The Original Polynomial Function

Oct 13, 2025 by talynt.org 1 views

Introduction

Polynomial graphs are a fundamental concept in mathematics, used to represent the behavior of polynomial functions. These graphs can be used to visualize the behavior of a function, including its maximum and minimum values, as well as its end behavior. In this article, we will explore how the graph of a polynomial function changes when a new term is added to the function.

The Original Polynomial Function

The original polynomial function is given by:

y = 2x^6 + 9x^5 - 7x^3 - 1

This function has a degree of 6, which means that the highest power of x is 6. The coefficients of the terms are 2, 9, -7, and -1, respectively.

Adding a New Term

We are asked to add a new term, -3x^6, to the original polynomial function. The new function is given by:

y = 2x^6 + 9x^5 - 7x^3 - 1 - 3x^6

Simplifying the New Function

To simplify the new function, we can combine like terms. The two terms with the highest power of x are 2x^6 and -3x^6. Combining these terms, we get:

y = -x^6 + 9x^5 - 7x^3 - 1

Understanding the End Behavior

The end behavior of a polynomial function is determined by the leading term, which is the term with the highest power of x. In this case, the leading term is -x^6. Since the coefficient of the leading term is negative, the ends of the graph will approach negative infinity.

Analyzing the Graph

To analyze the graph of the new function, we need to consider the behavior of the function as x approaches positive and negative infinity. As x approaches positive infinity, the term -x^6 dominates the function, causing the graph to approach negative infinity. Similarly, as x approaches negative infinity, the term -x^6 dominates the function, causing the graph to approach negative infinity.

Conclusion

In conclusion, when the term -3x^6 is added to the original polynomial function y = 2x^6 + 9x^5 - 7x^3 - 1, the ends of the graph will approach negative infinity. This is because the leading term of the new function is -x^6, which has a negative coefficient. Therefore, the correct answer is:

A. Both ends of the graph will approach negative infinity.

Key Takeaways

The end behavior of a polynomial function is determined by the leading term.
The leading term is the term with the highest power of x.
If the coefficient of the leading term is negative, the ends of the graph will approach negative infinity.

Real-World Applications

Understanding polynomial graphs and their end behavior has many real-world applications. For example, in physics, the end behavior of a polynomial function can be used to model the behavior of a physical system. In economics, the end behavior of a polynomial function can be used to model the behavior of a market.

Common Mistakes

When analyzing the end behavior of a polynomial function, it is common to make mistakes. One common mistake is to ignore the coefficient of the leading term. Another common mistake is to assume that the ends of the graph will approach positive infinity if the coefficient of the leading term is positive.

Tips and Tricks

To analyze the end behavior of a polynomial function, follow these tips and tricks:

Identify the leading term of the function.
Determine the coefficient of the leading term.
Use the coefficient to determine the end behavior of the function.

Conclusion

In conclusion, understanding polynomial graphs and their end behavior is a fundamental concept in mathematics. By analyzing the leading term of a polynomial function, we can determine the end behavior of the function. This knowledge has many real-world applications and is essential for anyone working in mathematics, physics, or economics.

Introduction

Polynomial graphs are a fundamental concept in mathematics, used to represent the behavior of polynomial functions. In our previous article, we explored how the graph of a polynomial function changes when a new term is added to the function. In this article, we will answer some frequently asked questions about polynomial graphs and their end behavior.

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

A: The end behavior of a polynomial function is determined by the leading term, which is the term with the highest power of x. If the coefficient of the leading term is positive, the ends of the graph will approach positive infinity. If the coefficient of the leading term is negative, the ends of the graph will approach negative infinity.

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

A: To determine the end behavior of a polynomial function, follow these steps:

Identify the leading term of the function.
Determine the coefficient of the leading term.
Use the coefficient to determine the end behavior of the function.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be written in the form:

y = ax^n + bx^(n-1) + ... + cx + d

where a, b, c, and d are constants, and n is a positive integer.

A rational function, on the other hand, is a function that can be written in the form:

y = f(x) / g(x)

where f(x) and g(x) are polynomial functions.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, follow these steps:

Determine the degree of the polynomial function.
Find the x-intercepts of the function.
Find the y-intercept of the function.
Use the x-intercepts and y-intercept to sketch the graph of the function.

Q: What is the difference between a polynomial function and a quadratic function?

A: A polynomial function is a function that can be written in the form:

y = ax^n + bx^(n-1) + ... + cx + d

where a, b, c, and d are constants, and n is a positive integer.

A quadratic function, on the other hand, is a function that can be written in the form:

y = ax^2 + bx + c

where a, b, and c are constants.

Q: How do I determine the vertex of a quadratic function?

A: To determine the vertex of a quadratic function, follow these steps:

Write the quadratic function in the form:

y = a(x - h)^2 + k

where (h, k) is the vertex of the function. 2. Use the formula:

h = -b / 2a

to find the x-coordinate of the vertex. 3. Use the formula:

k = f(h)

to find the y-coordinate of the vertex.

Q: What is the difference between a polynomial function and a linear function?

A: A polynomial function is a function that can be written in the form:

y = ax^n + bx^(n-1) + ... + cx + d

where a, b, c, and d are constants, and n is a positive integer.

A linear function, on the other hand, is a function that can be written in the form:

y = mx + b

where m and b are constants.

Conclusion

In conclusion, polynomial graphs are a fundamental concept in mathematics, used to represent the behavior of polynomial functions. By understanding the end behavior of a polynomial function, we can determine the behavior of the function as x approaches positive and negative infinity. This knowledge has many real-world applications and is essential for anyone working in mathematics, physics, or economics.

Key Takeaways

The end behavior of a polynomial function is determined by the leading term.
The leading term is the term with the highest power of x.
If the coefficient of the leading term is positive, the ends of the graph will approach positive infinity. If the coefficient of the leading term is negative, the ends of the graph will approach negative infinity.

Real-World Applications

Common Mistakes

Tips and Tricks

To analyze the end behavior of a polynomial function, follow these tips and tricks:

Identify the leading term of the function.
Determine the coefficient of the leading term.
Use the coefficient to determine the end behavior of the function.