Drill #92 Identify the maximum or minimum value of each quadratic function, then state the domain and range of each.

Drill #93 Identify the a.) max/min the b.) domain and range, and c.) roots (if any) of each graph: 1. 2. 3. Vert: (3/2, -1/5) Vert: (1, 0) Vert: (0, -1)

Drill #94 Identify the a.) max/min the b.) domain and range, and c.) roots (if any) of each graph: 1. 2. 3. Vert: (0, -1) Vert: (-2, 0) Vert: (2, 2)

Drill #95 Write a quadratic equation in standard form with the given root(s). 1. { -1, 6} 2. { ¾ , -½ } 3. { 3 }

6-2 Solving Quadratic Equations Objective: To estimate solutions to quadratic equations by graphing and to find the zeros of quadratic equations using the zero-product property.

(1.) zeros Definition: The x- coordinates where a function crosses the x- axis (the x- intercepts). These are all the values such that f(x) = 0. (2.) roots: The zeros of a function are also called roots of the function. They are values of x that satisfy

Solving a Quadratic Equation by Graphing* 1. Find the vertex (x = -b/2a) 2. Find 2 points on either side of the vertex. 3. Draw the parabola 4. Identify the points where the parabola crosses the x-axis. (if it is between two numbers, estimate the value) NOTE: If the parabola does not cross the x-axis then it has no zeroes (or roots)

Examples* Find the Zeros: Two solutions Ex1. Ex2.

Examples* Find the Zeros Ex3. Ex4.

Graphing on the TI-83/84 To enter the function: 1. Enter the equation into [y1 =] 2. To view graph [graph] 3. To adjust axes [window] Equation must be in

Study Guide Examples* Find the roots of each quadratic by graphing…

Writing Equations In Standard Form* If a quadratic equation has roots to write the equations in standard form: 1. Set up the equation 2. FOIL (Multiply) 3. Simplify (combine like terms) Example: SG 1, #1

Writing Equations in Standard Form (w/ Fractional Roots) If a quadratic has fractional roots… Before you FOIL 1. Find a common denominator for each factor 2. Multiply each side by the product of the denominators (cancel the denominators) Example: SG 2, #14

Writing Equations in Standard Form (w/ One Root) If a quadratic has one root… 1. Then the root is repeated… use the same root for x1 and x2 Example: Root = 6

Sketch a Graph* Sketch a graph of a quadratic with the following properties: Ex1: roots = {1, 4}; a > 0 Ex2: roots = {1, 4}; a < 0 Ex3: roots = { 2 }; a > 0 Ex4: roots = { }; a < 0

(3.) Zero Product Property** Definition: If the product of two numbers a(b) = 0 then either a = 0 or b = 0 or both Example: x (x – 1) = 0 x = 0 or x – 1 = 0 x = 1

Solving Quadratic Equations by Factoring* To solve a quadratic equation by factoring: 1. Group all the terms onto the same side of the equation ( ) 2. Factor the quadratic 3. Use the zero product property

Factoring Quadratic Equations Examples: Skills Practice: 7, 9, 11 Classwork 8, 10, 12

Solving Quadratic Equations* Examples: No Linear Term: No Constant: Quadratic Coefficient (a) = 1: Quadratic Coefficient (a) = 1