What Transformation Maps F ( X ) = 2 X F(x) = 2^x F ( X ) = 2 X To G ( X ) = 2 X − 8 G(x) = 2^{x-8} G ( X ) = 2 X − 8 ?A. Translates Left 8 Units B. Translates Up 8 Units

Mar 10, 2025 by ADMIN 172 views

What Transformation Maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ ?

Understanding the Problem

The problem asks us to determine the transformation that maps the function $f(x) = 2^x$ to the function $g(x) = 2^{x-8}$ . To solve this problem, we need to understand the concept of function transformations and how they affect the graph of a function.

Function Transformations

A function transformation is a change in the graph of a function that results from a change in the function's equation. There are several types of function transformations, including:

Vertical Stretch or Compression: This type of transformation changes the scale of the function's graph, either stretching or compressing it vertically.
Horizontal Stretch or Compression: This type of transformation changes the scale of the function's graph, either stretching or compressing it horizontally.
Reflection: This type of transformation reflects the function's graph across a horizontal or vertical line.
Translation: This type of transformation moves the function's graph up, down, left, or right.

Analyzing the Given Functions

The given functions are $f(x) = 2^x$ and $g(x) = 2^{x-8}$ . To determine the transformation that maps $f(x)$ to $g(x)$ , we need to analyze the differences between the two functions.

Differences Between the Functions

The main difference between the two functions is the exponent. In $f(x) = 2^x$ , the exponent is simply $x$ , while in $g(x) = 2^{x-8}$ , the exponent is $x-8$ . This difference in the exponent indicates that the graph of $g(x)$ is shifted to the right compared to the graph of $f(x)$ .

Determining the Transformation

Based on the analysis of the differences between the two functions, we can conclude that the transformation that maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ is a horizontal translation to the right by 8 units.

Justification

To justify this conclusion, let's consider the graph of $f(x) = 2^x$ . The graph of this function is an exponential curve that increases as $x$ increases. When we replace $x$ with $x-8$ in the equation, we are essentially shifting the graph to the right by 8 units. This is because the value of $x$ is now 8 units less than the original value, resulting in a shift to the right.

Conclusion

In conclusion, the transformation that maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ is a horizontal translation to the right by 8 units.

Answer

The correct answer is:

A. Translates right 8 units

Explanation

The transformation that maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ is a horizontal translation to the right by 8 units. This is because the exponent in $g(x)$ is $x-8$ , which is 8 units less than the exponent in $f(x)$ . As a result, the graph of $g(x)$ is shifted to the right compared to the graph of $f(x)$ .

Additional Information

It's worth noting that the transformation that maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ is a specific type of horizontal translation. This type of translation is often referred to as a shift or a translation.

Real-World Applications

Function transformations have many real-world applications in fields such as physics, engineering, and economics. For example, in physics, function transformations can be used to model the motion of objects under the influence of gravity or other forces. In engineering, function transformations can be used to design and optimize systems such as electrical circuits or mechanical systems. In economics, function transformations can be used to model the behavior of economic systems and make predictions about future trends.

Final Thoughts

In conclusion, the transformation that maps $f(x) = 2^x$ to $g(x) = 2^{x-8}$ is a horizontal translation to the right by 8 units. This type of transformation is an important concept in mathematics and has many real-world applications. By understanding function transformations, we can better analyze and model complex systems and make predictions about future trends.
Q&A: Function Transformations

Understanding Function Transformations

Function transformations are a fundamental concept in mathematics that can be used to analyze and model complex systems. In this article, we will answer some common questions about function transformations and provide examples to illustrate the concepts.

Q: What is a function transformation?

A: A function transformation is a change in the graph of a function that results from a change in the function's equation. There are several types of function transformations, including vertical stretch or compression, horizontal stretch or compression, reflection, and translation.

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

A: A vertical stretch or compression changes the scale of the function's graph, either stretching or compressing it vertically. A horizontal stretch or compression changes the scale of the function's graph, either stretching or compressing it horizontally.

Q: What is a reflection in function transformations?

A: A reflection in function transformations is a change in the graph of a function that results from reflecting the function's graph across a horizontal or vertical line. This can be thought of as "flipping" the graph over a line.

Q: What is a translation in function transformations?

A: A translation in function transformations is a change in the graph of a function that results from moving the function's graph up, down, left, or right. This can be thought of as "shifting" the graph to a new position.

Q: How do I determine the type of function transformation?

A: To determine the type of function transformation, you need to analyze the differences between the original function and the transformed function. Look for changes in the exponent, coefficient, or constant term.

Q: What are some common types of function transformations?

A: Some common types of function transformations include:

Vertical stretch or compression: This type of transformation changes the scale of the function's graph, either stretching or compressing it vertically.
Horizontal stretch or compression: This type of transformation changes the scale of the function's graph, either stretching or compressing it horizontally.
Reflection: This type of transformation reflects the function's graph across a horizontal or vertical line.
Translation: This type of transformation moves the function's graph up, down, left, or right.

Q: How do I apply function transformations to a function?

A: To apply a function transformation to a function, you need to replace the original function with the transformed function. For example, if you want to apply a vertical stretch to the function f(x) = 2x, you would replace f(x) with f(x) = 2(2x).

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

A: Function transformations have many real-world applications in fields such as physics, engineering, and economics. For example, in physics, function transformations can be used to model the motion of objects under the influence of gravity or other forces. In engineering, function transformations can be used to design and optimize systems such as electrical circuits or mechanical systems. In economics, function transformations can be used to model the behavior of economic systems and make predictions about future trends.

Q: How do I determine the type of function transformation that maps one function to another?

A: To determine the type of function transformation that maps one function to another, you need to analyze the differences between the two functions. Look for changes in the exponent, coefficient, or constant term.

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

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

Not analyzing the differences between the original function and the transformed function: Make sure to carefully analyze the differences between the two functions to determine the type of function transformation.
Not replacing the original function with the transformed function: Make sure to replace the original function with the transformed function to apply the function transformation correctly.
Not considering the context of the problem: Make sure to consider the context of the problem and the type of function transformation that is most appropriate.

Conclusion

In conclusion, function transformations are a fundamental concept in mathematics that can be used to analyze and model complex systems. By understanding function transformations, you can better analyze and model complex systems and make predictions about future trends. Remember to carefully analyze the differences between the original function and the transformed function, replace the original function with the transformed function, and consider the context of the problem.