What Is The End Behavior Of The Graph Of F ( X ) = X 5 − 8 X 4 + 16 X 3 F(x)=x^5-8x^4+16x^3 F ( X ) = X 5 − 8 X 4 + 16 X 3 ?A. F ( X ) → + ∞ F(x) \rightarrow +\infty F ( X ) → + ∞ As X → − ∞ X \rightarrow -\infty X → − ∞ ; F ( X ) → − ∞ F(x) \rightarrow -\infty F ( X ) → − ∞ As X → + ∞ X \rightarrow +\infty X → + ∞ B. $f(x) \rightarrow

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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 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 graph of the function $f(x)=x^5-8x^4+16x^3$.

Understanding the Degree of the Polynomial

The given function is a polynomial of degree 5, which means that the highest power of $x$ in the function is 5. When analyzing the end behavior of a polynomial function, the degree of the polynomial plays a crucial role. The degree of the polynomial determines the behavior of the function as $x$ approaches positive or negative infinity.

Leading Term Analysis

To determine the end behavior of the function, we need to analyze the leading term of the polynomial. The leading term is the term with the highest power of $x$. In this case, the leading term is $x^5$. The coefficient of the leading term is 1, which is positive.

End Behavior Analysis

Since the degree of the polynomial is odd (5) and the leading coefficient is positive (1), we can conclude that the function will approach positive infinity as $x$ approaches positive infinity. On the other hand, since the degree of the polynomial is odd and the leading coefficient is positive, we can also conclude that the function will approach negative infinity as $x$ approaches negative infinity.

Conclusion

In conclusion, the end behavior of the graph of $f(x)=x^5-8x^4+16x^3$ is that $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$ and $f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$. This is because the degree of the polynomial is odd and the leading coefficient is positive.

Final Answer

The final answer is A. $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$; $f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$.

Additional Information

It's worth noting that the end behavior of a polynomial function can be determined using the leading term and the degree of the polynomial. If the degree of the polynomial is even, the end behavior will be the same for both positive and negative infinity. If the degree of the polynomial is odd, the end behavior will be different for positive and negative infinity.

Example

Let's consider another example of a polynomial function: $f(x)=x^4-6x^3+9x^2$. In this case, the degree of the polynomial is even (4), and the leading coefficient is 1, which is positive. Therefore, the end behavior of the function is that $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$ and $f(x) \rightarrow +\infty$ as $x \rightarrow -\infty$.

Graphical Representation

The graph of the function $f(x)=x^5-8x^4+16x^3$ can be represented graphically as follows:

• As $x$ approaches positive infinity, the function approaches positive infinity.
• As $x$ approaches negative infinity, the function approaches negative infinity.

Mathematical Representation

The end behavior of the function can be represented mathematically as follows:

• As $x \rightarrow +\infty$, $f(x) \rightarrow +\infty$.
• As $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$.

Conclusion

In conclusion, the end behavior of the graph of $f(x)=x^5-8x^4+16x^3$ is that $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$ and $f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$. This is because the degree of the polynomial is odd and the leading coefficient is positive.

Final Answer

The final answer is A. $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$; $f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$.

References

• [1] Calculus, 6th edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Calculus, 3rd edition, James Stewart

Keywords

• End behavior
• Polynomial function
• Degree of the polynomial
• Leading term
• Leading coefficient
• Positive infinity
• Negative infinity

Tags

• #mathematics
• #calculus
• #algebra
• #polynomialfunctions
• #endbehavior
• #degreeofthepolynomial
• #leadingterm
• #leadingcoefficient
• #positiveinfinity
• #negativeinfinity
  

Introduction

In our previous article, we discussed the end behavior of the graph of the function $f(x)=x^5-8x^4+16x^3$. We concluded that the end behavior of the function is that $f(x) \rightarrow +\infty$ as $x \rightarrow +\infty$ and $f(x) \rightarrow -\infty$ as $x \rightarrow -\infty$. In this article, we will answer some frequently asked questions about the end behavior of polynomial functions.

Q1: What is the end behavior of a polynomial function with an even degree?

A1: 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 function will approach positive infinity as $x$ approaches positive or negative infinity. If the leading coefficient is negative, the function will approach negative infinity as $x$ approaches positive or negative infinity.

Q2: What is the end behavior of a polynomial function with an odd degree?

A2: 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 function will approach positive infinity as $x$ approaches positive infinity and negative infinity as $x$ approaches negative infinity. If the leading coefficient is negative, the function will approach negative infinity as $x$ approaches positive infinity and positive infinity as $x$ approaches negative infinity.

Q3: How do I determine the end behavior of a polynomial function?

A3: To determine the end behavior of a polynomial function, you need to analyze the degree of the polynomial and the leading coefficient. If the degree of the polynomial is even, the end behavior will be the same for both positive and negative infinity. If the degree of the polynomial is odd, the end behavior will be different for positive and negative infinity.

Q4: What is the significance of the leading term in determining the end behavior of a polynomial function?

A4: The leading term is the term with the highest power of $x$ in the polynomial function. The coefficient of the leading term determines the end behavior of the function. 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.

Q5: Can the end behavior of a polynomial function be determined using the graph of the function?

A5: Yes, the end behavior of a polynomial function can be determined using the graph of the function. As $x$ approaches positive or negative infinity, the graph of the function will approach positive or negative infinity, respectively.

Q6: What is the relationship between the degree of the polynomial and the end behavior of the function?

A6: The degree of the polynomial determines the end behavior of the function. If the degree of the polynomial is even, the end behavior will be the same for both positive and negative infinity. If the degree of the polynomial is odd, the end behavior will be different for positive and negative infinity.

Q7: Can the end behavior of a polynomial function be determined using the leading coefficient?

A7: Yes, the end behavior of a polynomial function can be determined using the leading coefficient. If the leading coefficient is positive, the function will approach positive infinity as $x$ approaches positive or negative infinity. If the leading coefficient is negative, the function will approach negative infinity as $x$ approaches positive or negative infinity.

Q8: What is the significance of the degree of the polynomial in determining the end behavior of a polynomial function?

A8: The degree of the polynomial determines the end behavior of the function. If the degree of the polynomial is even, the end behavior will be the same for both positive and negative infinity. If the degree of the polynomial is odd, the end behavior will be different for positive and negative infinity.

Q9: Can the end behavior of a polynomial function be determined using the graph of the function and the leading coefficient?

A9: Yes, the end behavior of a polynomial function can be determined using the graph of the function and the leading coefficient. As $x$ approaches positive or negative infinity, the graph of the function will approach positive or negative infinity, respectively, depending on the sign of the leading coefficient.

Q10: What is the relationship between the leading term and the end behavior of a polynomial function?

A10: The leading term is the term with the highest power of $x$ in the polynomial function. The coefficient of the leading term determines the end behavior of the function. 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.

Conclusion

In conclusion, the end behavior of a polynomial function can be determined using the degree of the polynomial and the leading coefficient. If the degree of the polynomial is even, the end behavior will be the same for both positive and negative infinity. If the degree of the polynomial is odd, the end behavior will be different for positive and negative infinity. The leading term is the term with the highest power of $x$ in the polynomial function, and the coefficient of the leading term determines the end behavior of the function.

Final Answer

The final answer is that the end behavior of a polynomial function can be determined using the degree of the polynomial and the leading coefficient.

References

• [1] Calculus, 6th edition, Michael Spivak
• [2] Algebra, 2nd edition, Michael Artin
• [3] Calculus, 3rd edition, James Stewart

Keywords

• End behavior
• Polynomial function
• Degree of the polynomial
• Leading term
• Leading coefficient
• Positive infinity
• Negative infinity

Tags

• #mathematics
• #calculus
• #algebra
• #polynomialfunctions
• #endbehavior
• #degreeofthepolynomial
• #leadingterm
• #leadingcoefficient
• #positiveinfinity
• #negativeinfinity