Question

identifying key features for graphing polynomial functions
given $f(x) = x^3 - 2x^2 - x + 2$.
determine the end behavior of $f(x)$.
$f(x) \to +\infty$ as $x \to -\infty$;
$f(x) \to -\infty$ as $x \to +\infty$
$f(x) \to -\infty$ as $x \to -\infty$;
$f(x) \to +\infty$ as $x \to +\infty$
$f(x) \to +\infty$ as $x \to -\infty$;
$f(x) \to +\infty$ as $x \to -\infty$
$f(x) \to -\infty$ as $x \to -\infty$;
$f(x) \to -\infty$ as $x \to +\infty$

Explanation:

Step 1: Recall the rule for end - behavior of polynomial functions

For a polynomial function of the form \( f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\), the end - behavior is determined by the leading term \( a_nx^n\), where \( n\) is the degree of the polynomial (the highest power of \( x\)) and \( a_n\) is the leading coefficient.
The degree \( n\) and the sign of the leading coefficient \( a_n\) determine the end - behavior:

• If \( n\) is odd:
• If \( a_n>0\), as \( x\to+\infty\), \( f(x)\to+\infty\) and as \( x\to-\infty\), \( f(x)\to-\infty\).
• If \( a_n < 0\), as \( x\to+\infty\), \( f(x)\to-\infty\) and as \( x\to-\infty\), \( f(x)\to+\infty\).
• If \( n\) is even:
• If \( a_n>0\), as \( x\to+\infty\) and \( x\to-\infty\), \( f(x)\to+\infty\).
• If \( a_n < 0\), as \( x\to+\infty\) and \( x\to-\infty\), \( f(x)\to-\infty\).

Step 2: Identify the leading term of the given polynomial

The given polynomial is \( f(x)=x^3-2x^2 - x + 2\). The leading term is \( x^3\), where the degree \( n = 3\) (which is odd) and the leading coefficient \( a_n=1>0\).

Step 3: Determine the end - behavior

Since \( n = 3\) (odd) and \( a_n = 1>0\), we have:
As \( x\to-\infty\), \( f(x)\to-\infty\) (because for an odd - degree polynomial with positive leading coefficient, when \( x\) approaches negative infinity, the function value approaches negative infinity) and as \( x\to+\infty\), \( f(x)\to+\infty\) (because for an odd - degree polynomial with positive leading coefficient, when \( x\) approaches positive infinity, the function value approaches positive infinity).

Looking at the options:

• The first option: \( f(x)\to+\infty\) as \( x\to-\infty\); \( f(x)\to-\infty\) as \( x\to+\infty\) is incorrect.
• The second option: \( f(x)\to-\infty\) as \( x\to-\infty\); \( f(x)\to+\infty\) as \( x\to+\infty\) is correct.
• The third option: \( f(x)\to+\infty\) as \( x\to-\infty\); \( f(x)\to+\infty\) as \( x\to-\infty\) is incorrect (also, the description has a typo, it should be \( x\to+\infty\) for the second part).
• The fourth option: \( f(x)\to-\infty\) as \( x\to-\infty\); \( f(x)\to-\infty\) as \( x\to+\infty\) is incorrect.

Answer:

The second option (the middle - left option: \( f(x)\to-\infty \) as \( x\to-\infty \); \( f(x)\to+\infty \) as \( x\to+\infty \))