# JOURNEY THROUGH QUADRATIC RELATIONS

### By: Bansari Shah

## What's Quadratics & How's it Useful

Basically the word quadratics comes from '__Quad'__ meaning square since the variables get squared ( ).

## Table of Contents

**Introduction**Key information about Parabola

- Basic Information Facts About Parabolas

Intro to Parabolas

- Key Features of Quadratic Relations

First & Second Differences

__Types of equations__

Vertex Form

Standard Form

Factored Form

__Vertex Form__

- Key Facts
- Transformations
- Finding An Equation When Given the Vertex
- Graphing from Vertex Form
- Isolating for x

__Standard Form__

- Solving Using Quadratic Equation
- Graphing from Standard Form
- Discriminant
- Completing the square: vertex to standard form

__Factored Form__

- Factors & zeros
- Expanding - factored form to standard form
- Common Factoring
- Simple Trinomial
- Complex Trinomial
- Difference of Squares
- Perfect Square
- Factoring by grouping

__Word Problems__

- Number Problems
- Optimization Problems
- Revenue Problems

__Reflection__

- Unit Analysis

__Helpful Links__

## Basic Important Facts About Parabolas

- Parabolas can open either
or__up____down__ - The
of a parabola is where the graph crosses the**zero****x-axis** - 'Zeros' can also be called "
" or "__x-intercepts____roots"__ - The
divides the parabola into**axis of symmetry**.**two equal halves** - The
of a parabola is the point where the__vertex__and the**axis of symmetry**. It is the**parabola meet**where the parabola is at the**point**or**minimum**value.**maximum** - The
is the value of the__optimal value__of the vertex.**y co-ordinate** - The
of a parabola is where the graph crosses the**y-intercept**.**y-axis**

## Key Features of Quadratic Relations

## First & Second Differences

## Types of Equations

## Vertex Form

## Standard Form

## Factored Form

## Vertex Form

## Key Facts

**vertex**

__(x,y) = (h,k)__

The **axis of symmetry** : __(x = h)__

The **optimal value** is __(y = k)__

__ Note:__ When writing down the vertex, remember that the h value is always the opposite. If the h value is positive then when you write it down it would be negative and if the h value is negative then when you write it down it would be positive.

## Transformations

When in a vertex form each letter is responsible for a transformation.

(__NOTE: if the 'a' value is negative, the vertex will be a maximum value & if the 'a' value is positive, the vertex will be a minimum value.)__

'h' moves the vertex of the parabola either left or right. When 'h' is negative move right and when its positive move left.

(__NOTE: it affects the horizontal translation)__

'k' moves the vertex of the parabola up or down. If the 'k' value is negative the parabola opens down. (__NOTE: affects the vertical translation. Also if the "a" value is negative there will be a reflection)__

## Finding An Equation When Given Vertex

## Graphing From Vertex Form

- Find the vertex, since the formula is given in vertex form. (h,k)
- Find the y-intercept. In order to find the y-intercept you need to plug in the value of x to be 0.
- Find the x-intercept(s). To find the x-intercept, you need to plug in the value of y to be 0. You can solve for "x" by using the quadratic formula.
- Graph the parabola using the points which you found in steps 1- 3.

## Isolating For x

## Standard Form

## Solving Using Quadratic Equation

## Graphing From Standard Form

## Discriminant

There are 3 possible solutions of a discriminant.

- When the discriminant is
__less than zero, there is no solution__. __W__hen the value of discriminant is__more than zero, there will be 2 solutions__.- When the discriminant is
__0, there will be only one solution__. Whether you add or subtract it from 'b', you will get the same answer.

## Solution 1

__When the discriminant is less than zero, there is absolute no solution.__## Solution 2

__When the value of the discriminant is more than zero, there will be 2 solutions.__## Solution 3

__When the discriminant is 0, there will be only one answer. Whether you add or subtract the 'b' you will get the same answer.__## Completing the Square : vertex to standard form

## Factored Form

## Factors & Zeros

## Expanding - factored form to standard form

__Factored Form to Standard Form__, you would expand the terms in the first bracket to the second bracket. Once you're done expanding, you're going to see that you have like terms so all you would do is add those like terms up accordingly to the signs.