Algebra 2 / Unit 3: Polynomial Functions
3.1 Polynomials and End Behavior
Notes
A polynomial is a sum of terms like $ax^n$ with whole-number exponents. Write it in standard form with exponents decreasing.
- The degree is the highest exponent. The leading coefficient is the number in front of that term.
- End behavior depends only on the leading term.
- Even degree: both ends go the same way (up if the leading coefficient is positive, down if negative).
- Odd degree: the ends go opposite ways (positive leading coefficient: down on the left, up on the right).
- Example: $f(x) = -x^4 + 3x$ has even degree and negative leading coefficient, so both ends go down.
Practice
Describe the end behavior of $f(x) = -2x^3 + x$.
Odd degree with a negative leading coefficient: up on the left, down on the right.
What are the degree and leading coefficient of $f(x) = 3x^2 - x^4 + 7$?
The highest power is $x^4$, and its coefficient is $-1$.
A polynomial has even degree and a positive leading coefficient. Its graph:
Like $y = x^2$, both ends go up.
Ready for more? Take the Unit 3 test.