Analyzing Graphs of Polynomial Functions

Q: How many turning points can a polynomial have?

At most one less than its degree: a degree-$n$ polynomial has at most $n - 1$ turning points.

Q: How many real zeros are possible?

At most $n$ for degree $n$. Some may be repeated or complex, so the real count can be lower.

The degree of a polynomial caps how wiggly its graph can be. A degree- $n$ polynomial has at most $n - 1$ turning points and at most $n$ real zeros.

These bounds let you rule out impossible graphs and estimate a polynomial's degree from its shape. They are limits, not exact counts.

Turning points

A turning point is where the graph changes from rising to falling or vice versa — a local peak or valley. A degree- $n$ polynomial has at most $n - 1$ of them. The curve below turns exactly once, at its single peak.

So a degree- $6$ polynomial can have at most $5$ turning points. It might have fewer, but never more.

Zeros and multiplicity

A degree- $n$ polynomial has at most $n$ real zeros ( $x$ -intercepts). Repeated factors create zeros with multiplicity, where the graph touches or flattens at the axis.

At an odd-multiplicity zero the graph crosses; at an even-multiplicity zero it touches and turns back. Counting with multiplicity, the total equals the degree.

Worked examples

Example 1: maximum turning points

A polynomial has degree $6$ . What is the greatest number of turning points its graph can have?

Turning points at most degree minus one

6 - 1

Simplify

5

Answer: $5$

Example 2: maximum zeros

At most how many real zeros can a degree-4 polynomial have?

At most the degree

4

Answer: $4$

Example 3: cross or touch at each zero

The graph of $p(x) = (x - 3)^2(x + 1)$ has zeros at $x = 3$ and $x = -1$ . Does it cross or touch the x-axis at each one?

The factor x minus 3 repeats twice, an even multiplicity

x = 3 \to \text{touches and turns back}

The factor x plus 1 appears once, an odd multiplicity

x = -1 \to \text{crosses}

Answer: Touches at $x = 3$ , crosses at $x = -1$

Try one yourself

Common questions

How many turning points can a polynomial have?

At most one less than its degree: a degree- $n$ polynomial has at most $n - 1$ turning points.

How many real zeros are possible?

At most $n$ for degree $n$ . Some may be repeated or complex, so the real count can be lower.

What is multiplicity?

How many times a factor repeats. Even multiplicity makes the graph touch the axis and turn; odd multiplicity makes it cross.

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Analyzing Graphs of Polynomial Functions

Turning points

Zeros and multiplicity

Worked examples

Example 1: maximum turning points

Example 2: maximum zeros

Example 3: cross or touch at each zero

Try one yourself

Common questions

Keep learning

Graphing Polynomial Functions & End Behavior

Roots & Zeros

Solving Polynomial Equations Algebraically