Question

end behavior
$x\to -\infty,y\to -\infty;x\to \infty,y\to \infty$
$x\to -\infty,y\to -\infty;x\to \infty,y\to -\infty$
$x\to -\infty,y\to \infty;x\to \infty,y\to \infty$
$x\to -\infty,y\to \infty;x\to \infty,y\to -\infty$

Explanation:

Step1: Analyze the leading coefficient and degree

For a polynomial function \( y = a_nx^n+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\).
If the degree \(n\) is odd:

• When \(a_n>0\), as \(x\to-\infty\), \(y\to-\infty\) and as \(x\to\infty\), \(y\to\infty\)
• When \(a_n < 0\), as \(x\to-\infty\), \(y\to\infty\) and as \(x\to\infty\), \(y\to-\infty\)

If the degree \(n\) is even:

• When \(a_n>0\), as \(x\to-\infty\) and \(x\to\infty\), \(y\to\infty\)
• When \(a_n < 0\), as \(x\to-\infty\) and \(x\to\infty\), \(y\to-\infty\)

The graph shown has an odd - degree polynomial (since the ends of the graph go in opposite directions). And from the graph, when \(x\to-\infty\), \(y\to-\infty\) and when \(x\to\infty\), \(y\to\infty\), which means the leading coefficient \(a_n>0\)

Answer:

\(x\to-\infty,y\to-\infty;x\to\infty,y\to\infty\) (the red box option)