Question

consider the polynomial function $h(x) = -2x^5 + 8x^4 - 2x^2 + 15$. what is the end behavior of the graph of $h$?

Explanation:

Step1: Identify degree and leading coefficient

The polynomial is $h(x) = -2x^5 + 8x^4 - 2x^2 + 15$. Degree $n=5$ (odd), leading coefficient $a=-2$ (negative).

Step2: Analyze as $x\to+\infty$

For odd degree, negative leading coefficient: as $x\to+\infty$, $-2x^5$ dominates, so $h(x)\to-\infty$.

Step3: Analyze as $x\to-\infty$

For odd degree, negative leading coefficient: as $x\to-\infty$, $-2x^5$ (negative times negative infinity) gives $h(x)\to+\infty$.

Answer:

As $x \to +\infty$, $h(x) \to -\infty$; as $x \to -\infty$, $h(x) \to +\infty$