What Is The End Behavior Of The Graph Of The Polynomial Function $f(x) = 2x^3 - 26x - 24$?A. As $x \rightarrow -\infty$, $y \rightarrow -\infty$ And As $x \rightarrow \infty$, $y \rightarrow -\infty$.B. As

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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 polynomial 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 polynomial function f(x) = 2x^3 - 26x - 24.

What is a Polynomial Function?

A polynomial function is a function that can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants and n is a non-negative integer. The degree of a polynomial function is the highest power of x in the function.

The Degree of the Polynomial Function

The degree of the polynomial function f(x) = 2x^3 - 26x - 24 is 3, since the highest power of x is 3.

The Leading Coefficient

The leading coefficient of a polynomial function is the coefficient of the term with the highest power of x. In the function f(x) = 2x^3 - 26x - 24, the leading coefficient is 2.

End Behavior of Polynomial Functions

The end behavior of a polynomial function is determined by the degree and the leading coefficient of the function. If the degree of the function is even, the end behavior is determined by the leading coefficient. If the degree of the function is odd, the end behavior is determined by the sign of the leading coefficient.

End Behavior of the Polynomial Function f(x) = 2x^3 - 26x - 24

Since the degree of the function f(x) = 2x^3 - 26x - 24 is odd, the end behavior is determined by the sign of the leading coefficient. The leading coefficient is 2, which is positive. Therefore, as x approaches positive or negative infinity, the function f(x) = 2x^3 - 26x - 24 approaches positive or negative infinity.

Conclusion

In conclusion, the end behavior of the polynomial function f(x) = 2x^3 - 26x - 24 is that as x approaches positive or negative infinity, the function approaches positive or negative infinity.

Answer

The correct answer is A. As x → -∞, y → -∞ and as x → ∞, y → ∞.

Example

Let's consider an example to illustrate the end behavior of the polynomial function f(x) = 2x^3 - 26x - 24. Suppose we want to find the end behavior of the function as x approaches 1000. We can plug in x = 1000 into the function and evaluate it.

f(1000) = 2(1000)^3 - 26(1000) - 24 = 2(1,000,000,000) - 26(1000) - 24 = 2,000,000,000 - 26,000 - 24 = 1,997,974,976

As we can see, the function f(x) = 2x^3 - 26x - 24 approaches a large positive value as x approaches 1000.

Graph of the Polynomial Function

The graph of the polynomial function f(x) = 2x^3 - 26x - 24 is a cubic curve that opens upward. The graph has a positive leading coefficient, which means that the function approaches positive infinity as x approaches positive or negative infinity.

Key Takeaways

  • The end behavior of a polynomial function is determined by the degree and the leading coefficient of the function.
  • If the degree of the function is even, the end behavior is determined by the leading coefficient.
  • If the degree of the function is odd, the end behavior is determined by the sign of the leading coefficient.
  • The graph of a polynomial function with a positive leading coefficient opens upward.

Final Thoughts

In conclusion, the end behavior of the polynomial function f(x) = 2x^3 - 26x - 24 is that as x approaches positive or negative infinity, the function approaches positive or negative infinity. This is because the degree of the function is odd and the leading coefficient is positive. The graph of the function is a cubic curve that opens upward, and the function approaches positive infinity as x approaches positive or 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 x approaches positive or negative infinity. It is determined by the degree and the leading coefficient of the 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 identify the degree and the leading coefficient of the function. If the degree is even, the end behavior is determined by the leading coefficient. If the degree is odd, the end behavior is determined by the sign of the leading coefficient.

Q3: What is the difference between the end behavior of even and odd degree polynomial functions?

A3: The end behavior of even degree polynomial functions is determined by the leading coefficient, while the end behavior of odd degree polynomial functions is determined by the sign of the leading coefficient.

Q4: How do I know if a polynomial function has an even or odd degree?

A4: To determine if a polynomial function has an even or odd degree, you need to identify the highest power of x in the function. If the highest power of x is even, the degree is even. If the highest power of x is odd, the degree is odd.

Q5: What is the significance of the leading coefficient in determining the end behavior of a polynomial function?

A5: The leading coefficient is the coefficient of the term with the highest power of x in the function. It determines the end behavior of the function if the degree is even. If the degree is odd, the sign of the leading coefficient determines the end behavior.

Q6: Can a polynomial function have a negative leading coefficient and still have a positive end behavior?

A6: No, a polynomial function cannot have a negative leading coefficient and still have a positive end behavior. The end behavior of a polynomial function is determined by the sign of the leading coefficient if the degree is odd.

Q7: How do I graph a polynomial function to visualize its end behavior?

A7: To graph a polynomial function, you can use a graphing calculator or a computer algebra system. The graph of a polynomial function with a positive leading coefficient opens upward, while the graph of a polynomial function with a negative leading coefficient opens downward.

Q8: Can a polynomial function have a horizontal asymptote and still have an end behavior?

A8: Yes, a polynomial function can have a horizontal asymptote and still have an end behavior. The horizontal asymptote is the horizontal line that the function approaches as x approaches positive or negative infinity.

Q9: How do I determine the horizontal asymptote of a polynomial function?

A9: To determine the horizontal asymptote of a polynomial function, you need to identify the degree and the leading coefficient of the function. If the degree is even, the horizontal asymptote is the line y = a_n, where a_n is the leading coefficient. If the degree is odd, the horizontal asymptote is the line y = 0.

Q10: Can a polynomial function have a vertical asymptote and still have an end behavior?

A10: Yes, a polynomial function can have a vertical asymptote and still have an end behavior. The vertical asymptote is the vertical line that the function approaches as x approaches a certain value.

Conclusion

In conclusion, the end behavior of a polynomial function is determined by the degree and the leading coefficient of the function. The leading coefficient determines the end behavior if the degree is even, while the sign of the leading coefficient determines the end behavior if the degree is odd. The graph of a polynomial function with a positive leading coefficient opens upward, while the graph of a polynomial function with a negative leading coefficient opens downward.