Quiz InstructionsPlease Remember You Can Use Desmos.Question 8The Function $f(x)=2^x$ Is Dilated To Become $g(x)=2^{\frac{1}{3} X}$. What Is The Effect On $f(x$\]?A. $f(x$\] Is Compressed Vertically By A Factor Of

Introduction

In mathematics, dilations are an essential concept that helps us understand how functions change when they are scaled or transformed. In this article, we will delve into the world of dilations and explore how they affect functions, specifically the function $f(x)=2^x$. We will examine the effect of dilating $f(x)$ to become $g(x)=2^{\frac{1}{3} x}$ and discuss the implications of this transformation.

What is a Dilation?

A dilation is a transformation that changes the size of a figure, but not its shape. In the context of functions, a dilation is a scaling factor that is applied to the input or output of the function. This scaling factor can be a positive or negative number, and it can be applied to the entire function or to a specific part of it.

Dilating the Function $f(x)=2^x$

The function $f(x)=2^x$ is a simple exponential function that grows rapidly as $x$ increases. When we dilate this function to become $g(x)=2^{\frac{1}{3} x}$, we are applying a scaling factor of $\frac{1}{3}$ to the input $x$. This means that the output of the function will be scaled by a factor of $\frac{1}{3}$ as well.

Effect of Dilation on $f(x)$

So, what is the effect of dilating $f(x)$ to become $g(x)=2^{\frac{1}{3} x}$? To understand this, let's consider the graph of the function $f(x)=2^x$. The graph of this function is an exponential curve that grows rapidly as $x$ increases.

When we dilate this function to become $g(x)=2^{\frac{1}{3} x}$, the graph of the function will be compressed vertically by a factor of $\frac{1}{3}$. This means that the output of the function will be scaled down by a factor of $\frac{1}{3}$, resulting in a flatter graph.

Visualizing the Dilation

To visualize the dilation, let's consider a specific point on the graph of $f(x)=2^x$. Suppose we choose the point $(1, 2)$, which is the value of the function when $x=1$. When we dilate this function to become $g(x)=2^{\frac{1}{3} x}$, the new value of the function at $x=1$ will be $g(1)=2^{\frac{1}{3} \cdot 1}=2^{\frac{1}{3}}$.

As we can see, the value of the function has been scaled down by a factor of $\frac{1}{3}$, resulting in a flatter graph.

Conclusion

In conclusion, dilating the function $f(x)=2^x$ to become $g(x)=2^{\frac{1}{3} x}$ results in a vertical compression of the graph by a factor of $\frac{1}{3}$. This means that the output of the function will be scaled down by a factor of $\frac{1}{3}$, resulting in a flatter graph.

Key Takeaways

• A dilation is a transformation that changes the size of a figure, but not its shape.
• Dilating a function applies a scaling factor to the input or output of the function.
• The function $f(x)=2^x$ is dilated to become $g(x)=2^{\frac{1}{3} x}$ by applying a scaling factor of $\frac{1}{3}$ to the input $x$.
• The effect of dilating $f(x)$ to become $g(x)=2^{\frac{1}{3} x}$ is a vertical compression of the graph by a factor of $\frac{1}{3}$.

Further Exploration

• Explore other types of dilations, such as horizontal dilations and dilations with negative scaling factors.
• Investigate how dilations affect other types of functions, such as linear and quadratic functions.
• Use graphing software or online tools to visualize the dilation of the function $f(x)=2^x$ and explore its effects on the graph.
  Quiz Instructions: Dilations in Functions =====================================================

Q&A: Dilations in Functions

Q1: What is a dilation in the context of functions?

A dilation is a transformation that changes the size of a figure, but not its shape. In the context of functions, a dilation is a scaling factor that is applied to the input or output of the function.

Q2: How does a dilation affect the graph of a function?

A dilation can affect the graph of a function in several ways, depending on the type of dilation and the direction of the dilation. A vertical dilation will change the height of the graph, while a horizontal dilation will change the width of the graph.

Q3: What is the effect of dilating the function $f(x)=2^x$ to become $g(x)=2^{\frac{1}{3} x}$?

The effect of dilating the function $f(x)=2^x$ to become $g(x)=2^{\frac{1}{3} x}$ is a vertical compression of the graph by a factor of $\frac{1}{3}$. This means that the output of the function will be scaled down by a factor of $\frac{1}{3}$, resulting in a flatter graph.

Q4: How can you visualize the dilation of a function?

You can visualize the dilation of a function by using graphing software or online tools to plot the original function and the dilated function. This will allow you to see the effect of the dilation on the graph of the function.

Q5: What are some common types of dilations?

Some common types of dilations include:

• Vertical dilations: These dilations change the height of the graph.
• Horizontal dilations: These dilations change the width of the graph.
• Dilation with negative scaling factor: This type of dilation will flip the graph over the x-axis or y-axis.

Q6: How can you determine the effect of a dilation on a function?

To determine the effect of a dilation on a function, you can use the following steps:

1. Identify the type of dilation (vertical, horizontal, or dilation with negative scaling factor).
2. Determine the direction of the dilation (in or out).
3. Apply the dilation to the function using the scaling factor.
4. Plot the original function and the dilated function to visualize the effect of the dilation.

Q7: What are some real-world applications of dilations?

Dilations have many real-world applications, including:

• Scaling up or down images or objects.
• Creating models or prototypes of objects.
• Designing buildings or structures.
• Creating special effects in movies or video games.

Q8: How can you use dilations to solve problems in mathematics?

Dilations can be used to solve problems in mathematics, including:

• Finding the equation of a dilated function.
• Graphing dilated functions.
• Solving problems involving dilations, such as finding the area or perimeter of a dilated shape.

Conclusion

In conclusion, dilations are an important concept in mathematics that can be used to solve problems and understand the behavior of functions. By understanding the effect of dilations on functions, you can use this knowledge to solve problems and create models or prototypes of objects.