# Quadratics - Unit 3

### By Amanda Persaud

## What Is a Quadratic Relationship?

A quadratic relationship links to a quadratic formula, which are used to calculate the height of a kicked ball, or such. Quadratic relationships has 2 as its highest degree;

__y = ax² + bx +c__would be a quadratic relationship's formula.## Quadratic Relationships vs. Linear Relationships

Quadratic relationships can be put into a chart; the same as a linear relationship. In a quadratic relationship, the first differences are not constant, but the second differences are. In a linear relationship, the first differences will turn out to be constant. The differences are found by putting the x coordinates and the corresponding y coordinates into seperate columns. Then, the first differences column is added, along with the second differences column. The first box in the first differences column should be ignored, and the first two boxes of the second differences column should be ignored.

Linear, Quadratic or Neither using First and Second Differences Tables

## Forms of Quadratic Equations

'**a**' represents the stretch factor (if there is one).

'**b**' and '**h**' represent the x coordinate of the parabola's vertex/the axis of symmetry.

'**c**' represents the parabola's y-intercept (if there is any).

'**k**' represents the y value of the vertex.

'**r**' and '**s**' represents the parabola's x-intercepts (if there are any).

__Vertex form:__ **y = a(x - h)² + k**

__X-intercept form:__ **y = a(x - r)(x - s)**

__Standard form:__ **0 = ax² + bx + c**

## Transformations

**horizontal translation to the**,

__right/left__by (__#__)**vertical translation**,

__up/down__(__#__)**vertical flip**,

**stretch by a factor of (**.

__#__)__Example:__

To get from **y = x****²** to **y = -2(x - 3)****² + 1**

- Horizontal translation to the right by 3
- Vertical translation up 1
- Vertical flip
- Stretch by a factor of 2

## Components of a Quadratic Graph

**6**key parts.

- Vertex
- X-intercept(s)/zero(s)
- Axis of symmetry
- Minimum/maximum
- Roots/solutions
- Y-intercept

## The Step Pattern

The step pattern works in a quadratic graph to find the next point if the stretch factor is given. The step pattern starts from the vertex of a parabola, and follows an "over 1, up/down 1, over 2, up/down 4, over 3 up/down 16" pattern. Up meaning that the stretch factor is positive and down meaning that the stretch factor is negative.

## Factoring and Expanding Quadratic Expressions

A quadratic expression in factored form could be: (x+2)(x+2). Factoring quadratic expressions turns the expression into x-intercept form. Quadratic expressions has

**five**types of factoring:**common factoring, perfect squares factoring, difference of squares factoring, factoring simple trinomials, and factoring complex trinomials**.Quadratic expressions could be expanded (simplified) by using the **distributive property**.

## Common

Factoring trinomials with a common factor

## Perfect Squares

Factoring perfect square trinomials

## Difference of Squares

Factoring difference of squares

## Simple Trinomials

factoring trinomials

## Complex Trinomials

Factoring trinomials with a non-1 leading coefficient by grouping

## Standard Form Equations

**(x + 4)(x + 3) = 0**

**x + 4 = 0 x + 3 = 0**

**x = -4 x = -3**

## Linking Quadratics

The three parts of quadratics link through many different ways. Factoring links to solving equations, as does solving equations to graphing.

## Graphing, Factoring and Solving

**y = x² + 7x + 12**can be factored into

**y = (x + 3)(x + 4).**

Solving goes into graphing as it helps to find certain points on a graph (e.g. x-intercepts, vertex, etc.).

## Solving Equations and Factoring

**5x**²

**+ 12x + 4 = 0**can be factored into

**(5x + 2)(x + 2) = 0**, which can then be solved as

**5x + 2 = 0**and

**x + 2 = 0**.