The addition of either $-x^8$ or $5 x^7$ will change the end behavior of $y=-2 x^7+5 x^6-24$. Explain how each of the added terms above would change the graph.
Answer (2)
Adding − x 8 makes the end behavior of the polynomial approach negative infinity on both sides, while adding 5 x 7 results in the polynomial approaching positive infinity on the right and negative infinity on the left.
;
Adding − x 8 changes the end behavior: as x → ± ∞ , y → − ∞ .
Adding 5 x 7 changes the end behavior: as x → ∞ , y → ∞ , and as x → − ∞ , y → − ∞ .
The original function y = − 2 x 7 + 5 x 6 − 24 has end behavior: as x → ∞ , y → − ∞ , and as x → − ∞ , y → ∞ .
Therefore, the addition of either term changes the end behavior of the original function, as described above. See explanation above.
Explanation
Analyzing the Original Function We are given the function y = − 2 x 7 + 5 x 6 − 24 and asked to describe how adding − x 8 or 5 x 7 changes its end behavior. The end behavior of a polynomial is determined by its leading term, which in this case is − 2 x 7 . This means that as x approaches infinity, y approaches negative infinity, and as x approaches negative infinity, y approaches infinity.
Adding − x 8 Let's consider the first case: adding − x 8 to the original function. The new function becomes y = − x 8 − 2 x 7 + 5 x 6 − 24 . The leading term is now − x 8 . As x approaches infinity, y approaches negative infinity. As x approaches negative infinity, y also approaches negative infinity. Therefore, adding − x 8 changes the end behavior such that as x goes to both positive and negative infinity, y goes to negative infinity.
Adding 5 x 7 Now let's consider the second case: adding 5 x 7 to the original function. The new function becomes y = − 2 x 7 + 5 x 7 + 5 x 6 − 24 = 3 x 7 + 5 x 6 − 24 . The leading term is now 3 x 7 . As x approaches infinity, y approaches infinity. As x approaches negative infinity, y approaches negative infinity. Therefore, adding 5 x 7 changes the end behavior such that as x goes to infinity, y goes to infinity, and as x goes to negative infinity, y goes to negative infinity.
Concluding the Analysis In summary, adding − x 8 changes the end behavior so that y approaches negative infinity as x approaches both positive and negative infinity. Adding 5 x 7 changes the end behavior so that y approaches infinity as x approaches infinity and y approaches negative infinity as x approaches negative infinity.
Examples
Understanding end behavior is crucial in fields like physics and engineering, where polynomial functions model various phenomena. For example, predicting the trajectory of a projectile or analyzing the stability of a structure often involves examining the end behavior of related polynomial equations. By knowing how different terms affect the end behavior, engineers can design safer and more efficient systems.
