To determine the end behavior of the polynomial function [tex]\(f(x) = 2x^3 - 26x - 24\)[/tex], we analyze how the function behaves as [tex]\(x\)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(+\infty\)[/tex].
1. Leading Term Identification:
The leading term in the polynomial [tex]\(f(x) = 2x^3 - 26x - 24\)[/tex] is [tex]\(2x^3\)[/tex]. This term determines the end behavior of the polynomial because it has the highest power of [tex]\(x\)[/tex].
2. Degree and Coefficient Analysis:
- The function [tex]\(f(x)\)[/tex] is a cubic polynomial (degree 3).
- The leading coefficient is [tex]\(2\)[/tex], which is positive.
3. End Behavior Determination:
- For polynomials of odd degree (such as cubic polynomials), and with a positive leading coefficient, the end behavior is as follows:
- As [tex]\(x \rightarrow -\infty\)[/tex], the function value [tex]\(f(x) \rightarrow -\infty\)[/tex]. This is because when [tex]\(x\)[/tex] is large and negative, [tex]\(x^3\)[/tex] will be very large and negative, and multiplying by the positive coefficient 2 will still give a large negative value.
- As [tex]\(x \rightarrow \infty\)[/tex], the function value [tex]\(f(x) \rightarrow \infty\)[/tex]. This is because when [tex]\(x\)[/tex] is large and positive, [tex]\(x^3\)[/tex] will be very large and positive, and multiplying by the positive coefficient 2 will give a large positive value.
Therefore, the end behavior of the graph of the polynomial function [tex]\(f(x) = 2x^3 - 26x - 24\)[/tex] is:
- As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(y \rightarrow -\infty\)[/tex].
- As [tex]\(x \rightarrow \infty\)[/tex], [tex]\(y \rightarrow \infty\)[/tex].
So, the correct answer is:
As [tex]\(x \rightarrow -\infty\)[/tex], [tex]\(y \rightarrow -\infty\)[/tex] and as [tex]\(x \rightarrow \infty\)[/tex], [tex]\(y \rightarrow \infty\)[/tex].
