Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = 5x^3 + 5x^2 - 245 - 245x$
answer

• 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 Leading Term

The leading term of \( f(x) = 5x^3 + 5x^2 - 245 - 245x \) is \( 5x^3 \). The degree is 3 (odd) and the leading coefficient is 5 (positive).

Step2: Apply End Behavior Rules

For a polynomial with odd degree and positive leading coefficient:

• As \( x \to \infty \), \( x^3 \to \infty \), so \( 5x^3 \to \infty \), thus \( y \to \infty \).
• As \( x \to -\infty \), \( x^3 \to -\infty \), so \( 5x^3 \to -\infty \), thus \( y \to -\infty \).

Answer:

as \( x \to \infty, y \to \infty \) and as \( x \to -\infty, y \to -\infty \) (the third option)