Let's analyze the given polynomial and understand how the addition of a term impacts its behavior.
The initial polynomial is:
[tex]\[ y = 2x^5 + 9x^5 - 7x^3 - 1 \][/tex]
Firstly, we combine like terms:
[tex]\[ y = (2x^5 + 9x^5) - 7x^3 - 1 \][/tex]
[tex]\[ y = 11x^5 - 7x^3 - 1 \][/tex]
Now, we add the term [tex]\(-3x^6\)[/tex] to the polynomial:
[tex]\[ y = -3x^6 + 11x^5 - 7x^3 - 1 \][/tex]
This polynomial is now:
[tex]\[ y = -3x^6 + 11x^5 - 7x^3 - 1 \][/tex]
To understand the impact this term has on the behavior of the graph, we must consider the highest degree term, which determines the end behavior of the polynomial. The highest degree term here is [tex]\(-3x^6\)[/tex].
Next, we consider the following characteristics:
- Degree of the polynomial: 6 (even degree)
- Leading coefficient: [tex]\(-3\)[/tex] (negative)
For polynomials with an even degree and a negative leading coefficient, the behavior of the graph at the ends will be such that:
- As [tex]\( x \to +\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]
Hence, both ends of the graph will approach negative infinity.
Based on this analysis, we conclude that the correct statement describing the change in the graph of the given polynomial upon adding the term [tex]\(-3x^6\)[/tex] is:
[tex]\[ \text{Both ends of the graph will approach negative infinity.} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\text{Both ends of the graph will approach negative infinity.}} \][/tex]
