We want to estimate the coordinates of every turning point. A turning point is either a local maximum or minimum of the function. Let's look at the given graph to see where the turning points are.

As we can see above, the graph has three turning points at $(1, 0),$ $(2, - 2),$ and $(3, 0) .$ The points $(1, 0)$ and $(3, 0)$ correspond to local maximums and the point $(2, - 2)$ corresponds to local minimum.

Next, we will estimate the zeros of the function. A zero is the $x -$ coordinate of the point at which the graph intercepts the $x -$ axis. Let's consider the given graph.

The zeros occur at $x = 1$ and $x = 3 .$ Now, to find the least possible degree of the function, we will examine its end behavior.

We can see above that the end behavior is down and down. With this, we can make a conclusion about its degree.

The degree must be an even number .

Also, the graph has

3

turning points. Therefore, the degree of the function must be greater than or equal to

3 .

With this in mind, we conclude that the least possible degree of the polynomial is

4 .

Turning Points	$(1, 0.5),$ $(2, - 2),$ and $(3, 0)$
Local Maximums	$(1, 0)$ and $(3, 0)$
Local Minimum	$(2, - 2)$
Real Zeros	$x = 1$ and $x = 3$
Least Possible Degree	$4$

Exercise