To determine the end behavior of the polynomial function [tex]\( f(x) = -x^3 + x^2 - 4x + 2 \)[/tex], we need to look at the term with the highest degree, which is the term that dominates the behavior of the polynomial as [tex]\( x \)[/tex] approaches either positive or negative infinity.
The given polynomial function is:
[tex]\[ f(x) = -x^3 + x^2 - 4x + 2 \][/tex]
This is a cubic polynomial due to the highest power of [tex]\( x \)[/tex] being 3.
### Step-by-Step Solution:
1. Identify the Leading Term:
The leading term is the term with the highest power of [tex]\( x \)[/tex], which is [tex]\(-x^3\)[/tex].
2. Determine the Leading Coefficient:
The coefficient of the leading term [tex]\(-x^3\)[/tex] is [tex]\(-1\)[/tex].
3. Analyze the Leading Coefficient:
For cubic functions (polynomials of degree 3), the sign of the leading coefficient determines the end behavior.
- If the leading coefficient is positive (e.g., [tex]\( +x^3 \)[/tex]), as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex]. Also, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- If the leading coefficient is negative (e.g., [tex]\( -x^3 \)[/tex]), as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex]. Also, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
4. Apply the Leading Coefficient to Determine End Behavior:
Since our leading coefficient is [tex]\(-1\)[/tex] (negative):
- As [tex]\( x \to +\infty \)[/tex] (as [tex]\( x \)[/tex] goes to positive infinity), [tex]\( -x^3 \)[/tex] will dominate and [tex]\( -x^3 \to -\infty \)[/tex]. Therefore, [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] goes to negative infinity), [tex]\( -(-x)^3 = x^3 \)[/tex]. Therefore, [tex]\( -x^3 \to \infty \)[/tex]. Hence, [tex]\( f(x) \to +\infty \)[/tex].
### Conclusion:
As we have determined:
- The left end (as [tex]\( x \to -\infty \)[/tex]) goes up.
- The right end (as [tex]\( x \to +\infty \)[/tex]) goes down.
Therefore, the correct answer is:
B. The left end goes up and the right end goes down.
