Question

the function $f(x) = -x^3 + 2x^2 - x + 5$ is graphed on a coordinate grid. which statements accurately describe the end behavior of the graph of the function?

• as $x$ approaches negative infinity, $y$ approaches positive infinity. as $x$ approaches positive infinity, $y$ approaches negative infinity.
• as $x$ approaches negative infinity, $y$ approaches positive infinity. as $x$ approaches positive infinity, $y$ approaches positive infinity.
• as $x$ approaches negative infinity, $y$ approaches negative infinity. as $x$ approaches positive infinity, $y$ approaches negative infinity.
• as $x$ approaches negative infinity, $y$ approaches negative infinity. as $x$ approaches positive infinity, $y$ approaches positive infinity.

Explanation:

Step1: Identify Leading Term

The function is \( f(x) = -x^3 + 2x^2 - x + 5 \). The leading term is \( -x^3 \) (highest degree term).

Step2: Analyze End Behavior

For a polynomial, end behavior is determined by the leading term. The degree of the leading term is 3 (odd), and the leading coefficient is -1 (negative).

• When the degree is odd and the leading coefficient is negative:
• As \( x \to -\infty \), \( x^3 \to -\infty \), so \( -x^3 \to -(-\infty) = +\infty \) (so \( y \to +\infty \)).
• As \( x \to +\infty \), \( x^3 \to +\infty \), so \( -x^3 \to -\infty \) (so \( y \to -\infty \)).

Answer:

As \( x \) approaches negative infinity, \( y \) approaches positive infinity. As \( x \) approaches positive infinity, \( y \) approaches negative infinity. (The first option)