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Sagot :
When analyzing the end behavior of the polynomial function [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex], let's focus on the term with the highest power, which is [tex]\( -3x^3 \)[/tex].
For polynomial functions, the highest degree term usually determines the end behavior as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( +\infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]).
### Step-by-Step Solution:
1. Identify the Highest Degree Term:
- The function [tex]\( f(x) \)[/tex] is [tex]\( -3x^3 - x^2 + 1 \)[/tex].
- The highest degree term is [tex]\( -3x^3 \)[/tex].
2. Analyze the Leading Term:
- The term [tex]\( -3x^3 \)[/tex] has a degree of 3 (cubic term).
- The coefficient is [tex]\( -3 \)[/tex], which is negative.
3. Determine the End Behavior:
- As [tex]\( x \to +\infty \)[/tex]:
- The term [tex]\( -3x^3 \)[/tex] will dominate.
- Since the coefficient is negative, [tex]\( -3x^3 \)[/tex] will tend towards [tex]\( -\infty \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] will also tend towards [tex]\( -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex]:
- Again, the term [tex]\( -3x^3 \)[/tex] will dominate.
- Because [tex]\( x^3 \)[/tex] with a negative [tex]\( x \)[/tex] (i.e., [tex]\( x^3 \)[/tex] where [tex]\( x \)[/tex] is negative) results in a negative value, and multiplied by [tex]\( -3 \)[/tex] (a negative coefficient), the result will be positive.
- Thus, [tex]\( -3x^3 \)[/tex] will tend towards [tex]\( +\infty \)[/tex], and so will [tex]\( f(x) \)[/tex].
### Conclusion:
The end behavior of the function [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex] is:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
Therefore, any other cubic polynomial function with a negative leading coefficient will exhibit the same end behavior. For example, functions like [tex]\( g(x) = -2x^3 \)[/tex] or [tex]\( h(x) = -5x^3 + 2x \)[/tex] will share the same end behavior as [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex].
In summary, the end behavior of [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex] shows that as [tex]\( x \rightarrow +\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow +\infty \)[/tex]. This end behavior corresponds to any cubic function with a negative leading term.
For polynomial functions, the highest degree term usually determines the end behavior as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( +\infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]).
### Step-by-Step Solution:
1. Identify the Highest Degree Term:
- The function [tex]\( f(x) \)[/tex] is [tex]\( -3x^3 - x^2 + 1 \)[/tex].
- The highest degree term is [tex]\( -3x^3 \)[/tex].
2. Analyze the Leading Term:
- The term [tex]\( -3x^3 \)[/tex] has a degree of 3 (cubic term).
- The coefficient is [tex]\( -3 \)[/tex], which is negative.
3. Determine the End Behavior:
- As [tex]\( x \to +\infty \)[/tex]:
- The term [tex]\( -3x^3 \)[/tex] will dominate.
- Since the coefficient is negative, [tex]\( -3x^3 \)[/tex] will tend towards [tex]\( -\infty \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] will also tend towards [tex]\( -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex]:
- Again, the term [tex]\( -3x^3 \)[/tex] will dominate.
- Because [tex]\( x^3 \)[/tex] with a negative [tex]\( x \)[/tex] (i.e., [tex]\( x^3 \)[/tex] where [tex]\( x \)[/tex] is negative) results in a negative value, and multiplied by [tex]\( -3 \)[/tex] (a negative coefficient), the result will be positive.
- Thus, [tex]\( -3x^3 \)[/tex] will tend towards [tex]\( +\infty \)[/tex], and so will [tex]\( f(x) \)[/tex].
### Conclusion:
The end behavior of the function [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex] is:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
Therefore, any other cubic polynomial function with a negative leading coefficient will exhibit the same end behavior. For example, functions like [tex]\( g(x) = -2x^3 \)[/tex] or [tex]\( h(x) = -5x^3 + 2x \)[/tex] will share the same end behavior as [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex].
In summary, the end behavior of [tex]\( f(x) = -3x^3 - x^2 + 1 \)[/tex] shows that as [tex]\( x \rightarrow +\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow +\infty \)[/tex]. This end behavior corresponds to any cubic function with a negative leading term.
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