- The end behavior of a polynomial depends on its leading term \(a_n x^n\).
- Adding \(-x^8\) changes the degree of the polynomial to 8, which changes the end behavior.
- Adding \(5x^7\) results in a new leading term \(3x^7\), changing the sign of the coefficient and thus the end behavior.
- Other terms \((-3x^5, 1,000, -300)\) do not alter the leading term, so the end behavior remains constant.
- The terms that change the end behavior are \(\boxed{-x^8, 5x^7}\).
Explanation
- Understanding Polynomial End Behavior
The end behavior of a polynomial function \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\) is determined primarily by its leading term, \(a_n x^n\). The leading term tells us how the function behaves for very large positive and negative values of \(x\).
Specifically, for the polynomial \(y = -2x^7 + 5x^6 - 24\):
- The leading term is \(-2x^7\).
- The degree is \(7\), which is an odd number.
- The leading coefficient is \(-2\), which is a negative number.
Since the degree is odd and the leading coefficient is negative, the function approaches \(\infty\) as \(x \to -\infty\) and approaches \(-\infty\) as \(x \to \infty\). A term added to this polynomial will change its end behavior if and only if it alters the leading term in a way that changes the overall degree or the sign of the leading coefficient.
- Evaluating Each Term Addition
We examine each provided term by adding it to the original polynomial \(y = -2x^7 + 5x^6 - 24\) and identifying the new leading term.
- Adding \(-x^8\):
The new expression is \(y = -x^8 - 2x^7 + 5x^6 - 24\).
The new leading term is \(-x^8\). The degree is now 8, which is different from the original degree of 7. This change modifies the end behavior.
- Adding \(-3x^5\):
The new expression is \(y = -2x^7 + 5x^6 - 3x^5 - 24\).
The leading term remains \(-2x^7\). Since the degree and leading coefficient are unchanged, the end behavior remains the same.
- Adding \(5x^7\):
The new expression is \(y = (-2x^7 + 5x^7) + 5x^6 - 24 = 3x^7 + 5x^6 - 24\).
The new leading term is \(3x^7\). Although the degree remains 7, the leading coefficient has changed from -2 to 3. Since the sign of the leading coefficient has switched, the end behavior changes.
- Adding \(1,000\):
The new expression is \(y = -2x^7 + 5x^6 - 24 + 1000 = -2x^7 + 5x^6 + 976\).
The leading term remains \(-2x^7\). The end behavior does not change.
- Adding \(-300\):
The new expression is \(y = -2x^7 + 5x^6 - 24 - 300 = -2x^7 + 5x^6 - 324\).
The leading term remains \(-2x^7\). The end behavior does not change.
- Conclusion
Based on our analysis, adding \(-x^8\) changes the degree of the polynomial, and adding \(5x^7\) changes the sign of the leading coefficient. Both actions fundamentally alter the end behavior of the polynomial. The other terms, \(-3x^5\), \(1,000\), and \(-300\), do not change the degree or the sign of the leading coefficient, so they do not impact the end behavior. The terms that change the end behavior are \(-x^8\) and \(5x^7\).
Examples
In data modeling, polynomials are often used to represent growth trends. Understanding how changing coefficients affects the "long-run" prediction (end behavior) of these models is crucial. For instance, if a business analyst uses a polynomial model to forecast long-term revenue, adding a new variable (like a sudden unexpected cost) that changes the leading coefficient could flip their prediction from growth to decline. Recognizing these shifts helps in adjusting business strategies before it is too late.