Graphing Polynomial Functions: Zeros, Multiplicity and End Behavior

Zeros come straight from the factors

$$f(x) = (x + 2)(x - 1)(x - 3)$$

A product is zero when any factor is zero, so the zeros are $x = -2$, $x = 1$ and $x = 3$. Those are the $x$-intercepts.

• f(x)

The factor theorem guarantees this works both ways: every zero $a$ comes with a factor $x - a$, and every factor $x - a$ gives a zero $a$.

The $y$-intercept comes from setting $x = 0$:

$$f(0) = (2)(-1)(-3) = 6$$

Crossing or touching: multiplicity

Now take a cubic with a repeated factor:

$$g(x) = (x + 1)^2(x - 2)$$

The factor $x + 1$ appears twice, so the zero at $-1$ has multiplicity $2$.

• g(x)

At $x = -1$ the graph touches the axis and turns back. At $x = 2$ it crosses.

Multiplicity	At that zero, the graph
odd — $1$, $3$, …	crosses the $x$-axis
even — $2$, $4$, …	touches and turns back

The reason is the sign. Near $x = -1$, the factor $(x + 1)^2$ is a square, so it is never negative — it is positive on both sides of $-1$. The sign of $g$ is the same slightly left and slightly right of the zero, so the graph cannot pass through the axis there. An odd power like $(x - 2)$ changes sign at its zero, so the graph does.

End behavior: the leading term decides

Far from the origin, the highest-power term is so much larger than the rest that it decides everything. For $f(x) = x^3 - 2x^2 - 5x + 6$ at $x = 100$, the $x^3$ term is $1{,}000{,}000$ while the others together are about $-20{,}000$.

So the ends of the graph follow the leading term alone:

Leading term	Left end	Right end
even degree, positive coefficient	up	up
even degree, negative coefficient	down	down
odd degree, positive coefficient	down	up
odd degree, negative coefficient	up	down

A quadratic with $a > 0$ opens upward — both ends up — which is the even-degree, positive row. A cubic with a positive leading coefficient falls on the left and rises on the right, like both graphs above.

Even degree: both ends the same way. Odd degree: the ends go opposite ways. The sign of the leading coefficient decides which way.

Why the ends behave that way

For large $x$, a positive number to any power is positive, so every leading term with a positive coefficient sends the right end up.

On the left, $x$ is negative. An even power of a negative number is positive, so an even-degree graph goes up on the left too. An odd power stays negative, so an odd-degree graph goes down on the left.

That is the whole table, derived: the right end follows the coefficient’s sign, and the left end agrees with the right for even degree and disagrees for odd.

Sketching from the factors

Sketch $h(x) = -(x - 1)^2(x + 3)$.

Fact	Found from	Result
zeros	the factors	$1$ and $-3$
behavior at each zero	multiplicity	touches at $1$, crosses at $-3$
end behavior	leading term $-x^3$	up on the left, down on the right
$y$-intercept	$h(0)$	$-(1)(3) = -3$

Start at the upper left, come down and cross at $-3$, pass through $(0, -3)$, rise to touch the axis at $1$, then turn back down toward the lower right. Four facts, and the shape is fixed without plotting a single extra point.

A polynomial of degree $n$ has at most $n - 1$ turning points. A cubic has at most $2$, which is a check on any sketch: a cubic drawn with three turns is wrong.

Worked examples

Common mistakes

Practice problems