# Summary: Graphs of Polynomial Functions

## Key Concepts

- Polynomial functions of degree 2 or more are smooth, continuous functions.
- To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.
- Another way to find the
*x-*intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the*x*-axis. - The multiplicity of a zero determines how the graph behaves at the
*x*-intercepts. - The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
- The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
- The end behavior of a polynomial function depends on the leading term.
- The graph of a polynomial function changes direction at its turning points.
- A polynomial function of degree
*n*has at most*n*– 1 turning points. - To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most
*n*– 1 turning points. - Graphing a polynomial function helps to estimate local and global extremas.
- The Intermediate Value Theorem tells us that if [latex]f\left(a\right) \text{and} f\left(b\right)[/latex] have opposite signs, then there exists at least one value
*c*between*a*and*b*for which [latex]f\left(c\right)=0[/latex].

## Glossary

**global maximum**highest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all

*x*.

**global minimum**lowest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\le f\left(x\right)[/latex] for all

*x*.

**Intermediate Value Theorem**for two numbers

*a*and

*b*in the domain of

*f*, if [latex]a<b[/latex] and [latex]f\left(a\right)\ne f\left(b\right)[/latex], then the function

*f*takes on every value between [latex]f\left(a\right)[/latex] and [latex]f\left(b\right)[/latex]; specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the

*x*-axis

**multiplicity**the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], [latex]x=h[/latex] is a zero of multiplicity

*p*.

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- Revision and Adaptation.
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- College Algebra.
**Provided by:**OpenStax**Authored by:**Abramson, Jay et al..**Located at:**https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].