Graphing polynomial functions
Graphing Polynomial Functions
Graphing Polynomial Functions is a topic in Polynomial Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7c, which requires students to graph polynomial functions, identifying zeros, multiplicity, and end behavior.
Graphing a polynomial uses its end behavior (leading term) and its zeros, which cross (odd multiplicity) or touch (even multiplicity) the axis.
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 \(n\).
Degree and behavior:
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.
Even degree, negative lead: both ends go down.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to -\infty\) |
Set each factor to zero.
| \(x+2=0\) | \(\Rightarrow\) | \(x=-2\ \text{(single)}\) |
| \((x-1)^2=0\) | \(\Rightarrow\) | \(x=1\ \text{(double)}\) |
Odd multiplicity crosses; even touches.
| \(x=-2\) | \(:\) | \(\text{cross (mult. 1)}\) |
| \(x=1\) | \(:\) | \(\text{touch (mult. 2)}\) |
A degree-\(n\) polynomial has at most \(n-1\) turning points.
| \(n-1\) | \(=\) | \(5-1=4\) |
Common pitfalls
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.