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How to determine polynomial end behavior

The highest power controls the ends

At large positive or negative x, the leading term dominates a polynomial. An even degree gives the same direction at both ends; an odd degree gives opposite directions. The sign of the leading coefficient sets the right-hand direction.

Problem

Describe the left and right end behavior of each graph: (a) $3 x^{6} - 8 x^{2} - 4 x - 5$ ; (b) $2 x^{3} + 4 x^{2} + 7 x - 8$ ; (c) $x (x - 3) (5 x + 2)$ .

Answer

(a) Rises left and right. (b) Falls left, rises right. (c) Falls left, rises right.

Lower-degree terms do not change the directions far from the origin.

Step-by-step solution

1. Examine the first leading term

In (a), $3 x^{6}$ has even degree and positive coefficient. It tends to positive values on both the far left and far right, so both ends rise.

2. Examine the cubic

In (b), the leading term is $2 x^{3}$ . A positive odd-degree term is negative far to the left and positive far to the right: the graph falls, then rises.

3. Use the factored form

For (c), multiply only the leading pieces: $x$ times $x$ times $5 x$ gives $5 x^{3}$ . This is also positive and odd-degree, so it falls left and rises right.

More examples: leading terms in factored expressions

These additional problems were solved on 6 October 2026.

Example 4. Describe the end behavior of

f (x) = - 4 (x + \frac{1}{4})^{2} (x + 3)^{3} .

Multiply just the highest-power pieces:

- 4 \cdot x^{2} \cdot x^{3} = - 4 x^{5} .

As $x \to - \infty$ , the odd power $x^{5}$ is negative, so $- 4 x^{5} \to + \infty$ . As $x \to + \infty$ , $- 4 x^{5} \to - \infty$ . The graph rises on the left and falls on the right. Its far-end behavior matches $y = - 4 x^{5}$ .

Example 5. Describe the end behavior of

g (x) = \frac{1}{3} (3 x^{2} + 8)^{2} (x^{2} + 5) .

Remember to square the coefficient as well as the power:

\frac{1}{3} (3 x^{2})^{2} x^{2} = \frac{1}{3} \cdot 9 x^{4} \cdot x^{2} = 3 x^{6} .

The degree is even and the leading coefficient is positive, so $g (x) \to + \infty$ as $x \to - \infty$ and as $x \to + \infty$ . Both ends rise. The matching power function is $y = 3 x^{6}$ .

A common factored-form mistake

In $(3 x^{2} + 8)^{2}$ , the leading term is $9 x^{4}$ , not $3 x^{4}$ . Ignoring the squared coefficient gives the wrong comparison power function even if the degree happens to be correct.

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