What Is The End Behavior Of The Graph Of The Polynomial Function $y=7x^{12}-3x^8-9x^4$?A. As $x \rightarrow -\infty$, $y \rightarrow -\infty$ And As $x \rightarrow \infty$, $y \rightarrow -\infty$.B. As

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Introduction

When analyzing the behavior of a polynomial function, it is essential to understand its end behavior. The end behavior of a polynomial function refers to the behavior of the function as $x$ approaches positive or negative infinity. In this article, we will explore the end behavior of the polynomial function $y=7x^{12}-3x^8-9x^4$.

Understanding the Degree of the Polynomial

The degree of a polynomial function is the highest power of the variable in the function. In this case, the degree of the polynomial function $y=7x^{12}-3x^8-9x^4$ is 12. This means that the term with the highest power of $x$ is $7x^{12}$.

The Leading Term

The leading term of a polynomial function is the term with the highest power of the variable. In this case, the leading term is $7x^{12}$. The coefficient of the leading term is 7, which is a positive number.

End Behavior of the Polynomial Function

To determine the end behavior of the polynomial function, we need to consider the behavior of the function as $x$ approaches positive or negative infinity. Since the degree of the polynomial function is 12, which is even, the end behavior of the function will be determined by the leading term.

As $x \rightarrow -\infty$, $y \rightarrow \infty$

As $x$ approaches negative infinity, the term $7x^{12}$ will dominate the behavior of the function. Since the coefficient of the leading term is positive, the function will approach positive infinity as $x$ approaches negative infinity.

As $x \rightarrow \infty$, $y \rightarrow \infty$

As $x$ approaches positive infinity, the term $7x^{12}$ will also dominate the behavior of the function. Since the coefficient of the leading term is positive, the function will approach positive infinity as $x$ approaches positive infinity.

Conclusion

In conclusion, the end behavior of the polynomial function $y=7x^{12}-3x^8-9x^4$ is that as $x$ approaches negative infinity, $y$ approaches positive infinity, and as $x$ approaches positive infinity, $y$ also approaches positive infinity.

Final Answer

The final answer is that the end behavior of the polynomial function $y=7x^{12}-3x^8-9x^4$ is:

• As $x \rightarrow -\infty$, $y \rightarrow \infty$
• As $x \rightarrow \infty$, $y \rightarrow \infty$

This means that the correct answer is A.

Introduction

In our previous article, we explored the end behavior of the polynomial function $y=7x^{12}-3x^8-9x^4$. In this article, we will answer some frequently asked questions about the end behavior of polynomial functions.

Q: What is the end behavior of a polynomial function with an even degree?

A: The end behavior of a polynomial function with an even degree will be determined by the leading term. If the coefficient of the leading term is positive, the function will approach positive infinity as $x$ approaches positive or negative infinity. If the coefficient of the leading term is negative, the function will approach negative infinity as $x$ approaches positive or negative infinity.

Q: What is the end behavior of a polynomial function with an odd degree?

A: The end behavior of a polynomial function with an odd degree will be determined by the leading term. If the coefficient of the leading term is positive, the function will approach positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity. If the coefficient of the leading term is negative, the function will approach negative infinity as $x$ approaches positive infinity and positive infinity as $x$ approaches negative infinity.

Q: How do I determine the end behavior of a polynomial function?

A: To determine the end behavior of a polynomial function, you need to identify the degree of the function and the coefficient of the leading term. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: What is the difference between the end behavior and the behavior of a polynomial function?

A: The end behavior of a polynomial function refers to the behavior of the function as $x$ approaches positive or negative infinity. The behavior of a polynomial function refers to the behavior of the function at specific points or intervals.

Q: Can the end behavior of a polynomial function be determined by the graph of the function?

A: Yes, the end behavior of a polynomial function can be determined by the graph of the function. If the graph of the function approaches positive infinity as $x$ approaches positive or negative infinity, the end behavior is positive. If the graph of the function approaches negative infinity as $x$ approaches positive or negative infinity, the end behavior is negative.

Q: What is the significance of the end behavior of a polynomial function?

A: The end behavior of a polynomial function is significant because it can help us understand the behavior of the function at large values of $x$. This can be useful in a variety of applications, such as modeling population growth or chemical reactions.

Q: Can the end behavior of a polynomial function be changed by adding or subtracting terms?

A: Yes, the end behavior of a polynomial function can be changed by adding or subtracting terms. If a term with a higher degree is added or subtracted, the end behavior may change.

Q: How do I determine the end behavior of a polynomial function with multiple terms?

A: To determine the end behavior of a polynomial function with multiple terms, you need to identify the degree of the function and the coefficient of the leading term. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: Can the end behavior of a polynomial function be determined by the leading coefficient?

A: Yes, the end behavior of a polynomial function can be determined by the leading coefficient. If the leading coefficient is positive, the end behavior is positive. If the leading coefficient is negative, the end behavior is negative.

Q: What is the relationship between the end behavior and the degree of a polynomial function?

A: The end behavior of a polynomial function is related to the degree of the function. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: Can the end behavior of a polynomial function be determined by the graph of the function and the degree of the function?

A: Yes, the end behavior of a polynomial function can be determined by the graph of the function and the degree of the function. If the graph of the function approaches positive infinity as $x$ approaches positive or negative infinity, the end behavior is positive. If the graph of the function approaches negative infinity as $x$ approaches positive or negative infinity, the end behavior is negative.

Q: What is the significance of the degree of a polynomial function in determining the end behavior?

A: The degree of a polynomial function is significant in determining the end behavior because it determines whether the end behavior is positive or negative. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: Can the end behavior of a polynomial function be determined by the leading term and the degree of the function?

A: Yes, the end behavior of a polynomial function can be determined by the leading term and the degree of the function. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: What is the relationship between the end behavior and the leading term of a polynomial function?

A: The end behavior of a polynomial function is related to the leading term. If the leading term is positive, the end behavior is positive. If the leading term is negative, the end behavior is negative.

Q: Can the end behavior of a polynomial function be determined by the graph of the function and the leading term?

A: Yes, the end behavior of a polynomial function can be determined by the graph of the function and the leading term. If the graph of the function approaches positive infinity as $x$ approaches positive or negative infinity, the end behavior is positive. If the graph of the function approaches negative infinity as $x$ approaches positive or negative infinity, the end behavior is negative.

Q: What is the significance of the leading term in determining the end behavior of a polynomial function?

A: The leading term is significant in determining the end behavior of a polynomial function because it determines whether the end behavior is positive or negative. If the leading term is positive, the end behavior is positive. If the leading term is negative, the end behavior is negative.

Q: Can the end behavior of a polynomial function be determined by the degree of the function and the leading term?

A: Yes, the end behavior of a polynomial function can be determined by the degree of the function and the leading term. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: What is the relationship between the end behavior and the degree of the function and the leading term?

A: The end behavior of a polynomial function is related to the degree of the function and the leading term. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: Can the end behavior of a polynomial function be determined by the graph of the function, the degree of the function, and the leading term?

A: Yes, the end behavior of a polynomial function can be determined by the graph of the function, the degree of the function, and the leading term. If the graph of the function approaches positive infinity as $x$ approaches positive or negative infinity, the end behavior is positive. If the graph of the function approaches negative infinity as $x$ approaches positive or negative infinity, the end behavior is negative.

Q: What is the significance of the degree of the function, the leading term, and the graph of the function in determining the end behavior?

A: The degree of the function, the leading term, and the graph of the function are all significant in determining the end behavior of a polynomial function. The degree determines whether the end behavior is positive or negative, the leading term determines the sign of the end behavior, and the graph of the function determines the direction of the end behavior.

Q: Can the end behavior of a polynomial function be determined by the degree of the function, the leading term, and the graph of the function?

A: Yes, the end behavior of a polynomial function can be determined by the degree of the function, the leading term, and the graph of the function. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: What is the relationship between the end behavior and the degree of the function, the leading term, and the graph of the function?

A: The end behavior of a polynomial function is related to the degree of the function, the leading term, and the graph of the function. If the degree is even, the end behavior will be determined by the leading term. If the degree is odd, the end behavior will be determined by the leading term and the sign of the coefficient.

Q: Can the end behavior of a polynomial function be determined by the degree of the function, the leading term, the graph of the function, and the coefficient of the leading term?

A: Yes, the end behavior of a polynomial function can be determined by the degree of the function, the leading term, the graph of the function, and the coefficient of the leading term. If the degree is even, the end behavior will be determined by the leading term. If