzed the changes in the graph and understood the implications of this transformation. In this article, we will answer some frequently asked questions about the effect of dilation on $f(x) = 2^x$.

Q: What is dilation, and how does it affect the graph of a function?

A: Dilation is a transformation that involves scaling a figure by a certain factor. When a function is dilated, its graph is stretched or compressed in a specific direction. In the case of $f(x) = 2^x$, the dilation factor is $\frac{1}{3}$, which means that the input value $x$ is multiplied by $\frac{1}{3}$.

Q: What is the effect of dilation on the graph of $f(x) = 2^x$?

A: The effect of dilation on the graph of $f(x) = 2^x$ is a horizontal stretch. This means that the graph of $f(x)$ is stretched out horizontally, resulting in a wider graph.

Q: Why is the graph of $f(x) = 2^x$ stretched out horizontally?

A: The graph of $f(x) = 2^x$ is stretched out horizontally because the input value $x$ is multiplied by $\frac{1}{3}$. This means that the output value $2^x$ is also affected, resulting in a smaller value than $2^x$.

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

A: A horizontal stretch is a transformation that stretches a figure out horizontally, resulting in a wider graph. A vertical stretch, on the other hand, is a transformation that stretches a figure out vertically, resulting in a taller graph.

Q: Can dilation be applied to other types of functions?

A: Yes, dilation can be applied to other types of functions, including linear, quadratic, and polynomial functions. The effect of dilation on these functions will depend on the specific dilation factor and the type of function.

Q: How can I visualize the effect of dilation on a function?

A: You can visualize the effect of dilation on a function by using graphing software or a graphing calculator. These tools allow you to see the graph of a function before and after dilation, making it easier to understand the effect of the transformation.

Q: What are some real-world applications of dilation?

A: Dilation has many real-world applications, including architecture, engineering, and art. For example, architects use dilation to design buildings and bridges, while engineers use it to design machines and mechanisms. Artists also use dilation to create unique and interesting effects in their work.

In conclusion, dilation is a powerful transformation that can be used to change the graph of a function in a specific way. By understanding the effect of dilation on $f(x) = 2^x$, we can better appreciate the importance of this transformation in mathematics and its many real-world applications.

• Dilation is a transformation that involves scaling a figure by a certain factor.
• The effect of dilation on the graph of $f(x) = 2^x$ is a horizontal stretch.
• A horizontal stretch is a transformation that stretches a figure out horizontally, resulting in a wider graph.
• Dilation can be applied to other types of functions, including linear, quadratic, and polynomial functions.
• Dilation has many real-world applications, including architecture, engineering, and art.

For further reading on function transformations and dilation, we recommend the following resources:

• Khan Academy: Function Transformations
• Mathway: Dilation of Functions
• Wolfram Alpha: Function Dilation

• [1] Khan Academy. (n.d.). Function Transformations. Retrieved from https://www.khanacademy.org/math/algebra/x2f6b7f/x2f6b7f/x2f6b7f
• [2] Mathway. (n.d.). Dilation of Functions. Retrieved from https://www.mathway.com/subjects/Algebra/Transformations/Dilation
• [3] Wolfram Alpha. (n.d.). Function Dilation. Retrieved from https://www.wolframalpha.com/input/?i=function+dilation