# Quadratics

### Vertex Form~Standard Form~Factored Form

## Table of contents

- Terms a definitions
- First differences/ Second differences
- Vertex Form
- Standard form

- X-intercept- when the parabola meets with the x-axis
- Y-intercepts- when the parabola meets with the y-axis
- Zeros- the x value which makes the equation equal to zero
- Axis of symmetry- the vertical line which cut the parabola down in the middle
- Optimal value- The lowest or highest point of the parabola
- Vertex- the maximum or minimum point on the graph

## First differences/Second Differences

Really easy, just small mistakes will mess your work up. Always remember when finding differences

__always subtract bottom number from top__. If it is a first difference then it is a__linear relation (strait line)__, if it turns out as a second difference it is a__quadratic relation (curved lines)__. Examine this picture very well as it shown how to do all of## 1st diff.

if you find a relation in the first differences then it is a linear relation. always subtract the second number from first.

## first diff.

When you find a relation in the first diff. then it is linear. The way you know if you have a relation is if all the numbers are the same like in this picture all the twos. If they are different you would try doing second diff and see if you can find a relation there.

## 2nd diff.

If you find a relation in the 2nd differences then that means it is a quadratic relation.

## Neither?

if your find that there is no 1st diff or 2nd diff then that means it is neither no solution.

## Vertex form

This is the most easiest type of form used in quadratics. View picture below as it will tell you what the equation is

## Axis of symmetry, Optimal Value and transformations

Axis of symmetry

- splits the parabola in half

optimal value

- the minimum and maximum value

Transformation

the h value k value and a value

**All of this will be shown in a picture **

## What can we get from the vertex form

vertex- the coordinates of the parabola

## vertex h and k give us the x and y values. The h values sign it is carrying is always changed because of the ^2. which changes the sign it is carrying to the opposite one. | ## how do we graph the vertex now that we know how to find the vertex we can graph it by putting in the coordinates. If our vertex is (-7,5) we have to write down the transformations it takes: horizontal translation (x axis) 7 units left vertical translation (y axis) 5 units up | ## what about the variable a? The variable a from vertex form choose how wide or skinny the parabola would be. If it is a whole number like 1,2,3,4,17... it will be a stretch (Skinner). If it is a fraction or decimal like 1/2,0.5,3.5,3/4... then the parabola would be compressed (stretched,wider). Now we must write down this transformation like this. for ex. of a=2 then vertical stretch by a factor of 2 ex. if a=0.5 then compression by a factor of 0.5 |

## vertex

h and k give us the x and y values. The h values sign it is carrying is always changed because of the ^2. which changes the sign it is carrying to the opposite one.

## how do we graph the vertex

now that we know how to find the vertex we can graph it by putting in the coordinates. If our vertex is (-7,5) we have to write down the transformations it takes:

horizontal translation (x axis) 7 units left

vertical translation (y axis) 5 units up

## what about the variable a?

The variable a from vertex form choose how wide or skinny the parabola would be. If it is a whole number like 1,2,3,4,17... it will be a stretch (Skinner). If it is a fraction or decimal like 1/2,0.5,3.5,3/4... then the parabola would be compressed (stretched,wider). Now we must write down this transformation like this.

for ex. of a=2 then

vertical stretch by a factor of 2

ex. if a=0.5 then

compression by a factor of 0.5

Quadratic Functions Transformations and Vertex Form

## what if the a value is negative?

if the a value is negative then the whole parabola is flipped over, the right term used is reflected. The a value could be any number 2 or even 10000000000 but if it has a negative in front then it is flipped.

## How to use step pattern

To graph using vertex form you must use the step pattern which is very simple just small mistakes can mess you up. watch video below to learn the step pattern

## here is a video to help understand

Graphing Quadratics Using Step Patterns

## What to keep in mind in step pattern

Always multiply a by 1,3 and 5. also when plotting make sure you go left one and go up by how much the number is. Then do that to the left side.

## vertex form- Word problems

to do these word problems it requires all knowledge of vertex form here are examples of vertex form word problem

## factoring

we must know standard form for this

ax^2+bx+c=0

## factored form

multiplying binomials and special products

to multiply binomials and special products we must use "foil"

Algebra - FOIL Multiplying Binomials

## Common factoring

opposite of of foil, when we were doing foil we were expanding and then simplifying but here we will bring the expanded simplified form to its original form which is factored form

y=a(x-r)(x-s)

Common Factoring Tutorial

## simple trinomial facotring

Factoring the Simple Trinomial

## complex trinomial factoring

to do this we must learn the decomposition method

Factoring Complex Trinomials using Decomposition

## difference of suqares

## solving quadratics by factoring

finding zeros

## how to graph using factored form

## word problems

## standard form

standard form is

ax^2+bx+c=0

## completing the square

we know how to get from standard to factored and factored to standard but what about standard to vertex form, thats what completing the square does

Completing the Square - Solving Quadratic Equations

## quadratic formula

Using the Quadratic Formula

## reflection on a unit and assesments

the most favorite assessment of mine was the vertex form tips assignment we got to use desmos app, worked in partners to find the transformations of a vertex form graphed. i scored almost perfect on the partner section and on the individual i scored perfect. vertex unit was my favorite because it was the easiest and it brought my mark up and i learned alot about what parabolas and how we come across them in life a lot.