Let's analyze the given polynomial and the changes resulting from the addition of a new term:
The initial polynomial provided is:
[tex]\[ y = 8x^4 - 2x^3 + 5 \][/tex]
Now, we are adding a new term to this polynomial:
[tex]\[ y = 8x^4 - 2x^3 + 5 + 2x^5 \][/tex]
To determine how the graph changes, especially at the ends, we must consider which term dominates as [tex]\( x \)[/tex] grows very large in magnitude (either positively or negatively).
1. Original Polynomial Analysis:
The highest degree term in the original polynomial is [tex]\( 8x^4 \)[/tex]:
- For large positive [tex]\( x \)[/tex], [tex]\( 8x^4 \)[/tex] grows very large and positive.
- For large negative [tex]\( x \)[/tex], [tex]\( 8x^4 \)[/tex] also grows very large and positive since raising a negative number to an even power results in a positive value.
Therefore, the behavior at both ends of the original graph (as [tex]\( x \rightarrow \pm \infty \)[/tex]) would see both ends of the graph rise towards positive infinity.
2. New Polynomial Analysis After Addition:
With the new term [tex]\( 2x^5 \)[/tex] introduced, the highest degree term now changes:
- For large positive [tex]\( x \)[/tex], [tex]\( 2x^5 \)[/tex] grows very large and positive.
- For large negative [tex]\( x \)[/tex], [tex]\( 2x^5 \)[/tex] grows very large but negative (since raising a negative number to an odd power results in a negative value).
Therefore, the behavior at both ends of the graph, if we add [tex]\( 2x^5 \)[/tex], would change significantly. The new highest degree term, [tex]\( 2x^5 \)[/tex], dominates the behavior of the polynomial for both large positive and large negative [tex]\( x \)[/tex]:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex]
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex]
Thus, the ends of the graph will extend in opposite directions. This means that as [tex]\( x \)[/tex] goes to positive infinity, [tex]\( y \)[/tex] will go to positive infinity, and as [tex]\( x \)[/tex] goes to negative infinity, [tex]\( y \)[/tex] will go to negative infinity.
Therefore, the correct statement is:
"The ends of the graph will extend in opposite directions."
