Question

what is the end behavior of the graph of the polynomial function $f(x) = 3x^6 + 30x^5 + 75x^4$?
as $x \to -\infty$, $y \to \infty$ and as $x \to \infty$, $y \to \infty$.
as $x \to -\infty$, $y \to -\infty$ and as $x \to \infty$, $y \to \infty$.
as $x \to -\infty$, $y \to -\infty$ and as $x \to \infty$, $y \to -\infty$.
as $x \to -\infty$, $y \to \infty$ and as $x \to \infty$, $y \to -\infty$.

Explanation:

Step1: Identify the leading term

The polynomial function is \( f(x) = 3x^6 + 30x^5 + 75x^4 \). The leading term is \( 3x^6 \), which has a leading coefficient of \( 3 \) (positive) and a degree of \( 6 \) (even).

Step2: Determine end behavior from leading term

For a polynomial function, the end behavior is determined by the leading term. If the degree is even and the leading coefficient is positive:

• As \( x \to \infty \), \( x^n \) (where \( n \) is even) becomes very large positive, and multiplying by a positive leading coefficient keeps it positive. So \( y \to \infty \).
• As \( x \to -\infty \), \( (-x)^n = x^n \) (since \( n \) is even), so \( x^n \) is still very large positive, and multiplying by a positive leading coefficient keeps it positive. So \( y \to \infty \).

This matches the first option's description of the end behavior.

Answer:

The first option (As \( x \to -\infty \), \( y \to \infty \) and as \( x \to \infty \), \( y \to \infty \))