# Quadratics

### By:Puneet Guleria

## Intro

## Table of context

## Quadratics in vertex form

Key features of quadratic relations

Scatter plotLine/curve of best fit

Finite differences

-linear relation

-quadratic relation

-neither

Transformations of quadratics

-writing equations given the transformation

Graphing using step pattern

Finding equations given the vertex

## Quadratics in factored form

Common Factoring

Factoring Trinomials

Factoring Difference of Squares and Perfect Square Trinomials

Solving Quadratics by Factoring (finding the zeros)

Graphing Quadratics in Factored Form

## Quadratics in standard form

The Quadratic Formula

## Key features of quadratic relations

Axis of symmetry(where the parabola is divided equally)

Optimal value(the y coordinate of the vertex, can be maximum or minimum)

Zeros/Roots(the x-intercepts)

## Scatter plot

## Line and Curve of best fit

Step2: make the line/curve of best fit in a place where almost all points on both side of the line are equal. this is how to determine and make a line or curve of best fit.

## line of best fit

## Curve of best fit

## Using finite differences to determine if the relation is linear, quadratic, or neither

## Linear relation

## Quadratic relation

## Transformations

## writing equations given the trasformation

The steps

1- identify weather your parabola is opening up or down.

2- identify the stretch or compression(a)

3- identify the transformation left or right (h)

4- identify the transformation up or down (k)

5- finally put your variables into the equation y=a(x+3)^2+3

## Graphing using the step pattern

## Finding an equation given the vertex

steps

1- sub in your vertex into your equation y=a(x-h)^2+k (plug x value as your h but make your x the opposite sign(negative-positive or positive-negative) then for k plug in your y value the same as it is)

2- sub in one of your x intercepts into the equation

3- solve until you find the value of (a) which will complete the equation

4- plug in a to you original equation to complete the equation

## Finding x and why intercepts in vertex form

## multiplying binomials and special products

(x+5)^2---(x+5)(x+5)----x^2+5x+5x+25---x^2+10x+25 formula=a^2+@ab+c

(x+5)(x-5)---x^2+5x-5x-25----x^2-2 formula where equations are same but middle sign is different a^2-b^2

## common factoring

for example

6x^2 − 2x

whats common is 2 and x so the new term would be

2x(3x-1)

## factoring trinomials

Step 1 find two numbers that multiply to give ac (in other words a times c), and add to give b.

Step 2 Rewrite the middle with the new numbers to form a equation with 4 terms

Step 3 Factor the first two and last two terms separately

Step 4 put the common factors together

here is an example

2x^2 + 7x + 3 sum of (a)(c)=6 (6)(1)=6 6+1=7

2x2 + **6x + x** + 3

2x(x+3) + (x+3)

(2x+1)(x+3)

## factoring differences of squares

1 look for a gcf

2 square root the two numbers

3 determine if the the new numbers can be factored further

4 put numbers into factored form (x+a)(x-a)

Example: 18x^2-98y^2

2(9x^2-49^2)

square root the middle terms

2(3x-7y)(3x+7y)

## factoring perfect square trinomials

## factoring overview

## solving quadratics by factoring

so the first step is to get one side to be zero- x^2-10x=-16---x^2-10x+16=0

next find two numbers that multiply to give ac (in other words a times c), and add to give b-

(x-8)=0 (x-2)=0

then solve for zeros-- x-8=0--- x=8 x-2=0--- x=2

## Graphing Quadratics in Factored Form

## completing the squares

**Step 1**Divide all terms by

**a**(the coefficient of

**x2**).