Answer :
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.
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.
