Determining the Translation

Oct 13, 2025 by windailys.com 110 views

Introduction

Exponential functions are a fundamental concept in mathematics, and understanding their graphs is crucial for various applications in science, engineering, and economics. In this article, we will explore the graphs of two exponential functions, $f(x)=\left(\frac{2}{3}\right)^x$ and $g(x)=\left(\frac{2}{3}\right)^{x-2}$ , and determine the translation that transformed the parent function $f(x)$ to $g(x)$ .

The Parent Function

The parent function $f(x)=\left(\frac{2}{3}\right)^x$ is an exponential function with a base of $\frac{2}{3}$ . This function has a limited domain and range, and its graph is a decreasing curve that approaches the x-axis as $x$ approaches infinity.

f(x) = (2/3)^x

The Transformed Function

The transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is also an exponential function with a base of $\frac{2}{3}$ . However, this function has been shifted two units to the right compared to the parent function $f(x)$ .

g(x) = (2/3)^(x-2)

Determining the Translation

To determine the translation that transformed the parent function $f(x)$ to $g(x)$ , we need to analyze the difference between the two functions. By comparing the two functions, we can see that the only difference is the exponent, which is $x-2$ in the transformed function $g(x)$ .

f(x) = (2/3)^x
g(x) = (2/3)^(x-2)

This indicates that the transformed function $g(x)$ has been shifted two units to the right compared to the parent function $f(x)$ . Therefore, the translation that transformed the parent function $f(x)$ to $g(x)$ is a translation right 2 units.

Conclusion

In conclusion, the graph of the transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is a translation of the parent function $f(x)=\left(\frac{2}{3}\right)^x$ two units to the right. This translation is a fundamental concept in mathematics, and understanding it is crucial for various applications in science, engineering, and economics.

Key Takeaways

The parent function $f(x)=\left(\frac{2}{3}\right)^x$ is an exponential function with a base of $\frac{2}{3}$ .
The transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is also an exponential function with a base of $\frac{2}{3}$ , but it has been shifted two units to the right compared to the parent function $f(x)$ .
The translation that transformed the parent function $f(x)$ to $g(x)$ is a translation right 2 units.

References

[1] Exponential Functions
[2] Graphs of Exponential Functions
[3] Translations of Exponential Functions
The Graphs of Exponential Functions: Understanding Translations ===========================================================

Q&A: Understanding the Graphs of Exponential Functions

Q: What is the parent function $f(x)=\left(\frac{2}{3}\right)^x$ ?

A: The parent function $f(x)=\left(\frac{2}{3}\right)^x$ is an exponential function with a base of $\frac{2}{3}$ . This function has a limited domain and range, and its graph is a decreasing curve that approaches the x-axis as $x$ approaches infinity.

Q: What is the transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ ?

A: The transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is also an exponential function with a base of $\frac{2}{3}$ . However, this function has been shifted two units to the right compared to the parent function $f(x)$ .

Q: How do we determine the translation that transformed the parent function $f(x)$ to $g(x)$ ?

A: To determine the translation that transformed the parent function $f(x)$ to $g(x)$ , we need to analyze the difference between the two functions. By comparing the two functions, we can see that the only difference is the exponent, which is $x-2$ in the transformed function $g(x)$ .

Q: What is the translation that transformed the parent function $f(x)$ to $g(x)$ ?

A: The translation that transformed the parent function $f(x)$ to $g(x)$ is a translation right 2 units.

Q: What are some key takeaways from this article?

A: Some key takeaways from this article are:

The parent function $f(x)=\left(\frac{2}{3}\right)^x$ is an exponential function with a base of $\frac{2}{3}$ .
The transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is also an exponential function with a base of $\frac{2}{3}$ , but it has been shifted two units to the right compared to the parent function $f(x)$ .
The translation that transformed the parent function $f(x)$ to $g(x)$ is a translation right 2 units.

Q: What are some further reading resources on exponential functions and their graphs?

A: Some further reading resources on exponential functions and their graphs are:

Q: What are some references used in this article?

A: Some references used in this article are:

[1] Exponential Functions
[2] Graphs of Exponential Functions
[3] Translations of Exponential Functions

Conclusion

Key Takeaways

The parent function $f(x)=\left(\frac{2}{3}\right)^x$ is an exponential function with a base of $\frac{2}{3}$ .
The transformed function $g(x)=\left(\frac{2}{3}\right)^{x-2}$ is also an exponential function with a base of $\frac{2}{3}$ , but it has been shifted two units to the right compared to the parent function $f(x)$ .
The translation that transformed the parent function $f(x)$ to $g(x)$ is a translation right 2 units.

References

[1] Exponential Functions
[2] Graphs of Exponential Functions
[3] Translations of Exponential Functions