What Is The End Behavior Of The Graph Of The Polynomial Function $y=10x^9-4x$?A. As $x \rightarrow -\infty$, $y \rightarrow \infty$ And As $x \rightarrow \infty$, $y \rightarrow \infty$.B. As $x

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Introduction

When analyzing the behavior of a polynomial function, it's essential to understand its end behavior. The end behavior of a polynomial function is 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 $y=10x^9-4x$.

Understanding the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial. In the given function $y=10x^9-4x$, the highest power of the variable $x$ is 9. This means that the degree of the polynomial is 9.

Even or Odd Degree Polynomial

To determine the end behavior of the polynomial, we need to determine whether the degree of the polynomial is even or odd. If the degree is even, the end behavior will be the same for both positive and negative infinity. If the degree is odd, the end behavior will be different for positive and negative infinity.

In the given function $y=10x^9-4x$, the degree is 9, which is an odd degree. This means that the end behavior will be different for positive and negative infinity.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power of the variable. In the given function $y=10x^9-4x$, the leading coefficient is 10.

End Behavior of the Polynomial

To determine the end behavior of the polynomial, we need to consider the sign of the leading coefficient and the degree of the polynomial. If the degree is odd and the leading coefficient is positive, the end behavior will be:

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

If the degree is odd and the leading coefficient is negative, the end behavior will be:

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

In the given function $y=10x^9-4x$, the degree is 9, which is an odd degree, and the leading coefficient is 10, which is positive. Therefore, the end behavior of the polynomial will be:

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

Conclusion

In conclusion, the end behavior of the polynomial function $y=10x^9-4x$ is that as $x$ approaches negative infinity, $y$ approaches negative infinity, and as $x$ approaches positive infinity, $y$ approaches positive infinity.

Frequently Asked Questions

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

A: The end behavior of a polynomial function is the behavior of the function as $x$ approaches positive or negative infinity.

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

A: To determine the end behavior of a polynomial function, you need to consider the degree of the polynomial and the sign of the leading coefficient.

Q: What is the degree of the polynomial function $y=10x^9-4x$?

A: The degree of the polynomial function $y=10x^9-4x$ is 9.

Q: Is the degree of the polynomial function $y=10x^9-4x$ even or odd?

A: The degree of the polynomial function $y=10x^9-4x$ is odd.

Q: What is the leading coefficient of the polynomial function $y=10x^9-4x$?

A: The leading coefficient of the polynomial function $y=10x^9-4x$ is 10.

Q: What is the end behavior of the polynomial function $y=10x^9-4x$?

A: The end behavior of the polynomial function $y=10x^9-4x$ is that as $x$ approaches negative infinity, $y$ approaches negative infinity, and as $x$ approaches positive infinity, $y$ approaches positive infinity.

Introduction

In our previous article, we discussed the end behavior of the polynomial function $y=10x^9-4x$. We learned that the end behavior of a polynomial function is the behavior of the function as $x$ approaches positive or negative infinity. In this article, we will continue to explore the end behavior of polynomial functions and answer some frequently asked questions.

Q&A: 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 is the same for both positive and negative infinity. If the leading coefficient is positive, the end behavior will be:

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

If the leading coefficient is negative, the end behavior will be:

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

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 is different for positive and negative infinity. If the leading coefficient is positive, the end behavior will be:

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

If the leading coefficient is negative, the end behavior will be:

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

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

A: To determine the end behavior of a polynomial function, you need to consider the degree of the polynomial and the sign of the leading coefficient.

Q: What is the degree of the polynomial function $y=3x^4-2x^2+1$?

A: The degree of the polynomial function $y=3x^4-2x^2+1$ is 4, which is an even degree.

Q: What is the leading coefficient of the polynomial function $y=3x^4-2x^2+1$?

A: The leading coefficient of the polynomial function $y=3x^4-2x^2+1$ is 3.

Q: What is the end behavior of the polynomial function $y=3x^4-2x^2+1$?

A: The end behavior of the polynomial function $y=3x^4-2x^2+1$ is that as $x$ approaches negative infinity, $y$ approaches negative infinity, and as $x$ approaches positive infinity, $y$ approaches positive infinity.

Q: What is the degree of the polynomial function $y=x^3-2x^2+1$?

A: The degree of the polynomial function $y=x^3-2x^2+1$ is 3, which is an odd degree.

Q: What is the leading coefficient of the polynomial function $y=x^3-2x^2+1$?

A: The leading coefficient of the polynomial function $y=x^3-2x^2+1$ is 1.

Q: What is the end behavior of the polynomial function $y=x^3-2x^2+1$?

A: The end behavior of the polynomial function $y=x^3-2x^2+1$ is that as $x$ approaches negative infinity, $y$ approaches positive infinity, and as $x$ approaches positive infinity, $y$ approaches negative infinity.

Conclusion

In conclusion, the end behavior of a polynomial function is determined by the degree of the polynomial and the sign of the leading coefficient. If the degree is even, the end behavior will be the same for both positive and negative infinity. If the degree is odd, the end behavior will be different for positive and negative infinity.

Frequently Asked Questions

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

A: The end behavior of a polynomial function is the behavior of the function as $x$ approaches positive or negative infinity.

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

A: To determine the end behavior of a polynomial function, you need to consider the degree of the polynomial and the sign of the leading coefficient.

Q: What is the degree of the polynomial function $y=3x^4-2x^2+1$?

A: The degree of the polynomial function $y=3x^4-2x^2+1$ is 4.

Q: What is the leading coefficient of the polynomial function $y=3x^4-2x^2+1$?

A: The leading coefficient of the polynomial function $y=3x^4-2x^2+1$ is 3.

Q: What is the end behavior of the polynomial function $y=3x^4-2x^2+1$?

A: The end behavior of the polynomial function $y=3x^4-2x^2+1$ is that as $x$ approaches negative infinity, $y$ approaches negative infinity, and as $x$ approaches positive infinity, $y$ approaches positive infinity.

Q: What is the degree of the polynomial function $y=x^3-2x^2+1$?

A: The degree of the polynomial function $y=x^3-2x^2+1$ is 3.

Q: What is the leading coefficient of the polynomial function $y=x^3-2x^2+1$?

A: The leading coefficient of the polynomial function $y=x^3-2x^2+1$ is 1.

Q: What is the end behavior of the polynomial function $y=x^3-2x^2+1$?

A: The end behavior of the polynomial function $y=x^3-2x^2+1$ is that as $x$ approaches negative infinity, $y$ approaches positive infinity, and as $x$ approaches positive infinity, $y$ approaches negative infinity.