End behavior of a fifth degree polynomial

For large $|x|$ the leading term $-4x^5$ outweighs everything else, so it alone decides the end behavior. Its coefficient is $-4$, which is negative, and its degree is 5, which is odd.

An odd power keeps the sign of $x$: $x^5$ is huge and positive as $x\to \infty$ and huge and negative as $x\to -\infty$. Multiplying by $-4$ flips both. So as $x\to -\infty$, $-4x^5\to \infty$, and as $x\to \infty$, $-4x^5\to -\infty$.

That is choice B: negative leading term, odd degree, the graph rises on the left and falls on the right. A quick check: $p(-10)=-4(-100000)+300+1=400301$, large and positive, and $p(10)=-400000+301=-399699$, large and negative.

Each choice

• A. The $3x^2$ term is positive with an even power, and reading the end behavior off it gives this choice. That term is not the leading term; the degree is set by $-4x^5$, so both the sign and the parity are wrong here.
• B (correct). The leading term $-4x^5$ is negative with odd degree, so the outputs go to $\infty$ on the left and to $-\infty$ on the right.
• C. The odd degree is read correctly, but the sign of $-4$ was missed. With a positive leading coefficient the limits described here would be true; the negative coefficient flips them.
• D. The sign and the degree are both read correctly, but the limits do not follow from them. This is the end behavior of a positive odd-degree polynomial like $x^5$. Multiplying by a negative number reverses both directions.