Algebra 2
Polynomial functions
Graphing polynomial functions
20 practice questions
0 video lessons
Theory + worked examples
Theory
A polynomial's graph is shaped by three features:
- End behavior from the leading term (degree and sign).
- Zeros from the factors; the multiplicity decides cross vs touch.
- Turning points: at most \(n-1\) for degree \(n\).
Odd multiplicity crosses the axis; even multiplicity touches and turns around.
Single zero crosses; double zero touches.
What to read for the graph.
Degree and behavior:
\[\text{degree } n\Rightarrow \le n-1 \text{ turning points}\]
Even degree: ends match; odd degree: ends differ.
How to graph a polynomial
- Find the end behavior from the leading term.
- Find the zeros and their multiplicities.
- Mark cross/touch at each zero.
- Sketch a smooth curve with at most \(n-1\) turns.
Example 1 — End behavior
Describe the end behavior of \(f(x)=-2x^4+\dots\).
Solution
Even degree, negative lead: both ends go down.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to -\infty\) |
Example 2 — Zeros from factors
Find the zeros of \(f(x)=(x+2)(x-1)^2\).
Solution
Set each factor to zero.
| \(x+2=0\) | \(\Rightarrow\) | \(x=-2\ \text{(single)}\) |
| \((x-1)^2=0\) | \(\Rightarrow\) | \(x=1\ \text{(double)}\) |
Example 3 — Cross or touch
At each zero of \(f(x)=(x+2)(x-1)^2\), does the graph cross or touch?
Solution
Odd multiplicity crosses; even touches.
| \(x=-2\) | \(:\) | \(\text{cross (mult. 1)}\) |
| \(x=1\) | \(:\) | \(\text{touch (mult. 2)}\) |
Example 4 — Turning points
At most how many turning points does a degree-5 polynomial have?
Solution
A degree-\(n\) polynomial has at most \(n-1\) turning points.
| \(n-1\) | \(=\) | \(5-1=4\) |
Common pitfalls
Even multiplicity touches; odd crosses — don't mix them up.
End behavior is the leading term only, not the constant.
At most \(n-1\) turning points, not \(n\).
Frequently asked questions
What sets a polynomial's end behavior?
The leading term — its degree and sign.
What does multiplicity tell you?
Odd multiplicity means the graph crosses the axis; even means it touches.
How many turning points can a degree-\(n\) polynomial have?
At most \(n-1\).
How do you find the zeros?
Set each factor equal to zero.
More in Polynomial functions