Noboa7idn
Answered

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Which statement describes how the graph of the given polynomial would change if the term [tex]2x^5[/tex] is added?

[tex]y = 8x^4 - 2x^3 + 5[/tex]

A. Both ends of the graph will approach negative infinity.
B. The ends of the graph will extend in opposite directions.
C. Both ends of the graph will approach positive infinity.
D. The ends of the graph will approach zero.

Sagot :

GinnyAnsweridn
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."
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