Quadratic Functions Chapter 5

5.1 Introduction to Quadratic Functions A quadratic function is any function that can be written in the form: f(x) = ax2 + bx +c

Identify a, b, and c for the following quadratics: f(x) = x2 + 3x +7 a = b = c = f(x) = 3x2 + 4x – 12 a = b = c = f(x) = -x2 – 2x + 10 a = b = c = f(x) = 20 – 4x – 2x2 a = b = c =

The graph of a quadratic is a parabola. There are 2 types: Opens up Opens down Vertex Axis of Symmetry Vertex Axis of Symmetry

Let’s take a look at the graphs of 1-4 Let’s take a look at the graphs of 1-4. State whether it opens up or down. f(x) = x2 + 3x +7 up or down? f(x) = 3x2 + 4x – 12 up or down? f(x) = -x2 – 2x + 10 up or down? f(x) = 20 – 4x – 2x2 up or down? What do the “up” ones have in common? What do the “down” ones have in common?

Maximum and Minimum Values if a > 0 the parabola opens up, and the y-coordinate of the vertex is the minimum value. Ex. (1.) f(x) = x2 + 3x +7 a = 1 opens up find the vertex vertex: min.: if a < 0 the parabola opens down, and the y-coordinate of the vertex is the maximum value. Ex. (2.) f(x) = -x2 + 4x – 12 a = -1  opens down vertex: max.:

5.2 Intro to Solving Quadratic Functions Solve x2 = 9 for x therefore x = 3 or x = -3 If x2 = a and a ≥ 0, then

Properties of Square Roots: Product Property of Square Roots: if a ≥ 0 and b ≥ 0 : Quotient Property of Square Roots:

Practice Problems Solve 4x2 + 13 = 253 and give exact solutions.

Practice Problems Solve 9(x – 2)2 = 121

Pythagorean Theorem If ∆ABC is a right triangle with the right angle, then a2 + b2 = c2 . Leg Hypotenuse

Practice Problems Find the unknown length in each right triangle. Give answers to the nearest tenth. x = 2.5 y = 5.1 z

Practice Problems Find the unknown length in each right triangle. Give answers to the nearest tenth. q = 4.0 p r = 8.2