# Quadratic Star BY: AKASH K.

### Learn everything to know about Quadratics!

## What is a Quadratic Relation?

## Real World Examples of Parabolas

## LISTEN TO MUSIC WHILE LEARNING?

## Table of Contents

**Introduction**

- Analyzing Quadratics
- Graphed Quadratics- Parabolas
- Transformations

**Types of Equations**

- Factored Form
- Standard Form
- Vertex Form

**Vertex Form**

- Key Components
- Finding the Equation
- Graphing From Vertex Form

- Chart Base Parabola

- From an equation

**Factored Form**

- Key Components
- Expanding
- Common Factoring
- Common Factoring Binomials
- Factoring By Grouping
- Factoring Simple Trinomials
- Factoring Complex Trinomials
- Special Cases

- Perfect Squares

- Difference of Squares

- Graphing Factored Form

**Standard Form**

- Solving Using Quadratic Formula
- Discriminant
- Factoring Standard Form (Standard Form To Factored Form)
- Completing The Square (Standard Form To Vertex Form)
- Graphing Standard Form

**Types of Math Problems**

- Motion problems
- Optimization (fencing problems)
- Geometric problems
- Economic problems
- Number problems

- Connections Between Topics
- Reflection of the Unit

## Introduction

## Analyzing Quadratics

## Graphed Quadratics- Parabolas

Parabolas can be drawn when the vertex is found and the step pattern is found

X Intercept- Where the parabola meets the x axis, can also be called a zero or root (x=_)

Y Intercept- Where the parabola meets the y axis (y=_)

Axis of Symmetry- The x value of the vertex (x=_)

Optimal Value- The y value of vertex, also considered the maximum or minimum value (y=_)

Vertex- The crossing point of the axis of symmetry and the optimal value (x,y)

Parabolas can be graphed using "

## Transformations

A Quadratic Equation (Vertex Form): y=a(x-h)^2+k

The 'a' value changes the vertical stretch or compression of the parabola and it can change which way the opening would be. If the a value is 1 or lower it would be a vertical compression and if it were 1 or higher it would be considered a vertical stretch. Also, when the number is a positive it opens up from the vertex, but when it is negative, it reflects off the vertex and opens downwards.

The'-h' translates the parabola horizontally. If it was negative, it would shift the vertex to the right and if it were positive, it would shift the vertex to the left.

The 'k' is used to translate the vertex vertically (up or down). If 'k' was to be positive it would translate the vertex up and if it were negative, it would translate the vertex down.

## Type of Equations

## Factored Form

## Standard Form

ax²+bx+c

## Vertex Form

## Vertex Form

## Key Components

Some key components in the equation:

'a' is the vertical stretch of the parabola

'y' and 'x' are points on the parabola

'h' is the horizontal shift of the vertex (the x coordinate)

'k' is the vertical shift of the parabola (the y coordinate)

If the equation has a negative sign in front of 'a' than this means the parabola is reflected vertically off the vertex (opens down)

## Finding the Equation

## Graphing From Vertex Form

## Chart Base Parabola

## From an Equation

When graphing from vertex form you can use the 'k' and 'h' values to figure out the vertex's location on the graph. As you know, the change in 'h' makes it horizontally shift and the change in 'k' makes it vertically shift.

## Factored Form

**Factored form**, the product of a constant and two linear terms

## Key Componenets

__Key points__

When the (a) value changes the zeroes do not change

When the (a) value changes the axis of symmetry do not change

When the (a) value changes the optimal value does change

## Expanding

## Common Factoring

## Common Factoring Binomials

## Factoring By Grouping

## Factoring Simple Trinomials

The indication of the trinomial to notify the person the it is a simple trinomial is if :

- The leading coefficient is 1
- It is in the form shown above

When factoring simple trinamials you would end up with two brackets and have two numbers that will not only multiply to get the 'c' value, but will add to get the 'b' value

A step by step example is shown below

## Factoring Complex Trinomials

The indication of the trinomial to notify the person the it is a complex trinomial is if :

- The leading coefficient is 2 or greater
- It is in the form shown above

When factoring complex trinamials you would end up with two brackets and have two numbers that will not only multiply to get the 'c' value, but will add to get the 'b' value

A step by step example is shown below:

## Perfect Squares

Form: a² + 2ab +b²

Factored: (a + b)²

## Difference of Squares

Factored: (a + b)(a - b)

Are Trinomials that have is wrote out as a binomial of a square subtracting another square. The factored for is the number subtracted by the other number and then multiplied by the number adding to the other number ('product' of 'sum' and 'difference)

## Graphing Factored Form

**Information You Need:**

**X-Intercepts**

You can get this by solving for the X's

Example: y=-(x-2)(x+4)

*Each bracket has an x-intercept

0=(x-2) *solve for x

2=x

0=(x+4) *solve for x

-4=x

So, the two x-intercepts are 2 and -4

**Axis of Symmetry**

The axis of symmetry is located in the middle of the x-intercepts

You can get this by adding the two values and dividing them by two

1. Add two values

-4+2

=-2

2. Divide the value by 2

-2/2

=-1

So, the axis of symmetry is -1

**The Optimal Value**

You can obtain this by substituting the axis of symmetry into the vertex form equation

1. Sub in the axis of symmetry value into the vertex form equation

Axis of symmetry - x=(-1)

y=(-1-2)(-1+4)

2. Solve for the y value

y=(-1-2)(-1+4)

y=(-3)(3)

y=(-9)

So, the optimal value is (-9)

This means the vertex is (-1,-9)

**The Final Parabola**

Use the 'a' value to determine the step pattern to complete the parabolas

Over 1 Down 1x1 = 1

Over 2 Down 4x1 = 4

Over 3 Down 9x1 = 9

## Standard Form

## Solving Using The Quadratic Formula

From the standard for equation, (ax²+bx+c), you will use the 'a', 'b', and 'c' values to be subbed into the equation

This formula is recommended if the equation is not easily factor-able

## Discriminant

The Discriminant shows us if there is 1, 2, or no x-intercepts in the parabola

If the discriminant is equal to a real number greater than 0, then there will be two roots

If the discriminant is equal to 0, than there will be one root

If the discriminant is a negative number, then there will be no roots

## Factoring Standard Form (Standard Form To Factored Form)

*Refer Back To "Factoring Simple Trinomials" and "Factoring Complex Trinomials" For Help

## Completing The Square (Standard Form To Vertex Form)

This comes in handy when needing to find the vertex for graphing, axis of symmetry, and optimal value

This is called completing the square because when the equation is layed out with algebra tiles, you must make find a way to make the product tiles a square

An example is shown below:

## Graphing Standard Form

- The vertex
- The zeros

__The Vertex__

**Axis of symmetry**

In order to get the vertex, you must use a formula which gets you the axis of symmetry:

x=-b/2a

**EXAMPLE**: y=x²+3x+2

a=1 b= 3

x= -b/2a

x= -3/2(1)

**x= -1.5**

**The Optimal Value**

In order to get the optimal value you must plug in the axis of symmety for the 'x' value in the original equation

y=x²+3x+2

y=(-1.5)²+3(-1.5)+2

y= 2.25-4.5+2

**y= -0.25**

So, the vertex is (-1.5, -0.25)

__The Zeros__

## Types of Math Problems

## Motion Problems

**Some key features to look for:**

**The X-Axi****s**

The x-axis usually refers to the time of the objects flight or the vertical distance the object travels

**The Y-Axis**

The y-axis usually refers to the height/altitude of the object from the ground

**The x-intercepts**

The x-intercepts in these problems usually refer to the points where the object is at ground level

**The Optimal Value**

The optimal value usually refers to the point where the object is at its highest altitude

**The Axis of Symmetry**

The Axis of Symmetry usually refers to the time during flight where the object is at its highest altitude

## Optimization (fencing problems)

## Geometric problems

## Economic Problems

## Number Problems

## Connections Between Topics

__Equation and graphing__

All of the forms of equations connect to the parabolas of graphs depending on their values. You can get out the x intercepts of any of the equations and they directly relate to the zeros of the graphs and can describe to you how horizontally stretched the parabola can be. Also the equations can all relate to the vertex of the parabola from various methods. This can tell you the maximum/minimum and axis of symmetry of the parabola. Furthermore, the equations can show the stretch or compression of the parabola and altogether this shows how the equations relate and give a sense of how a parabola might look.

__Discriminant and Zeros__

The discriminant in the quadratic formula can directly relates to the number of zeros there are in a graph. For example if your discriminant equals 0 you know there is on solution (one zero) and if there's one zero that means the vertex of the parabola is on x and is opening up or down.

__Equation and Relationship__

When graphing you would use the relation ship between x and y because you must show the line between the relation of the x axis and the y axis. An equation is used to find the solutions. These solutions are for figuring out the value of 'x'.

## Reflection of the Unit

After sometime, I started slacking off and procrastinating with my work. I felt over confident and this cause me to become lazy and effortless with my work. feel this is my down fall and I realized this when I later got bad results on my "Quadratics 2 Test".