PolyNomials Definition : A polynomial function is a function that can be expressed in the form: p(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 Where a n, a n-1, …, a 1, a 0 are real numbers, a n ≠ 0, the exponents are non-negative integers Definition: The degree of a polynomial is largest exponent of x. ✔ ✔ ✔ ✔ The degree is 2 The degree is 1 The degree is 0 The degree is 3
Quadratic functions 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) = ax 2 + 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.
The graph of a quadratic polynomial is called a parabola. Parabolas p(x) = ax 2 + bx + c a > 0a < 0 vertex Axis of Symmetry Axis of Symmetry
How does the graph of a quadratic function change as we change a, b, and c ? Parabolas a decreases from 1 towards 0
How does the graph of a quadratic function change as we change a, b, and c ? Parabolas a increases from 1 to 10
How does the graph of a quadratic function change as we change a, b, and c ? Parabolas c increases from 0 to 2
How does the graph of a quadratic function change as we change a, b, and c ? Parabolas c decreases from 0 to -2
Definition: 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. Standard form
Summary General Form: Vertex: Axis of symmetry: standard Form: Vertex: Axis of symmetry: 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 – 0.5x 2 How much should the company spend on advertising to maximize profits?