- Identify the leading term \(a_n x^n\) of each polynomial to determine its long-term growth pattern.
- For \(f_1(x) = 18x^{16} + 8x^9 - 19x^2 + 3x - 14\), since the degree is even (\(n=16\)) and the leading coefficient is positive (\(a_n=18\)), the function approaches \(\infty\) at both ends.
- For \(f_2(x) = 17x^{25} + 10x^9 - 17x^3 - 6x - 3\), since the degree is odd (\(n=25\)) and the leading coefficient is positive (\(a_n=17\)), the function approaches \(\infty\) as \(x \rightarrow \infty\) and \(-\infty\) as \(x \rightarrow -\infty\).
- The end behaviors are: \(f_1(x) \rightarrow \infty, \infty\); \(f_2(x) \rightarrow \infty, -\infty\).
- \(\boxed{1. (\infty, \infty); 2. (\infty, -\infty)}\)
Explanation
- Analyzing End Behavior Principles
We are tasked with determining the end behavior of two given polynomials as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\).
- The first polynomial is \(f_1(x) = 18x^{16} + 8x^9 - 19x^2 + 3x - 14\).
- The second polynomial is \(f_2(x) = 17x^{25} + 10x^9 - 17x^3 - 6x - 3\).
The end behavior of a polynomial \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\) is governed by its leading term, \(a_n x^n\), where \(n\) is the degree of the polynomial and \(a_n\) is the leading coefficient. As \(|x|\) becomes very large, the term with the highest power dominates the expression, meaning that the behavior of \(f(x)\) as \(x \rightarrow \pm\infty\) mimics the behavior of its leading term \(a_n x^n\).
- Evaluating the First Polynomial
For \(f_1(x) = 18x^{16} + 8x^9 - 19x^2 + 3x - 14\), the degree \(n=16\) is even and the leading coefficient \(a_{16} = 18\) is positive.
For an even degree polynomial with a positive leading coefficient, the graph opens upwards at both ends.
- As \(x \rightarrow \infty\), \(x^{16} \rightarrow \infty\), so \(18x^{16} \rightarrow \infty\).
- As \(x \rightarrow -\infty\), \(x^{16} \rightarrow \infty\) (since the power is even), so \(18x^{16} \rightarrow \infty\).
Therefore, for \(f_1(x)\), as \(x \rightarrow \infty, y \rightarrow \infty\) and as \(x \rightarrow -\infty, y \rightarrow \infty\).
- Evaluating the Second Polynomial
For \(f_2(x) = 17x^{25} + 10x^9 - 17x^3 - 6x - 3\), the degree \(n=25\) is odd and the leading coefficient \(a_{25} = 17\) is positive.
For an odd degree polynomial with a positive leading coefficient, the graph behaves like the parent function \(y = x^n\).
- As \(x \rightarrow \infty\), \(x^{25} \rightarrow \infty\), so \(17x^{25} \rightarrow \infty\).
- As \(x \rightarrow -\infty\), \(x^{25} \rightarrow -\infty\) (since the power is odd), so \(17x^{25} \rightarrow -\infty\).
Therefore, for \(f_2(x)\), as \(x \rightarrow \infty, y \rightarrow \infty\) and as \(x \rightarrow -\infty, y \rightarrow -\infty\).
- Conclusion and Final Answers
Combining our findings:
- For \(f_1(x) = 18x^{16} + 8x^9 - 19x^2 + 3x - 14\):
As \(x \rightarrow \infty, y \rightarrow \infty\)
As \(x \rightarrow -\infty, y \rightarrow \infty\)
- For \(f_2(x) = 17x^{25} + 10x^9 - 17x^3 - 6x - 3\):
As \(x \rightarrow \infty, y \rightarrow \infty\)
As \(x \rightarrow -\infty, y \rightarrow -\infty\)
These results represent the end behavior of the given polynomials.
Examples
Understanding the end behavior of polynomials is essential in fields like physics and economics for predicting long-term trends. For example, if you are modeling the growth of a population or the long-term cost of a project using a polynomial function, determining the end behavior helps you understand whether the model will grow indefinitely, decline, or stabilize as the variable (time or scale) increases towards infinity. By analyzing the highest power and its coefficient, you can quickly determine if your project's costs will explode in the future or if a biological population will hit a ceiling.