To determine the end behavior of the polynomial function [tex]\( f(x) = -3x^2 + 5x - 2 \)[/tex], we need to examine how the function behaves as [tex]\( x \)[/tex] approaches both negative infinity ([tex]\(-\infty\)[/tex]) and positive infinity ([tex]\(+\infty\)[/tex]).
1. Identify the leading term:
The leading term in the polynomial [tex]\( f(x) = -3x^2 + 5x - 2 \)[/tex] is the term with the highest power of [tex]\( x \)[/tex], which is [tex]\(-3x^2\)[/tex].
2. Analyze the end behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \to -\infty \)[/tex], the term [tex]\(-3x^2 \)[/tex] will dominate the behavior of [tex]\( f(x) \)[/tex].
- Since the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-3\)[/tex]), the term [tex]\(-3x^2 \)[/tex] will tend to [tex]\(-\infty \)[/tex] as [tex]\( x \)[/tex] becomes very large in magnitude (whether positive or negative).
Therefore, [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
3. Analyze the end behavior as [tex]\( x \to +\infty \)[/tex]:
Similarly, as [tex]\( x \to +\infty \)[/tex], the term [tex]\(-3x^2 \)[/tex] will also dominate the behavior of [tex]\( f(x) \)[/tex].
- Again, because the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-3\)[/tex]), the term [tex]\(-3x^2 \)[/tex] will tend to [tex]\(-\infty \)[/tex] as [tex]\( x \)[/tex] becomes very large.
Therefore, [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to +\infty \)[/tex].
4. Conclusion:
Based on this analysis, we can fill in the blanks for the end behavior of the polynomial function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) \to -\infty \quad \text{as} \quad x \to -\infty
\][/tex]
[tex]\[
f(x) \to -\infty \quad \text{as} \quad x \to +\infty
\][/tex]
Hence, the end behavior of [tex]\( f(x) = -3x^2 + 5x - 2 \)[/tex] is summarized as follows:
[tex]\[
f(x) \to -\infty \quad \text{as} \quad x \to -\infty
\][/tex]
[tex]\[
f(x) \to -\infty \quad \text{as} \quad x \to +\infty
\][/tex]
