Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?

$f(x) = 25x + 5x^3 + 40x^2 - 250$

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 $x \to \infty, y \to \infty$

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Explanation:

Step1: Identify Leading Term

The function \( f(x) = 25x + 5x^3 + 40x^2 - 250 \) has the leading term \( 5x^3 \) (highest degree term).

Step2: Analyze Leading Term Degree and Coefficient

The degree of the leading term (\( 5x^3 \)) is 3 (odd), and the coefficient is 5 (positive).
For a polynomial with an odd - degree leading term and positive coefficient:

• When \( x \to \infty \), we consider the behavior of \( x^3 \). As \( x\) gets very large positive, \( x^3\) is positive, and multiplying by 5 (positive) keeps it positive. So \( y = 5x^3+\cdots\to\infty\) as \( x\to\infty\).
• When \( x \to -\infty \), we consider the behavior of \( x^3\). As \( x\) gets very large negative, \( x^3=(-|x|)^3=-|x|^3\) (negative), and multiplying by 5 (positive) gives \( 5x^3=- 5|x|^3\) (negative). So \( y = 5x^3+\cdots\to-\infty\) as \( x\to-\infty\).

Answer:

as \( x \to -\infty \), \( y \to -\infty \) and as \( x \to \infty \), \( y \to \infty \)