Question

which statement describes how the graph of the given polynomial would change if the term $-3x^6$ is added?
$y = 2x^6 + 9x^5 - 7x^3 - 1$
both ends of the graph will approach positive infinity.
the ends of the graph will approach zero.
both ends of the graph will approach negative infinity.
the ends of the graph will extend in opposite directions.

Explanation:

Step1: Find original leading term

Original polynomial: $y=2x^6 + 9x^5 -7x^3 -1$, leading term is $2x^6$.

Step2: Add new term to get new leading term

Add $-3x^6$: New leading term = $2x^6 + (-3x^6) = -x^6$.

Step3: Analyze end behavior of new polynomial

Degree of $-x^6$ is even, leading coefficient is negative. For even degree, both ends behave same; negative coefficient means both ends approach $-\infty$.

Answer:

C. Both ends of the graph will approach negative infinity.