Which Statement Describes How The Graph Of The Given Polynomial Would Change If The Term $2x^5$ Is Added To $y = 8x^4 - 2x^3 + 5$?A. Both Ends Of The Graph Will Approach Negative Infinity.B. The Ends Of The Graph Will Extend In

Introduction

Polynomial graphs are a fundamental concept in mathematics, used to represent the relationship between variables in a polynomial equation. When a term is added to a polynomial, it can significantly alter the graph's behavior. In this article, we will explore how the graph of the given polynomial would change if the term $2x^5$ is added to $y = 8x^4 - 2x^3 + 5$.

The Original Polynomial

The original polynomial is given by $y = 8x^4 - 2x^3 + 5$. This polynomial has a degree of 4, which means it has four terms with different powers of $x$. The graph of this polynomial will have a certain shape, with a specific behavior at the ends.

Adding the New Term

When the term $2x^5$ is added to the original polynomial, the new polynomial becomes $y = 8x^4 - 2x^3 + 5 + 2x^5$. This new term has a degree of 5, which is higher than the original polynomial. This addition will significantly alter the graph's behavior.

Analyzing the New Polynomial

To understand how the graph of the new polynomial will change, we need to analyze the behavior of the new term $2x^5$. This term has a positive coefficient, which means it will contribute to the growth of the polynomial as $x$ increases.

Behavior at the Ends

As $x$ approaches positive infinity, the term $2x^5$ will dominate the polynomial, causing the graph to extend in the positive direction. This is because the power of $x$ in the new term is higher than the power of $x$ in the original polynomial.

Behavior at the Ends (Continued)

As $x$ approaches negative infinity, the term $2x^5$ will also dominate the polynomial, causing the graph to extend in the negative direction. This is because the power of $x$ in the new term is odd, which means it will change sign as $x$ changes sign.

Conclusion

In conclusion, the graph of the new polynomial will extend in both the positive and negative directions as $x$ approaches infinity. This is because the term $2x^5$ has a higher degree than the original polynomial, causing it to dominate the behavior of the graph at the ends.

The Correct Answer

The correct answer is B. The ends of the graph will extend in both directions.

Additional Analysis

To further analyze the behavior of the graph, we can consider the following:

• Even and Odd Functions: The term $2x^5$ is an odd function, which means it will change sign as $x$ changes sign. This will cause the graph to extend in both the positive and negative directions.
• Degree of the Polynomial: The degree of the new polynomial is 5, which is higher than the original polynomial. This will cause the graph to extend in both directions as $x$ approaches infinity.
• Leading Term: The leading term of the new polynomial is $2x^5$, which has a positive coefficient. This will cause the graph to extend in the positive direction as $x$ increases.

Conclusion (Continued)

In conclusion, the graph of the new polynomial will extend in both the positive and negative directions as $x$ approaches infinity. This is because the term $2x^5$ has a higher degree than the original polynomial, causing it to dominate the behavior of the graph at the ends.

Final Answer

The final answer is B. The ends of the graph will extend in both directions.

References

• [1] "Polynomial Graphs" by Math Open Reference
• [2] "Polynomial Functions" by Khan Academy
• [3] "Graphing Polynomials" by Purplemath

Additional Resources

• [1] "Polynomial Graphs" by Wolfram Alpha
• [2] "Polynomial Functions" by MIT OpenCourseWare
• [3] "Graphing Polynomials" by Mathway
  Polynomial Graphs Q&A =========================

Q: What is the degree of the polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$?

A: The degree of the polynomial is 5, which is the highest power of $x$ in the polynomial.

Q: How does the term $2x^5$ affect the graph of the polynomial?

A: The term $2x^5$ causes the graph to extend in both the positive and negative directions as $x$ approaches infinity. This is because the power of $x$ in the new term is odd, which means it will change sign as $x$ changes sign.

Q: What is the behavior of the graph at the ends?

A: As $x$ approaches positive infinity, the graph will extend in the positive direction. As $x$ approaches negative infinity, the graph will extend in the negative direction.

Q: Is the polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$ an even or odd function?

A: The polynomial is an odd function because the term $2x^5$ is an odd function.

Q: What is the leading term of the polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$?

A: The leading term of the polynomial is $2x^5$, which has a positive coefficient.

Q: How does the degree of the polynomial affect its behavior at the ends?

A: The degree of the polynomial affects its behavior at the ends by causing the graph to extend in both the positive and negative directions as $x$ approaches infinity.

Q: What is the significance of the term $2x^5$ in the polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$?

A: The term $2x^5$ is significant because it causes the graph to extend in both the positive and negative directions as $x$ approaches infinity.

Q: How does the polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$ compare to the polynomial $y = 8x^4 - 2x^3 + 5$?

A: The polynomial $y = 8x^4 - 2x^3 + 5 + 2x^5$ has a higher degree than the polynomial $y = 8x^4 - 2x^3 + 5$, which means it will have a different behavior at the ends.

Q: What is the final answer to the original question?

A: The final answer is B. The ends of the graph will extend in both directions.

Q: What are some additional resources for learning about polynomial graphs?

A: Some additional resources for learning about polynomial graphs include:

• [1] "Polynomial Graphs" by Wolfram Alpha
• [2] "Polynomial Functions" by MIT OpenCourseWare
• [3] "Graphing Polynomials" by Mathway

Q: What are some common mistakes to avoid when working with polynomial graphs?

A: Some common mistakes to avoid when working with polynomial graphs include:

• Not considering the degree of the polynomial
• Not analyzing the behavior of the graph at the ends
• Not identifying the leading term of the polynomial

Q: How can I practice working with polynomial graphs?

A: You can practice working with polynomial graphs by:

• Graphing polynomials using a graphing calculator or software
• Analyzing the behavior of the graph at the ends
• Identifying the leading term of the polynomial
• Comparing the behavior of different polynomials