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Describe the end behavior of the following function:

[tex]\( F(x) = 2x^4 + x^3 \)[/tex]

A. The graph of the function starts high and ends high.

B. The graph of the function starts high and ends low.

C. The graph of the function starts low and ends high.

D. The graph of the function starts low and ends low.

Sagot :

GinnyAnswer
To determine the end behavior of the function [tex]\( F(x) = 2x^4 + x^3 \)[/tex], let's analyze the dominant term and understand how the function behaves as [tex]\( x \)[/tex] approaches positive and negative infinity.

1. The leading term in the function [tex]\( F(x) = 2x^4 + x^3 \)[/tex] is [tex]\( 2x^4 \)[/tex]. This is because [tex]\( x^4 \)[/tex] grows much faster than [tex]\( x^3 \)[/tex] as [tex]\( x \)[/tex] becomes very large in magnitude.
2. Consider the behavior as [tex]\( x \to \infty \)[/tex] (i.e., as [tex]\( x \)[/tex] becomes very large and positive):
- The term [tex]\( 2x^4 \)[/tex] dominates, and since [tex]\( x^4 \)[/tex] is positive when [tex]\( x \)[/tex] is positive, [tex]\( 2x^4 \)[/tex] is also positive.
- Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( 2x^4 + x^3 \)[/tex] will be dominated by the positive [tex]\( 2x^4 \)[/tex], and the function [tex]\( F(x) \to \infty \)[/tex].
3. Next, consider the behavior as [tex]\( x \to -\infty \)[/tex] (i.e., as [tex]\( x \)[/tex] becomes very large and negative):
- Again, the term [tex]\( 2x^4 \)[/tex] dominates because of its higher power compared to [tex]\( x^3 \)[/tex].
- Since [tex]\( x^4 \)[/tex] is always positive, regardless of whether [tex]\( x \)[/tex] is negative or positive, [tex]\( 2x^4 \)[/tex] remains positive even when [tex]\( x \)[/tex] is very large negative.
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( 2x^4 + x^3 \)[/tex] is dominated by the positive [tex]\( 2x^4 \)[/tex], and the function [tex]\( F(x) \to \infty \)[/tex].

From this analysis, we can summarize the end behavior of the function [tex]\( F(x) = 2x^4 + x^3 \)[/tex] as follows:
- As [tex]\( x \to \infty \)[/tex], [tex]\( F(x) \to \infty \)[/tex] (the function goes towards positive infinity).
- As [tex]\( x \to -\infty \)[/tex], [tex]\( F(x) \to \infty \)[/tex] (the function goes towards positive infinity).

This tells us that the graph of the function starts low when [tex]\( x \)[/tex] is very negative (due to polynomial changes in sign effects) and ends high when [tex]\( x \)[/tex] is positive since both cases drive the function towards positive infinity.

Therefore, the correct answer is:
C. The graph of the function starts low and ends high.
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