Quadratic Functions

p(x) = an xn + an-1 xn-1 + … + a1 x + a0 PolyNomials Definition: A polynomial function is a function that can be expressed in the form: p(x) = an xn + an-1 xn-1 + … + a1 x + a0 Where an , an-1 , … , a1 , a0 are real numbers, an ≠ 0, the exponents are non-negative integers ✔ The degree is 2 ✔ The degree is 1 ✔ The degree is 0 ✔ The degree is 3 Definition: The degree of a polynomial is largest exponent of x.

Quadratic functions Definition: p(x) = ax2 + bx + c A polynomial of degree 0 is called a constant function. A polynomial of degree 1 is called a linear function. Definition: A degree 2 polynomial function is called a quadratic function. The general form a quadratic function is p(x) = ax2 + bx + c where a, b, and c are real numbers with a ≠ 0. Quadratic functions are incredibly important functions that show up everywhere in the real world.

Parabolas p(x) = ax2 + bx + c a > 0 a < 0 The graph of a quadratic polynomial is called a parabola. p(x) = ax2 + bx + c Axis of Symmetry vertex Axis of Symmetry vertex a > 0 a < 0

Parabolas a decreases from 1 towards 0 How does the graph of a quadratic function change as we change a, b, and c? a decreases from 1 towards 0

Parabolas a increases from 1 to 10 How does the graph of a quadratic function change as we change a, b, and c? a increases from 1 to 10

Parabolas c increases from 0 to 2 How does the graph of a quadratic function change as we change a, b, and c? c increases from 0 to 2

Parabolas c decreases from 0 to -2 How does the graph of a quadratic function change as we change a, b, and c? c decreases from 0 to -2

Standard form Definition: p(x) = a(x – h)2 + k The standard form of a quadratic function is p(x) = a(x – h)2 + k Where (h, k) is the vertex of its graph and a ≠ 0.

Summary General Form: standard Form: Vertex: Axis of symmetry: Vertex: Parabola opens up Parabola opens down

problems Find the vertex and the x-intercepts of the following functions:

problems Find the quadratic function with the indicated vertex that passing though the given point: 1. Vertex: (2,3) Point: (0,2) 2. Vertex: (-2,-2) Point: (-1,0) 3. Vertex: (6,6) Point: (1/2, 3/4)

problems The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) that the company spends on advertising according to the model: P(x) = 230 + 20x – 0.5x2 How much should the company spend on advertising to maximize profits?