What Is The End Behavior Of The Graph Of The Polynomial Function F ( X ) = 2 X 3 − 26 X − 24 F(x) = 2x^3 - 26x - 24 F ( X ) = 2 X 3 − 26 X − 24 ?A. As X → − ∞ , Y → − ∞ X \rightarrow -\infty, Y \rightarrow -\infty X → − ∞ , Y → − ∞ And As X → ∞ , Y → − ∞ X \rightarrow \infty, Y \rightarrow -\infty X → ∞ , Y → − ∞ .B. As $x \rightarrow

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Introduction

When analyzing the behavior of a polynomial function, it's essential to understand the end behavior, which refers to the behavior of the function as the input values approach positive or negative infinity. In this article, we will explore the end behavior of the polynomial function f(x)=2x3−26x−24f(x) = 2x^3 - 26x - 24.

What is End Behavior?

End behavior is a crucial concept in algebra and calculus, as it helps us understand the long-term behavior of a function. It's particularly important when dealing with polynomial functions, as they can exhibit different behaviors as the input values approach infinity.

Types of End Behavior

There are two types of end behavior:

  • Asymptotic behavior: This refers to the behavior of the function as the input values approach positive or negative infinity. In other words, it describes the long-term behavior of the function.
  • Horizontal asymptote: This is a horizontal line that the function approaches as the input values approach positive or negative infinity.

Determining End Behavior

To determine the end behavior of a polynomial function, we need to examine the leading term, which is the term with the highest degree. In the case of the function f(x)=2x3−26x−24f(x) = 2x^3 - 26x - 24, the leading term is 2x32x^3.

Leading Term and End Behavior

The leading term determines the end behavior of the function. If the leading term is positive, the function will approach positive infinity as the input values approach positive infinity. If the leading term is negative, the function will approach negative infinity as the input values approach positive infinity.

Analyzing the Function

Let's analyze the function f(x)=2x3−26x−24f(x) = 2x^3 - 26x - 24. The leading term is 2x32x^3, which is positive. This means that as the input values approach positive infinity, the function will approach positive infinity.

End Behavior of the Function

To determine the end behavior of the function, we need to examine the behavior of the function as the input values approach positive and negative infinity.

As x→∞,y→∞x \rightarrow \infty, y \rightarrow \infty

As the input values approach positive infinity, the function will approach positive infinity. This is because the leading term 2x32x^3 dominates the behavior of the function as the input values approach positive infinity.

As x→−∞,y→−∞x \rightarrow -\infty, y \rightarrow -\infty

As the input values approach negative infinity, the function will approach negative infinity. This is because the leading term 2x32x^3 dominates the behavior of the function as the input values approach negative infinity.

Conclusion

In conclusion, the end behavior of the polynomial function f(x)=2x3−26x−24f(x) = 2x^3 - 26x - 24 is that as x→∞,y→∞x \rightarrow \infty, y \rightarrow \infty and as x→−∞,y→−∞x \rightarrow -\infty, y \rightarrow -\infty. This is because the leading term 2x32x^3 dominates the behavior of the function as the input values approach positive and negative infinity.

Final Answer

The final answer is:

A. As x→−∞,y→−∞x \rightarrow -\infty, y \rightarrow -\infty and as x→∞,y→∞x \rightarrow \infty, y \rightarrow \infty.

This is the correct answer because the leading term 2x32x^3 dominates the behavior of the function as the input values approach positive and negative infinity.

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Q1: What is the end behavior of a polynomial function?

A1: The end behavior of a polynomial function refers to the behavior of the function as the input values approach positive or negative infinity. It's a crucial concept in algebra and calculus, as it helps us understand the long-term behavior of a function.

Q2: How do I determine the end behavior of a polynomial function?

A2: To determine the end behavior of a polynomial function, you need to examine the leading term, which is the term with the highest degree. If the leading term is positive, the function will approach positive infinity as the input values approach positive infinity. If the leading term is negative, the function will approach negative infinity as the input values approach positive infinity.

Q3: What is the difference between asymptotic behavior and horizontal asymptote?

A3: Asymptotic behavior refers to the behavior of the function as the input values approach positive or negative infinity. Horizontal asymptote, on the other hand, is a horizontal line that the function approaches as the input values approach positive or negative infinity.

Q4: How do I determine the horizontal asymptote of a polynomial function?

A4: To determine the horizontal asymptote of a polynomial function, you need to examine the leading term and the degree of the polynomial. If the degree of the polynomial is even, the horizontal asymptote is the ratio of the leading coefficient to the degree of the polynomial. If the degree of the polynomial is odd, there is no horizontal asymptote.

Q5: Can a polynomial function have multiple horizontal asymptotes?

A5: No, a polynomial function cannot have multiple horizontal asymptotes. However, a polynomial function can have a horizontal asymptote and a slant asymptote.

Q6: What is a slant asymptote?

A6: A slant asymptote is a line that the function approaches as the input values approach positive or negative infinity. It's a line that is not horizontal, but rather has a slope.

Q7: How do I determine the slant asymptote of a polynomial function?

A7: To determine the slant asymptote of a polynomial function, you need to divide the polynomial by the leading term. The quotient will be the slant asymptote.

Q8: Can a polynomial function have no horizontal or slant asymptote?

A8: Yes, a polynomial function can have no horizontal or slant asymptote. This occurs when the degree of the polynomial is even and the leading coefficient is zero.

Q9: How do I determine the end behavior of a rational function?

A9: To determine the end behavior of a rational function, you need to examine the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, the function will approach positive or negative infinity as the input values approach positive or negative infinity. If the degree of the numerator is less than the degree of the denominator, the function will approach zero as the input values approach positive or negative infinity.

Q10: Can a rational function have a horizontal or slant asymptote?

A10: Yes, a rational function can have a horizontal or slant asymptote. This occurs when the degree of the numerator is less than the degree of the denominator.

Conclusion

In conclusion, understanding the end behavior of polynomial functions is crucial in algebra and calculus. By examining the leading term and the degree of the polynomial, you can determine the end behavior of the function. Additionally, you can determine the horizontal and slant asymptotes of a polynomial function by dividing the polynomial by the leading term.

Final Answer

The final answer is:

A polynomial function can have a horizontal or slant asymptote, but it cannot have multiple horizontal asymptotes. The end behavior of a polynomial function is determined by the leading term and the degree of the polynomial.