Question

determine the end behavior of the polynomial function:
$y = -x^6 + 4x^8 - 3x^{10} + 2x - 6$

a. as $x \to -\infty$, $f(x) \to -\infty$ and as $x \to +\infty$, $f(x) \to -\infty$.

b. as $x \to -\infty$, $f(x) \to -\infty$ and as $x \to +\infty$, $f(x) \to +\infty$.

c. as $x \to -\infty$, $f(x) \to \infty$ and as $x \to +\infty$, $f(x) \to \infty$.

d. as $x \to -\infty$, $f(x) \to \infty$ and as $x \to +\infty$, $f(x) \to -\infty$.

Explanation:

Step1: Identify leading term

The leading term is $-3x^{10}$ (highest degree term).

Step2: Analyze degree and coefficient

Degree is 10 (even), coefficient is $-3$ (negative).

Step3: Determine end behavior

For even degree, $x\to\pm\infty$ gives $x^{10}\to+\infty$. Multiply by $-3$: $-3x^{10}\to-\infty$.

Answer:

A. As $x
ightarrow -\infty, f(x)
ightarrow -\infty$ and as $x
ightarrow +\infty, f(x)
ightarrow -\infty$.