A polynomial graph shows how a polynomial function behaves for different input values.
Important features include roots, y-intercept, turning points, intervals of increase or decrease,
multiplicity of roots, and end behavior.
1. Polynomial function
A polynomial function has the form:
\[
\begin{aligned}
P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.
\end{aligned}
\]
The highest power \(n\) is the degree, and \(a_n\) is the leading coefficient.
2. Roots and x-intercepts
A root is a value that makes the polynomial equal zero:
\[
\begin{aligned}
P(r)=0.
\end{aligned}
\]
On the graph, real roots appear as x-intercepts. If \(r\) is a root, then \(x-r\) is a factor:
\[
\begin{aligned}
P(x)=(x-r)Q(x).
\end{aligned}
\]
3. Multiplicity of roots
If a factor appears more than once, the root has multiplicity greater than 1.
For example:
\[
\begin{aligned}
P(x)=(x-2)^3(x+1)
\end{aligned}
\]
has root \(x=2\) with multiplicity \(3\), and root \(x=-1\) with multiplicity \(1\).
Multiplicity affects the graph:
- Odd multiplicity: the graph usually crosses the x-axis.
- Even multiplicity: the graph usually touches and turns around at the x-axis.
- Higher multiplicity: the graph becomes flatter near the root.
4. y-intercept
The y-intercept is found by setting \(x=0\):
\[
\begin{aligned}
P(0)=a_0.
\end{aligned}
\]
Therefore, the y-intercept is:
\[
\begin{aligned}
(0,a_0).
\end{aligned}
\]
5. Derivative preview
The derivative gives the slope of the graph:
\[
\begin{aligned}
P'(x)=\frac{d}{dx}P(x).
\end{aligned}
\]
If:
\[
\begin{aligned}
P(x)=x^3-6x^2+11x-6,
\end{aligned}
\]
then:
\[
\begin{aligned}
P'(x)=3x^2-12x+11.
\end{aligned}
\]
6. Turning points
Turning points usually occur where:
\[
\begin{aligned}
P'(x)=0.
\end{aligned}
\]
A turning point may be a local maximum or a local minimum.
A polynomial of degree \(n\) can have at most \(n-1\) turning points.
7. Increasing and decreasing intervals
The sign of the derivative tells whether the graph is increasing or decreasing:
\[
\begin{aligned}
P'(x)>0 &\quad \text{means increasing},\\
P'(x)<0 &\quad \text{means decreasing}.
\end{aligned}
\]
Critical points split the number line into intervals. Testing the sign of \(P'(x)\)
on each interval gives increasing and decreasing behavior.
8. End behavior
End behavior is controlled by the degree and leading coefficient.
9. Worked example
Analyze:
\[
\begin{aligned}
P(x)=x^3-6x^2+11x-6.
\end{aligned}
\]
Factor the polynomial:
\[
\begin{aligned}
x^3-6x^2+11x-6
&=
(x-1)(x-2)(x-3).
\end{aligned}
\]
Therefore, the roots are:
\[
\begin{aligned}
x=1,\quad x=2,\quad x=3.
\end{aligned}
\]
The derivative is:
\[
\begin{aligned}
P'(x)=3x^2-12x+11.
\end{aligned}
\]
Solve \(P'(x)=0\) to find turning-point candidates:
\[
\begin{aligned}
3x^2-12x+11=0.
\end{aligned}
\]
The polynomial has degree \(3\) and positive leading coefficient, so the left end falls and the right end rises.
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Confusing roots with turning points.
- Forgetting that repeated roots can touch the x-axis instead of crossing it.
- Using only the constant term to describe the whole graph.
- Ignoring the leading coefficient when describing end behavior.
- Assuming every critical point is automatically a local maximum or minimum.
- Forgetting that a degree \(n\) polynomial has at most \(n-1\) turning points.
Key idea: roots explain x-intercepts, derivatives explain turning points, and degree plus leading coefficient explains end behavior.
