Quadratics Relationships

By : Anmol Saggu

What is Quadratics

The name Quadratic came from the word "quad" meaning square, because the variable gets squared (like x²).

Where are Parabola used in daily life

parabolas are used a lot in daily life. These are some of examples where parabolas are used.
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Table of Contents

1 Introduction

- Analyzing quadratics (second differences)

- Introduction to parabolas

-Transformation



2 Types of equations

- Vertex form

- Factored form

- Standard form



3 Factored form

- Common factoring

- Factoring simple trinomials

- Factoring complex trinomials

- Special cases

  • Difference of squares
  • Perfect square

- Factoring by grouping

- Graphing factored form


4 Vertex form

- To find an equation when only given the vertex

- Isolating for x

- Graphing vertex form



5 Standard form

- Completing the square (standard to vertex)

- Graphing from standard form

- Solving using quadratic equation

- Discriminant

Analyzing quadratics (second differences)

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If the first difference is same then it's linear equation but, if the second differences are same then it's quadratic equation.

If none of them are same then it's neither

Introduction to Parabolas

A parabola is basically a graphed version of a quadratic equation. When u graph the equation, the shape looks like a Curve. Which in Math is called a Parabola.

A Parabola can either open upwards or downwards depending on the A value. If the A value is a positive number then the parabola will open upwards and if the A value is a negative number then the parabola will open downwards.

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Main components of a parabola


Vertex - where the axis of symmetry and parabola meet

The axis of symmetry - divides the parabola into two equal halves

Optimal value - the y co-ordinate of the vertex/ highest or lowest point

X- intercepts - the zeroes

Y- intercept - when the graph crosses the y-axis/ y ≤ optimal value or y ≥ optimal value

vertex

A vertex is a maximum or minimum value of a parabola

X- intercepts/Zeros

Zeros is an another name for X- intercepts of a Parabola. The easiest form to find zeros is the factored form.

optimal value

Optimal value is an another name for y-intercept of a parabola. Also known as a maxmimum value or height of a parabola

Axis of Symmetry

Axis of Symmetry (AOS) is basically taking average of the two numbers or in this case taking average of the two x-intercepts.

Formula of AOS : A+B/2

step pattern

In this video it briefly explains how to graph parabolas using step pattern
Graphing Quadratics Using Step Patterns

Desmos!! It's a great app to check your parabola answers on your electronic device

quadratic Transformations

Horizontal Translation - It is a type of translation in which the parabola can be moved up or down depending on the value of "K"


Vertical Translation: It is a type of translation in which the parabola can be moves right or left depending on the value of "H"


Vertical Stretch/Compression: vertical stretch is when the parabola stretches or expands. it is only possible if the "A" value is bigger than 1. Vertical compression is when the parabola gets thin and small. That happens when the "A" value is less than 1.


Vertical Reflection: A vertical reflection is when the parabola flips or reflects vertically

3 Different Types of Quadratic Forms

1.Vertex Form - y= a(x-h)^2+k

2.Standard Form - y=a(x-r) (x-s)

3.Factored Form - ax^2+bx+c

Expanding (factored form)

Learning how to expand

Graphing parabola's in vertex form

Graphing parabolas in vertex form

Graphing Parabola's from Standard form

Graphing Parabolas in Standard Form

Graphing parabola's from factored form

Graphing Parabolas in Factored Form y=a(x-r)(x-s)

common factoring

this is the easiest factoring method
Factoring Trinomials (A quadratic Trinomial) by Trial and Error

factoring simple trinomials

Factoring trinomials with a leading 1 coefficient

factoring complex trinomials

Factoring Complex Trinomials - Organized Trial and Error

converting vertex and factored form to standard form

this video below explains the connections between these 3 quadratic equations : vertex form, factored form and standard form.
Discriminant of Quadratic Equations

Completing the Squares

Quadratic Formula

Quadratic formula can only be used for quadratic equations
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watch this video below to learn how to solve quadratic equations using quadratic formula

Discriminant

discriminant is basically the number in the square root in the quadratic equation.
Discriminant of Quadratic Equations

word problems

word problem #1
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word problem #2
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word problem #3
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word problem #4
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You can also watch this video to understand the steps more clearly incase you misunderstood something
This video explain some of the similar word problems we did in this unit and on tests
How to Solve Word Problems Using Quadratic Equations

Reflection of the unit!!

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In the past 6 weeks, in this unit of Quadratics i learned a lot of new stuff such as quadratic equations, graphing the equation which then look like a parabola. I also learned how to find the vertex of a parabola, AOS, x-intercepts, etc. I also learned about the quadratic formula, completing the square, prefect square, etc. To be honest all these topics are not that hard its just that you need a lot of practice specially with word problems. For me the application part is holding down my mark, so i started to practice application part such as word problems a lot, so that i can boost up my mark. Due to all the practice i did and did my homework everyday today on my third unit test of quadratics i was easily able to solve all the word problems except some mistakes. Final suggestion : MATH IS A SUBJECT WHICH YOU CANNOT UNDERSTAND WITHOUT PRACTICE. The more practice you do for math the more better it is for you.