Describe The Transformation That Maps The Translations From The Parent Function, $f(x)=2^x$, To The Function $g(x)=2^x-9$. State Any Changes To The Domain, Range, Intercepts, Asymptotes, And End Behavior.

Introduction

In mathematics, transformations of functions are essential to understand the behavior and characteristics of a function. A transformation is a change in the function's graph, which can be achieved by applying various operations such as shifting, scaling, and reflecting. In this article, we will discuss the transformation that maps the parent function $f(x)=2^x$ to the function $g(x)=2^x-9$. We will analyze the changes to the domain, range, intercepts, asymptotes, and end behavior of the function.

Parent Function: $f(x)=2^x$

The parent function $f(x)=2^x$ is an exponential function with a base of 2. This function has a characteristic "S" shape, where the value of the function increases rapidly as the input value increases. The domain of the function is all real numbers, and the range is all positive real numbers.

Transformation: $g(x)=2^x-9$

To transform the parent function $f(x)=2^x$ to the function $g(x)=2^x-9$, we need to apply a vertical shift of 9 units down. This means that for every input value of $x$, the output value of $g(x)$ will be 9 units less than the output value of $f(x)$.

Changes to the Domain

The domain of the function $g(x)=2^x-9$ remains the same as the domain of the parent function $f(x)=2^x$, which is all real numbers. The transformation does not affect the domain of the function.

Changes to the Range

The range of the function $g(x)=2^x-9$ is also the same as the range of the parent function $f(x)=2^x$, which is all positive real numbers. However, since the function is shifted 9 units down, the minimum value of the function is now -9, rather than 0.

Changes to the Intercepts

The $y$-intercept of the function $g(x)=2^x-9$ is -9, which is 9 units less than the $y$-intercept of the parent function $f(x)=2^x$, which is 0. The $x$-intercept of the function $g(x)=2^x-9$ remains the same as the $x$-intercept of the parent function $f(x)=2^x$, which is undefined.

Changes to the Asymptotes

The function $g(x)=2^x-9$ has a horizontal asymptote at $y=-9$, which is the same as the horizontal asymptote of the parent function $f(x)=2^x$. The function does not have any vertical asymptotes.

Changes to the End Behavior

The end behavior of the function $g(x)=2^x-9$ is the same as the end behavior of the parent function $f(x)=2^x$. As $x$ approaches infinity, the value of the function approaches infinity, and as $x$ approaches negative infinity, the value of the function approaches 0.

Conclusion

In conclusion, the transformation that maps the parent function $f(x)=2^x$ to the function $g(x)=2^x-9$ is a vertical shift of 9 units down. This transformation affects the range and intercepts of the function but does not change the domain, asymptotes, or end behavior.

Key Takeaways

• The domain of the function $g(x)=2^x-9$ remains the same as the domain of the parent function $f(x)=2^x$.
• The range of the function $g(x)=2^x-9$ is all positive real numbers, but the minimum value of the function is now -9.
• The $y$-intercept of the function $g(x)=2^x-9$ is -9, which is 9 units less than the $y$-intercept of the parent function $f(x)=2^x$.
• The function $g(x)=2^x-9$ has a horizontal asymptote at $y=-9$ and does not have any vertical asymptotes.
• The end behavior of the function $g(x)=2^x-9$ is the same as the end behavior of the parent function $f(x)=2^x$.
  Q&A: Transformation of Functions $f(x)=2^x$ to $g(x)=2^x-9$ ===========================================================

Frequently Asked Questions

Q: What is the transformation that maps the parent function $f(x)=2^x$ to the function $g(x)=2^x-9$?

A: The transformation that maps the parent function $f(x)=2^x$ to the function $g(x)=2^x-9$ is a vertical shift of 9 units down.

Q: How does the transformation affect the domain of the function?

A: The transformation does not affect the domain of the function. The domain of the function $g(x)=2^x-9$ remains the same as the domain of the parent function $f(x)=2^x$, which is all real numbers.

Q: How does the transformation affect the range of the function?

A: The transformation affects the range of the function. The range of the function $g(x)=2^x-9$ is all positive real numbers, but the minimum value of the function is now -9.

Q: What is the effect of the transformation on the intercepts of the function?

A: The transformation affects the intercepts of the function. The $y$-intercept of the function $g(x)=2^x-9$ is -9, which is 9 units less than the $y$-intercept of the parent function $f(x)=2^x$, which is 0. The $x$-intercept of the function $g(x)=2^x-9$ remains the same as the $x$-intercept of the parent function $f(x)=2^x$, which is undefined.

Q: What is the effect of the transformation on the asymptotes of the function?

A: The transformation does not affect the asymptotes of the function. The function $g(x)=2^x-9$ has a horizontal asymptote at $y=-9$, which is the same as the horizontal asymptote of the parent function $f(x)=2^x$. The function does not have any vertical asymptotes.

Q: What is the effect of the transformation on the end behavior of the function?

A: The transformation does not affect the end behavior of the function. The end behavior of the function $g(x)=2^x-9$ is the same as the end behavior of the parent function $f(x)=2^x$. As $x$ approaches infinity, the value of the function approaches infinity, and as $x$ approaches negative infinity, the value of the function approaches 0.

Q: Can I apply other transformations to the function $f(x)=2^x$?

A: Yes, you can apply other transformations to the function $f(x)=2^x$. Some common transformations include horizontal shifts, vertical stretches, and reflections. However, the specific transformation you apply will affect the characteristics of the function.

Q: How can I graph the function $g(x)=2^x-9$?

A: You can graph the function $g(x)=2^x-9$ by applying a vertical shift of 9 units down to the graph of the parent function $f(x)=2^x$. This will give you the graph of the function $g(x)=2^x-9$.

Q: What are some real-world applications of the function $g(x)=2^x-9$?

A: The function $g(x)=2^x-9$ has many real-world applications, including modeling population growth, chemical reactions, and financial investments. The function can also be used to model the growth of bacteria, the spread of diseases, and the behavior of financial markets.

Q: Can I use the function $g(x)=2^x-9$ to model a specific real-world scenario?

A: Yes, you can use the function $g(x)=2^x-9$ to model a specific real-world scenario. For example, you can use the function to model the growth of a population of bacteria, the spread of a disease, or the behavior of a financial market. However, you will need to adjust the parameters of the function to fit the specific scenario you are modeling.