Which Description Applies To The Graph Of Y = 7 ( 4 ) − 9 − 2 Y=7(4)^{-9}-2 Y = 7 ( 4 ) − 9 − 2 ?A. The Graph Is A Translation 9 Units Left And 5 Units Down Of The Graph Of Y = 7 ( 4 ) 2 Y=7(4)^2 Y = 7 ( 4 ) 2 . The Graph Has An Asymptote At Y = 2 Y=2 Y = 2 . The Domain Is All Real Numbers, And The

Introduction

In mathematics, graphing functions is a crucial aspect of understanding their behavior and properties. The graph of a function can provide valuable insights into its domain, range, asymptotes, and other characteristics. In this article, we will delve into the graph of the function $y=7(4)^{-9}-2$ and analyze its properties.

The Function $y=7(4)^{-9}-2$

The given function is $y=7(4)^{-9}-2$. To understand its graph, we need to break down the function into its components. The function consists of two parts: $7(4)^{-9}$ and $-2$. The first part is an exponential function with a base of 4 and an exponent of -9, while the second part is a constant.

Exponential Functions

Exponential functions have the form $y=a^x$, where $a$ is the base and $x$ is the exponent. In the given function, the base is 4 and the exponent is -9. When the exponent is negative, the function is in the form $y=a^{-x}$, which can be rewritten as $y=\frac{1}{a^x}$.

Rewriting the Function

Using the property of negative exponents, we can rewrite the function as $y=7\left(\frac{1}{4^9}\right)-2$. This can be further simplified to $y=\frac{7}{4^9}-2$.

Properties of the Function

Now that we have rewritten the function, let's analyze its properties.

Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, the domain is all real numbers, as there are no restrictions on the input values.

Range

The range of a function is the set of all possible output values. To determine the range, we need to consider the behavior of the function as the input values approach positive and negative infinity.

Asymptotes

An asymptote is a line that the graph of a function approaches as the input values approach positive or negative infinity. In the case of the given function, we can see that the graph approaches the line $y=-2$ as the input values approach positive or negative infinity.

Translation

The graph of the function $y=7(4)^{-9}-2$ can be obtained by translating the graph of the function $y=7(4)^2$ 9 units left and 5 units down.

Conclusion

In conclusion, the graph of the function $y=7(4)^{-9}-2$ has a domain of all real numbers, a range of all real numbers except -2, and an asymptote at $y=-2$. The graph can be obtained by translating the graph of the function $y=7(4)^2$ 9 units left and 5 units down.

Discussion

The graph of a function can provide valuable insights into its properties and behavior. In this article, we analyzed the graph of the function $y=7(4)^{-9}-2$ and discussed its domain, range, asymptotes, and translation.

Key Takeaways

• The graph of the function $y=7(4)^{-9}-2$ has a domain of all real numbers.
• The graph of the function $y=7(4)^{-9}-2$ has a range of all real numbers except -2.
• The graph of the function $y=7(4)^{-9}-2$ has an asymptote at $y=-2$.
• The graph of the function $y=7(4)^{-9}-2$ can be obtained by translating the graph of the function $y=7(4)^2$ 9 units left and 5 units down.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Functions" by Khan Academy

Further Reading

• "Exponential Functions: A Comprehensive Guide" by Math Is Fun
• "Graphing Functions: A Step-by-Step Guide" by Purplemath
  Q&A: Understanding the Graph of $y=7(4)^{-9}-2$ =====================================================

Introduction

In our previous article, we analyzed the graph of the function $y=7(4)^{-9}-2$ and discussed its properties, including its domain, range, asymptotes, and translation. In this article, we will answer some frequently asked questions about the graph of this function.

Q: What is the domain of the function $y=7(4)^{-9}-2$?

A: The domain of the function $y=7(4)^{-9}-2$ is all real numbers. This means that the function is defined for all possible input values.

Q: What is the range of the function $y=7(4)^{-9}-2$?

A: The range of the function $y=7(4)^{-9}-2$ is all real numbers except -2. This means that the function can take on any value except -2.

Q: What is the asymptote of the function $y=7(4)^{-9}-2$?

A: The asymptote of the function $y=7(4)^{-9}-2$ is the line $y=-2$. This means that as the input values approach positive or negative infinity, the graph of the function approaches the line $y=-2$.

Q: How can I graph the function $y=7(4)^{-9}-2$?

A: To graph the function $y=7(4)^{-9}-2$, you can start by graphing the function $y=7(4)^2$. Then, translate the graph 9 units left and 5 units down to obtain the graph of the function $y=7(4)^{-9}-2$.

Q: What is the relationship between the graph of $y=7(4)^{-9}-2$ and the graph of $y=7(4)^2$?

A: The graph of $y=7(4)^{-9}-2$ is a translation of the graph of $y=7(4)^2$ 9 units left and 5 units down.

Q: Can I use the graph of $y=7(4)^{-9}-2$ to determine the value of the function at a specific input value?

A: Yes, you can use the graph of $y=7(4)^{-9}-2$ to determine the value of the function at a specific input value. Simply look at the graph and find the point that corresponds to the input value you are interested in.

Q: How can I use the graph of $y=7(4)^{-9}-2$ to understand the behavior of the function as the input values approach positive or negative infinity?

A: You can use the graph of $y=7(4)^{-9}-2$ to understand the behavior of the function as the input values approach positive or negative infinity by looking at the asymptote of the graph. As the input values approach positive or negative infinity, the graph approaches the asymptote.

Conclusion

In this article, we answered some frequently asked questions about the graph of the function $y=7(4)^{-9}-2$. We discussed the domain, range, asymptotes, and translation of the graph, as well as how to graph the function and use the graph to understand the behavior of the function.

Key Takeaways

• The domain of the function $y=7(4)^{-9}-2$ is all real numbers.
• The range of the function $y=7(4)^{-9}-2$ is all real numbers except -2.
• The asymptote of the function $y=7(4)^{-9}-2$ is the line $y=-2$.
• The graph of $y=7(4)^{-9}-2$ is a translation of the graph of $y=7(4)^2$ 9 units left and 5 units down.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Graphing Functions" by Khan Academy

Further Reading

• "Exponential Functions: A Comprehensive Guide" by Math Is Fun
• "Graphing Functions: A Step-by-Step Guide" by Purplemath