Polynomial graphs

When $x$ is huge, $x^{3}$ dwarfs everything else in $x^{3} - 5 x + 2$ . So the ends of the graph follow the leading term alone.

y = \frac{1}{4} (x + 3) x (x - 3)

has zeros at

- 3

, 0 and 3. Odd degree: its ends go opposite ways.

Work through it

Degree 4 is even, so the ends match.Same direction

The sign is negative.Both down

A product is zero if a factor is. $x = 0, 2, - 5$

Reading $(x + 5)$ as a zero at 5.

$x + 5 = 0$ when $x = - 5$ .

The short version
Far out, only the leading term matters. Each factor

(x - r)

gives a zero at

r

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