### By: Hashim Mohamed

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

- Axis of symmetry

- Optimal value

Standard Form: y=ax^2+bx+c

-axis of symmetry

-Optimal value

- Using completing the sq to turn into vertex

-Factoring

-Common

-Simple

-Complex

-Perfect Squares

- Difference of squares

## PARABOLAS

Parabolas are a curve on a graph, you can use parabolas to graph a distance from a soccer ball that is kicked, or even throwing a stone.

Here are some important terms to remember, this is what makes 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 DIFFERENCES

If the first differences are the same it'll create a linear relationship. If the first differences aren't the same but the second differences are it's a quadratic relationship.

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

## EQUATION FORMS

There are three different forms you'll need to know.

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

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

Standard form: ax2+bx+c=0

## VERTEX FORM

Using the vertex form is one of the equations to graph a parabola. Below are the transformations and key words, and step pattern.

Ex. y= -2(x-2)^2 +3

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.

## GRAPHING VERTEX FORM

I will show you how to graph the example above y=-2(x-2)^2 +3 using our step pattern rule.

## FINDING AN EQUATION FOR A PARABOLA

You can find an equation for a parabola when you have its vertex, and a point. For example, find an equation for the parabola with vertex (2.6) that passes through the point (5,3). Using my transformation guide above this will be easy to solve. The answer is below with steps provided.

## FACTORED FORM

In factored form a(x-r) (x-s) you can expand or simplify your question, or graph it

## STANDARD FORM

With standard form you can factor a question given to you in many ways.

## COMMON FACTORING AND EXPANDING/SIMPLIFYING

Common factoring is generally used in the standard form.

## SIMPLE TRINOMIAL FACTORING

Simple trinomials are written in three terms where x=1 in the equation y=ax^2+bx+c. You basically have to make sure that you find two numbers that multiply to the last term (c) and add to the middle term (bx).

You can use trial and error which is a method known for finding the terms by taking a guess and seeing if its correct. We'll use that in complex factoring as well.

## COMPLEX TRINOMIAL FACTORING

Complex trinomial factoring is similar to simple trinomial factoring but, x doesn't = 1 in the equation y=ax^2 +bx +c. You may still use trial and error.

## PERFECT OF SQUARES & DIFFERENCE OF SQUARES

Perfect Squares must follow the formula of (a+b) (a+b).

Difference of squares must follow the formula of (a-b) (a+b).

To see if its a difference of squares you must check for:

1. If the term is a square

2. If there is a - in between the two numbers.

## COMPLETING THE SQUARE AND TURNING IT INTO VERTEX FORM

Completing the square