Integrated Math 3 / Unit 1: Polynomial Functions
1.1 Polynomial Basics and End Behavior
Notes
A polynomial function is a sum of terms $a x^n$ with whole-number exponents. The degree is the highest exponent and the leading coefficient is the coefficient of that term.
- Example: $f(x) = -2x^3 + 5x - 1$ has degree 3 and leading coefficient $-2$.
- End behavior depends only on the leading term.
- Even degree, positive leading coefficient: both ends rise. Even degree, negative: both ends fall.
- Odd degree, positive leading coefficient: falls left, rises right. Odd degree, negative: rises left, falls right.
- A polynomial of degree $n$ has at most $n - 1$ turning points.
Practice
What is the end behavior of $f(x) = -2x^3 + 5x - 1$?
Odd degree with a negative leading coefficient: as $x \to -\infty$, $f \to +\infty$, and as $x \to \infty$, $f \to -\infty$.
What are the degree and leading coefficient of $f(x) = (x + 1)(x^2 - 3)(2x - 5)$?
The degrees add: $1 + 2 + 1 = 4$. The leading coefficients multiply: $1 \cdot 1 \cdot 2 = 2$.
What is the greatest number of turning points a degree 5 polynomial can have?
A degree $n$ polynomial has at most $n - 1$ turning points, so $5 - 1 = 4$.
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