Use The Leading Coefficient Test To Determine Whether { Y \rightarrow \infty $}$ Or { Y \rightarrow -\infty $}$ As { X \rightarrow \infty $}$ For The Function.${ Y = 7x^6 - X^2 + 13 } A S \[ As \[ A S \[ X \rightarrow

Introduction

When analyzing the behavior of a polynomial function as the input variable approaches positive or negative infinity, we often rely on the leading coefficient test to determine whether the function approaches positive or negative infinity. This test is particularly useful for polynomial functions of degree greater than or equal to 1. In this article, we will explore how to apply the leading coefficient test to determine the end behavior of the function ${ y = 7x^6 - x^2 + 13 }$ as ${ x \rightarrow \infty }$.

Understanding the Leading Coefficient Test

The leading coefficient test is a simple yet powerful tool for determining the end behavior of a polynomial function. The test states that if a polynomial function ${ f(x) }$ has a degree ${ n }$ and a leading coefficient ${ a_n }$, then:

• If ${ a_n > 0 }$ and ${ n }$ is even, then ${ f(x) \rightarrow \infty }$ as ${ x \rightarrow \infty }$.
• If ${ a_n > 0 }$ and ${ n }$ is odd, then ${ f(x) \rightarrow \infty }$ as ${ x \rightarrow \infty }$.
• If ${ a_n < 0 }$ and ${ n }$ is even, then ${ f(x) \rightarrow -\infty }$ as ${ x \rightarrow \infty }$.
• If ${ a_n < 0 }$ and ${ n }$ is odd, then ${ f(x) \rightarrow -\infty }$ as ${ x \rightarrow \infty }$.

Applying the Leading Coefficient Test to the Given Function

Now that we have a solid understanding of the leading coefficient test, let's apply it to the given function ${ y = 7x^6 - x^2 + 13 }$. The leading term of this function is ${ 7x^6 }$, which has a degree of 6 and a leading coefficient of 7. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Visualizing the End Behavior of the Function

To gain a better understanding of the end behavior of the function, let's visualize its graph. As ${ x }$ approaches positive infinity, the function ${ y = 7x^6 - x^2 + 13 }$ will continue to increase without bound. Similarly, as ${ x }$ approaches negative infinity, the function will also increase without bound. This is because the leading term ${ 7x^6 }$ dominates the behavior of the function as ${ x }$ approaches infinity.

Conclusion

In conclusion, the leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. By identifying the leading coefficient and the degree of the leading term, we can use the leading coefficient test to determine whether the function approaches positive or negative infinity as the input variable approaches positive or negative infinity. In this article, we applied the leading coefficient test to the function ${ y = 7x^6 - x^2 + 13 }$ and concluded that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Example Problems

Problem 1

Determine the end behavior of the function ${ y = 2x^4 - 3x^2 + 1 }$ as ${ x \rightarrow \infty }$.

Solution

The leading term of this function is ${ 2x^4 }$, which has a degree of 4 and a leading coefficient of 2. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Problem 2

Determine the end behavior of the function ${ y = -x^3 + 2x^2 - 3 }$ as ${ x \rightarrow \infty }$.

Solution

The leading term of this function is ${ -x^3 }$, which has a degree of 3 and a leading coefficient of -1. Since the degree of the leading term is odd and the leading coefficient is negative, we can conclude that the function approaches negative infinity as ${ x \rightarrow \infty }$.

Practice Problems

Problem 1

Determine the end behavior of the function ${ y = 5x^5 - 2x^3 + 1 }$ as ${ x \rightarrow \infty }$.

Problem 2

Determine the end behavior of the function ${ y = -x^4 + 3x^2 - 2 }$ as ${ x \rightarrow \infty }$.

Problem 3

Determine the end behavior of the function ${ y = 2x^6 - 3x^4 + 1 }$ as ${ x \rightarrow \infty }$.

Problem 4

Determine the end behavior of the function ${ y = -x^5 + 2x^3 - 3 }$ as ${ x \rightarrow \infty }$.

Problem 5

Determine the end behavior of the function ${ y = 5x^4 - 2x^2 + 1 }$ as ${ x \rightarrow \infty }$.

Solutions

Problem 1

The leading term of this function is ${ 5x^5 }$, which has a degree of 5 and a leading coefficient of 5. Since the degree of the leading term is odd and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Problem 2

The leading term of this function is ${ -x^4 }$, which has a degree of 4 and a leading coefficient of -1. Since the degree of the leading term is even and the leading coefficient is negative, we can conclude that the function approaches negative infinity as ${ x \rightarrow \infty }$.

Problem 3

The leading term of this function is ${ 2x^6 }$, which has a degree of 6 and a leading coefficient of 2. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Problem 4

The leading term of this function is ${ -x^5 }$, which has a degree of 5 and a leading coefficient of -1. Since the degree of the leading term is odd and the leading coefficient is negative, we can conclude that the function approaches negative infinity as ${ x \rightarrow \infty }$.

Problem 5

The leading term of this function is ${ 5x^4 }$, which has a degree of 4 and a leading coefficient of 5. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Conclusion

In conclusion, the leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. By identifying the leading coefficient and the degree of the leading term, we can use the leading coefficient test to determine whether the function approaches positive or negative infinity as the input variable approaches positive or negative infinity. In this article, we applied the leading coefficient test to several example problems and concluded that the functions approach positive or negative infinity as ${ x \rightarrow \infty }$.

Introduction

The leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. However, it can be a bit confusing to apply, especially for those who are new to the concept. In this article, we will answer some of the most frequently asked questions about the leading coefficient test, providing clarity and insight into this important mathematical concept.

Q: What is the leading coefficient test?

A: The leading coefficient test is a method used to determine the end behavior of a polynomial function. It involves identifying the leading coefficient and the degree of the leading term, and then using this information to determine whether the function approaches positive or negative infinity as the input variable approaches positive or negative infinity.

Q: How do I identify the leading coefficient and the degree of the leading term?

A: To identify the leading coefficient and the degree of the leading term, you need to look at the polynomial function and identify the term with the highest degree. The coefficient of this term is the leading coefficient, and the degree of this term is the degree of the leading term.

Q: What if the polynomial function has multiple terms with the same degree?

A: If the polynomial function has multiple terms with the same degree, you need to combine these terms to form a single term. The coefficient of this combined term is the leading coefficient, and the degree of this combined term is the degree of the leading term.

Q: What if the leading coefficient is negative?

A: If the leading coefficient is negative, the function will approach negative infinity as the input variable approaches positive or negative infinity.

Q: What if the degree of the leading term is even?

A: If the degree of the leading term is even, the function will approach positive infinity as the input variable approaches positive or negative infinity.

Q: What if the degree of the leading term is odd?

A: If the degree of the leading term is odd, the function will approach negative infinity as the input variable approaches positive or negative infinity.

Q: Can I use the leading coefficient test for all types of functions?

A: No, the leading coefficient test is only applicable to polynomial functions. It is not applicable to rational functions, trigonometric functions, or other types of functions.

Q: What if the polynomial function has a degree of 0?

A: If the polynomial function has a degree of 0, it is a constant function, and the leading coefficient test is not applicable.

Q: Can I use the leading coefficient test to determine the end behavior of a function at a specific point?

A: No, the leading coefficient test is used to determine the end behavior of a function as the input variable approaches positive or negative infinity. It is not used to determine the end behavior of a function at a specific point.

Q: How do I apply the leading coefficient test to a function with multiple variables?

A: To apply the leading coefficient test to a function with multiple variables, you need to identify the leading coefficient and the degree of the leading term in each variable. Then, you need to use the leading coefficient test for each variable separately.

Conclusion

In conclusion, the leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. By understanding the leading coefficient test and how to apply it, you can gain a deeper understanding of the behavior of polynomial functions and make more informed decisions in a variety of mathematical and real-world contexts.

Practice Problems

Problem 1

Determine the end behavior of the function ${ y = 2x^4 - 3x^2 + 1 }$ as ${ x \rightarrow \infty }$.

Problem 2

Determine the end behavior of the function ${ y = -x^3 + 2x^2 - 3 }$ as ${ x \rightarrow \infty }$.

Problem 3

Determine the end behavior of the function ${ y = 5x^5 - 2x^3 + 1 }$ as ${ x \rightarrow \infty }$.

Problem 4

Determine the end behavior of the function ${ y = -x^4 + 3x^2 - 2 }$ as ${ x \rightarrow \infty }$.

Problem 5

Determine the end behavior of the function ${ y = 2x^6 - 3x^4 + 1 }$ as ${ x \rightarrow \infty }$.

Solutions

Problem 1

The leading term of this function is ${ 2x^4 }$, which has a degree of 4 and a leading coefficient of 2. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Problem 2

The leading term of this function is ${ -x^3 }$, which has a degree of 3 and a leading coefficient of -1. Since the degree of the leading term is odd and the leading coefficient is negative, we can conclude that the function approaches negative infinity as ${ x \rightarrow \infty }$.

Problem 3

The leading term of this function is ${ 5x^5 }$, which has a degree of 5 and a leading coefficient of 5. Since the degree of the leading term is odd and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Problem 4

The leading term of this function is ${ -x^4 }$, which has a degree of 4 and a leading coefficient of -1. Since the degree of the leading term is even and the leading coefficient is negative, we can conclude that the function approaches negative infinity as ${ x \rightarrow \infty }$.

Problem 5

The leading term of this function is ${ 2x^6 }$, which has a degree of 6 and a leading coefficient of 2. Since the degree of the leading term is even and the leading coefficient is positive, we can conclude that the function approaches positive infinity as ${ x \rightarrow \infty }$.

Conclusion

In conclusion, the leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. By understanding the leading coefficient test and how to apply it, you can gain a deeper understanding of the behavior of polynomial functions and make more informed decisions in a variety of mathematical and real-world contexts.