Question

determine the end behavior of the graph of the given polynomial function.
y = 2x^7 - 5x^2 + 7x + 6
choose the correct answer below
\bigcirc up and down
\bigcirc up and up
\bigcirc down and up
\bigcirc down and down

Explanation:

Step1: Identify degree and leading coefficient

The polynomial is \( y = 2x^7 - 5x^2 + 7x + 6 \). The degree (highest power of \( x \)) is 7 (odd), and the leading coefficient (coefficient of \( x^7 \)) is 2 (positive).

Step2: Apply end - behavior rules

For a polynomial with odd degree:

• If the leading coefficient is positive, as \( x

ightarrow+\infty \), \( y
ightarrow+\infty \) (graph goes up), and as \( x
ightarrow-\infty \), \( y
ightarrow-\infty \) (graph goes down). So the end - behavior is down (as \( x
ightarrow-\infty \)) and up (as \( x
ightarrow+\infty \)).

Answer:

down and up