Graphing Polynomial Functions & End Behavior

The two tails of a polynomial graph — its end behavior — are decided entirely by the degree (even or odd) and the sign of the leading coefficient.

Knowing those two facts lets you predict which way each end points before plotting a single interior point. It is the fastest sketch you can make.

The four cases

Even degree: both tails go the same direction — up if the leading coefficient is positive, down if negative. Think of a parabola or a W.

Odd degree: the tails go opposite directions. Positive leading coefficient falls on the left and rises on the right; negative does the reverse.

The simplest even-degree graph is the parabola $y = x^2$ below: both tails rise together because the leading coefficient is positive.

For $f(x) = -3x^5 + x^2 - 4$ : degree $5$ is odd and the leading coefficient $-3$ is negative, so the graph rises on the left and falls on the right.

Only the leading term matters for end behavior; the smaller-degree terms bend the middle but not the far tails.

Describe the end behavior of $f(x) = -3x^5 + x^2 - 4$ .

Odd degree, negative leading coefficient

\text{tails go opposite ways}

Negative flips the standard odd shape

\text{up-left, down-right}

Answer: Rises on the left, falls on the right

Which way do the tails of $f(x) = 2x^4$ point?

Even degree, positive lead

\text{both tails same direction, up}

Answer: Both tails rise

Describe the end behavior of $f(x) = -x^4 + 6x^2$ .

Even degree, negative leading coefficient

\text{both tails same direction}

The negative sign points that direction down

\text{down-left, down-right}

Answer: Both tails fall

What determines end behavior?

Only the degree (even or odd) and the sign of the leading coefficient. The other terms do not affect the far tails.

How do even and odd differ?

Even-degree graphs have both tails pointing the same way; odd-degree graphs have tails pointing opposite ways.

Does the leading coefficient's size matter?

For end-behavior direction, only its sign matters. Its size affects steepness, not which way the tails point.

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