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Quadratic Portfolio

Before figuring out quadratics and what there all about we need to know some terms that are used all the time. A fun and simple way to remember them is This crossword below.

Quadratic Vocabulary

Download it and try it. :)

Now what is an Quadratic? it is just a pretty big word for an almost simple branch of algebra. Quadratics can be defined as an expression or equation having a square. For example: x² + 3x + 5 = 0. The exponent above the first x term indicates a quadratic. Therefore x² = 49 would also be a quadratic. The three methods most commonly used to solve quadratic  formula are factoring, completing the square, and the  quadratic  formula.

Now before figuring out how to solve a quadratic we need to be able to understand its key properties. Lets look at this graph below.

 Key Features of Quadratic Relations:

vertex and divides the parabola into 2 equal parts (also called the maximum value or the minimum value )
 * ** Vertex **: the coordinates of the minimum/maximum point
 * ** Axis of symmetry **:the vertical line passing through the
 * ** x-intercepts (zeros) **: found by setting y=0 and solving for x
 * ** y-­intercept **: found by setting x=0 and solving for y
 * ** optimal value **: the y ­coordinate of the vertex
 * ** Equation **: y = ax ²+bx+c where a indicates the direction of opening and c represents the y ­intercept



Now in the case of needing to solve Quadratic equations Factoring is the simplest form of solving a  quadratic  equation. When the  quadratic  equation is in its standard form, it is easy to visualize if the constants //a //, //b //, and //c. //A simple way to complete factoring is by decomposition found below. It could be used to figure out the roots or answers of the equation.



If i do not make any sense check out this youtube video i'm sure it will help :). media type="youtube" key="P5Q5SvuVJnk" height="390" width="480"

Think your ready? try your factoring skills in this fun game!! Factoring Game

Another way we could solve quadratic equations is by comp....comp...completing the Square!!!!

Why would need to complete the square though :S? Well it is an easy way to figure the vertex form of an equation Y=a(x-h)^2 + k ﻿and then you will be able to figure out maximums and minimus of a parabola or the vertices (plural of vertex).

media type="youtube" key="xGOQYTo9AKY?rel=0" height="390" width="480"

Now check out this prezzi i made about Quadratics in real life not too amazing but it clarifies some things :)!!

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Now some things you need to know or remember about quadratics.

1. When i graph a quadratic the things i need to know are the....
 * The vertex
 * the axis of symmetry
 * the domain and range
 * the zeros
 * the roots



2. When solving a problem the i need to remember.....
 * start off with a let statement.
 * read carefully and make sure to add the equation in the proper form.
 * complete equation and double check answer
 * end with a let statement and any needed answers

3. In quadratics applications the zeros are important because they help you solve the equation itself. The xeros are where the parabola cross the x-axis. When zeros are known the value of a could be determined.



How do i find the zeros? Quadratic functions can only have 0, 1, or 2 zeros. You can determine the number of zeros either by graphing or by analyzing the function. For a quadratic equation ax ²+bx+c=0 the value of the discriminant could be used.Use this table. and leading into the second key idea below, the zeros can be determined by the location of the vertex relative to the x-axis, and the direction of the opening.
 * = Value of the discriminant ||= <span style="font-family: Arial,Helvetica,sans-serif;">number of zeros/solutions ||
 * b 2 ﻿-4ac>0 ||= 2 ||
 * b 2 ﻿-4ac=0 ||= 1 ||
 * b 2 ﻿-4ac<0 ||= 0 ||
 * if a>0, and the vertex is above the x-axis, there are no zeros.
 * if a>0, and the vertex is below the x- axis, there are tow zeros.
 * if a<0,and the vertex is above the x-axis, there are two zeros.
 * if a<0, and the vertex is below the x-axis, there are no zeros.
 * if the vertex is on the x-axis, there is one zero.

Check out the math textbook page 184 for examples.

4.In quadratic applications, the vertex gives me information about<span style="font-family: Calibri,sans-serif; line-height: 14px;"> the maximum or minimum value of a funtction. //a// determines the size and direction of the parabola. The larger the value of a the steeper the parabola is. If a is positive the parabola opens upward and vice versa.

<span style="color: #333333; font-family: Arial,Helvetica,sans-serif;"> How can you find the maximum/minimum of a quadratic function? You can find the maximum/minimum of a quadratic function by completing the square. Check out my earlier explanation.
 * > **General Form**
 * y = x2 + 2x + 8** ||> **Standard Form (Completing the Square)**
 * Shortcut: y = a(x + b/2a ) 2 - b2/4a + c** ||
 * > y = x2 + 2x + 8 ||> [[image:http://staff.argyll.epsb.ca/jreed/math20p/quadraticFunctions/images/completingSquare_01.gif width="251" height="44"]] ||
 * > y = (x2 + 2x ) + 8 ||> [[image:http://staff.argyll.epsb.ca/jreed/math20p/quadraticFunctions/images/completingSquare_02.gif width="251" height="64" align="left"]] ||
 * > y = (x2 + 2x ** + 1 - 1 ) ** + 8 ||> [[image:http://staff.argyll.epsb.ca/jreed/math20p/quadraticFunctions/images/completingSquare_03.gif width="253" height="64"]] ||
 * > y = **(x2 + 2x + 1)** - 1 + 8 ||> [[image:http://staff.argyll.epsb.ca/jreed/math20p/quadraticFunctions/images/completingSquare_04.gif width="256" height="65"]] ||
 * > y = **(x + 1)2** + 7 ||> [[image:http://staff.argyll.epsb.ca/jreed/math20p/quadraticFunctions/images/completingSquare_05.gif width="285" height="75" align="center"]] ||