# Quadratics

### By: Gurjot Uppal

## Table of Content

__Introduction__- Intro to Parabolas
- First and Second Differences

*Equation Forms*

- Factored Form
- Standard Form
- Vertex Form

**Vertex Form:***y=a(x-h)^2 +k*

- Axis of Symmetry
- Optimal Value
- Transformations
- Step Pattern
- X-int or zeroes
- Finding an equation for a parabola

**Factored Form:**** **y=a(x-r) (x-s)

X-int or zeroes

Optimal value

- Axis symmetry

__Standard Form:__*y=ax^2+bx+c*

- Quadratic Formula
- axis of symmetry
- Optimal value
- Using completing the sq to turn into vertex
- Factoring
- Common
- Simple
- Complex
- Perfect Squares
- Difference of squares

## Parabolas

Important terms to remember, these are what make up a Parabola

Axis of Symmetry: It's in the middle of your parabola. It helps to find your vertex.

Vertex Point: This is the curve of the parabola. It's the direction of where it faces.

Optimal Value: This is the value of your vertex.

## First and Second Difference

Below is a chart of an example of the first and second differences.

## Types of equations

Vertex form: h= a(x-h)2 +k

Factored form: a(x-r) (x-s)

Standard form: ax2+bx+c=0

## Vertex form

Ex. y= -4(x-2)^2 +6

Vertical Reflection: If it's a - the parabola will open downwards. If it's positive it will open upwards.

Vertical Stretch: This effects the step pattern instead of doing the basic step pattern routine it will double instead look below.

Horizontal Translation: This moves the parabola left or right. It's also the part of the vertex.

Vertical Translation: This moves the parabola up or down. It's also a part of the vertex.

## Optimal Value

## Transformations

## How to find X-intercepts

y=a(x-h)^2+k

y=3(x+1)^2-108

0=3(x+1)^2-108

+108 +108

108=3(x+1)^2

divide 108 and 3 by 3

36=(x+1)^2

square root 36 and it will cancel out the squared because what you do to one side of the equation you have to do to the other.

6=(x+1)

-1 -1

5=x

By setting y=0 it helped us find the x-intercepts

## Step Pattern

## Factored form y= a(x-r)(x-s)

## x-intercepts (r and s)

The equation for factored form is y= a(x-r)(x-s). With the R and S given you already have x but you have to change the sign to the opposite given. For example: y= (x-6)(x+7) you are going to switch the negative to and positive and positive to a negative (x-6) x=6 (x+7) x= -7. This is one rule you need to do.

## Axis of symmetry (x=(r+s)/2)

To find the axis of symmetry you need to add R and S and then divide them by 2. Thats how you will find the axis of symmetry for factored form.

## Optimal value (sub in)

## Standard form y =ax^2+bx+c

## Zeros

## Axis of symmetry

## Completing the square

y= (-4x^2-8x)-1

(b/2)^2

(8/2)^2 = 16

y= (-4x^2-8x+16-16)-1

y= (-4x^2-8x+16)-16-1

y= (-4x^2-8x+16)-17

y= (-2x+4)^2-17

This is one example of turning standard form into vertex form