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

Vertex 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 Standard Form: Vertex Form: Vertex : Axis of symmetry: Parabola opens up Parabola opens down

Finding the vertex  

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

Practice 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)

Quadratic Equations and Problem Solving A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: How long does it take for the diver to hit the surface of the water? seconds

Quadratic Equations and Problem Solving The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden? The width is w. The length is w+5. feet feet

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?