Drill #92

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

244)(.2

542)(.12

2

xxxf

xxxf

Drill #93Identify 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 #94Identify 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

02 cbxax

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

0

)(2

2

cbxax

cbxaxxf

Study Guide Examples*

Find the roots of each quadratic by graphing…

Writing Equations In Standard Form*

If a quadratic equation has rootsto write the equations in standard form:1. Set up the equation

2. FOIL (Multiply)3. Simplify (combine like terms)

Example: SG 1, #1

21, xx

0))(( 21 xxxx

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) = 0x = 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

02 cbxax

0))(( 21 xxxx

0

0

2

1

xx

xx

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

092 x

xx 32

2142 xx

0532 2 xx