- Polynomial end behavior is dominated by the leading term \(a_n x^n\).
- For \(f(x) = -6x^{20} + 17x^{12} + 6x^4 - 10x + 11\), the leading term \(-6x^{20}\) has an even degree and a negative coefficient, so \(y \to -\infty\) as \(x \to \infty\) and \(y \to -\infty\) as \(x \to -\infty\).
- For \(f(x) = -13x^{15} - 17x^{12} + 17x^3 + 12x - 8\), the leading term \(-13x^{15}\) has an odd degree and a negative coefficient, so \(y \to -\infty\) as \(x \to \infty\) and \(y \to \infty\) as \(x \to -\infty\).
- The final results for the boxes are: Polynomial 3: \(\boxed{-\infty, -\infty}\), Polynomial 4: \(\boxed{-\infty, \infty}\).
Explanation
- Understanding Polynomial End Behavior
The end behavior of a polynomial function \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) is determined by its leading term \(a_n x^n\). As the magnitude of \(x\) becomes very large, the term with the highest degree grows much faster than all the other terms, meaning the behavior of the entire polynomial at the extremes matches the behavior of its leading term. We look at the degree \(n\) and the leading coefficient \(a_n\) to determine this behavior.
- If \(n\) is even and \(a_n > 0\), the graph rises on both ends (\(y \to \infty\) as \(x \to \pm\infty\)).
- If \(n\) is even and \(a_n < 0\), the graph falls on both ends (\(y \to -\infty\) as \(x \to \pm\infty\)).
- If \(n\) is odd and \(a_n > 0\), the graph falls to the left and rises to the right (\(y \to -\infty\) as \(x \to -\infty\), \(y \to \infty\) as \(x \to \infty\)).
- If \(n\) is odd and \(a_n < 0\), the graph rises to the left and falls to the right (\(y \to \infty\) as \(x \to -\infty\), \(y \to -\infty\) as \(x \to \infty\)).
- Analyzing Polynomial 3
For the third polynomial \(f(x) = -6x^{20} + 17x^{12} + 6x^4 - 10x + 11\):
- The leading term is \(-6x^{20}\).
- The degree is \(20\), which is an even number.
- The leading coefficient is \(-6\), which is a negative number.
- Since the degree is even and the coefficient is negative, the function approaches negative infinity as \(x\) goes to either positive or negative infinity.
- Therefore, as \(x \to \infty, y \to -\infty\) and as \(x \to -\infty, y \to -\infty\).
- Analyzing Polynomial 4
For the fourth polynomial \(f(x) = -13x^{15} - 17x^{12} + 17x^3 + 12x - 8\):
- The leading term is \(-13x^{15}\).
- The degree is \(15\), which is an odd number.
- The leading coefficient is \(-13\), which is a negative number.
- Since the degree is odd and the coefficient is negative, the graph behaves like a downward-sloping line at its extremes. As \(x \to \infty\), the term \(-13x^{15}\) becomes very large and negative, so \(y \to -\infty\). As \(x \to -\infty\), the term \(x^{15}\) becomes very large and negative, and multiplying by \(-13\) results in a very large positive value, so \(y \to \infty\).
- Therefore, as \(x \to \infty, y \to -\infty\) and as \(x \to -\infty, y \to \infty\).
Examples
Understanding the end behavior of polynomials is crucial in fields like economics and engineering. For example, when modeling the profit of a large company over time, a polynomial might represent the total profit based on the amount of resources invested. By analyzing the end behavior, managers can predict whether profits will continue to rise, fall, or eventually stabilize if resources are scaled up indefinitely. Similar logic applies to modeling physical structures or population dynamics where extreme-case forecasting is essential for long-term planning.