2.1 Polynomial and Quadratic Functions

Polynomial Function:

Let n be a nonnegative integer and let a_n, a_n-1, . . . , a₂, a₁, a₀ be real numbers with a_n ≠ 0.

The function f(x) = a_nxⁿ + a_n-1x^n-1 + . . . + a₂x² + a₁x + a₀  is called a polynomial function of x with degree n.

Polynomial functions are classified by degree (the highest value for n where a_n is not equal to 0)

·      Constant function

o  f(x) = a, a ≠ 0;

o  Has a degree 0

·      Linear function

o  f(x) = mx + b, m ≠ 0

o  Has degree 1

Quadratic Function

·      Definition: Let a, b, and c b real number with a ≠ 0. Has degree 2. The function is f(x) = ax² + bx + c

·      Graph is a parabola, a special type of U-shaped curve

o  All parabolas are symmetric with respect to the axis of symmetry (axis of parabola)

·      Vertex is the point where the axis intersects the parabola

o  If a > 0, the vertex is the minimum point on the graph and the parabola opens upwards

o  If a < 0, the vertex is the maximum point on the graph and the parabola opens downwards

Standard Form of a Quadratic Function

·      Standard Form is f(x) = a(x-h)² + k

o   Especially convenient for sketching a parabola because it identifies vertex of the parabola as (h, k)

·      Identifying the Vertex of a Quadratic Function:

o  Describe the graph of f(x) = 2x² + 8x + 7 and identify the vertex

o  Solution:

§  f(x) = 2x² + 8x + 7

                = 2(x² + 4x) + 7

                = 2(x² + 4x + 4 – 4) + 7

·      Because b = 4, add and subtract (b/2)² = 4 within parentheses               

                = 2(x² + 4x + 4) – 2(4) + 7

                = 2(x + 2)² – 1

§  Vertex is (-2, -1)