Graphing polynomial functions

The zeros of a polynomial are the $x$-values where its graph touches or crosses the $x$-axis.
How the graph behaves at a zero depends on whether that zero is single, double, or higher.

At a single zero, the graph passes straight through the axis.
It comes from one factor like $(x + 2)$, and the curve is on one side of the axis before the zero and the other side after it.
This is the ordinary "crossing" behaviour.

At a double zero, the graph touches the axis and turns back without crossing.
It comes from a squared factor like $(x - 1)^2$, so the curve dips down to the axis and bounces back up (or peaks up and comes back down).
Changing direction right at the zero is the signature of a double zero.

So to read a polynomial graph, note where it meets the axis and how it behaves there.
A clean crossing signals a single zero; a touch-and-turn signals a double zero.
This lets you match a graph to its factored form, since each squared factor produces a bounce.

The graph below is a parabola.

Which polynomial does it represent?
(A) $(x + 1)(x - 2)$
(B) $(x - 1)(x + 2)$
The graph crosses the $x$-axis at $x = -1$ and $x = 2$, so its factors are $(x + 1)$ and $(x - 2)$.
Choice (A) has exactly these factors, so (A) is correct.

The graph below is a cubic.

Which polynomial does it represent?
(A) $(x + 3)(x - 2)^2$
(B) $(x + 3)(x - 2)$
The graph crosses at $x = -3$ but only touches and turns at $x = 2$, so $x = 2$ is a double zero that needs a squared factor.
Choice (A) has $(x - 2)^2$, so (A) is correct.

The graph below is a downward parabola.

Which polynomial does it represent?
(A) $-x(x - 3)$
(B) $x(x - 3)$
The graph crosses at $x = 0$ and $x = 3$ and opens downward, so the leading coefficient must be negative.
Choice (A) carries the negative sign, so (A) is correct.