Quadratic Unit Assignment

By:Tharakaa Rajakumar

Definition Of Quadratics:

The name Quadratics comes from the word: Quad meaning square because the variable gets squared. Quadratics is also used in our everyday life.

Table Of Contents

Introduction

1. Quadratic Relations
  • Second differences

2. Introduction To Parabola

  • Parabola's Labels
  • The Parabolic Definitions

3. Transformations Of Quadratics

  • The Base
  • The Step Pattern


Types Of Equations

1. Factored Form
  • Expanding/Distributive Property to Standard Form
  • Difference of Squares
  • Perfect Squares
  • Graphing Factored Form
  • Common Factoring
  • Factoring Simple Trinomials to Standard Form

2. Standard Form

  • Completing The Square Standard Form To Vertex Form
  • Grouping Standard form to Factored form
  • Complex trinomials standard form to factored form
  • Quadratic Formula
  • Discriminant


3. Vertex Form

  • Graphing Vertex Form
  • Solving For X
  • Solving For Y


Word Problems


  • Optimization questions


Connection To All The Quadratic Parts and Reflection

Types of Quadratic Equations

1. Vertex Form: y=a(x-h)² + k


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


3. Standard Form: y=ax² + bx + c

Quadratic Relations Using Second Differences

In order to know if it is a quadratic relationship you must have the second differences. This is how you make the chart to figure out if these points meet a quadratic relationship or a linear or neither. If it is a linear relationship it only needs the first differences that are constant. If there is a neither there is nothing that is constant in the first difference or the second difference.
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This is a quadratic relation because when i did the 1st differences the numbers weren't constant and so i did the second differences. When i did the second differences the numbers were constant. Therefore this is a quadratic relation.
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This is a linear relation because when i did the 1st differences the numbers were constant. Therefore i didn't have to do the second differences.
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Since the 1st and the 2nd differences don't have any constant number. It doesn't have a relationship therefore it is a neither.
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Defintion of Parabolic Terms

1. Vertex- The vertex can be the maximum or the minimum of the parabola depending on the direction of the opening. If the parabola open downward then it will have a maximum value (Highest Point). If the parabola opens upward then it will have a minimum value (Lowest Point).


2. Axis of Symmetry- Is the line that divides the parabola into two symmetric portions


3. X-intercepts- Points on the graph that intercept the x-axis.


4. Y-intercepts- Points on the graph that intercept the y-axis


5. Optimal Value- Highest y-value of a parabola.

The Step Pattern

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The Transformations

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Factored Form

Expanding/ Distributive Property

There are two ways to do Expanding In Factored Form to Standard Form in distributive property.

1. Simple distributive property is used when there's only a variable like x with no coefficients in front

2. FOIL in expanding for when there is a coefficient beside the variable x.

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The second way is using the acronym FOIL.3

F- First

O-Outer

I-Inner

L-Last

Ex.1 (6x +1) (x-3)

F- (6x) (x) = 6x²

O- (6x) (-3)= -18x

I- (1) (x) = x

L-(1) (-3) =-3


= 6x² -17x-3

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Perfect Squares

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Difference Of Squares

Basic Formula is: (a+b) (a-b)

a² - b² or (9x+8y) (9x-8y)

= 81x² - 72xy + 72xy - 64y²

= 81x² - 64y²

The (9x) multiplies with the other (9x) which becomes (81) and then the two (x's) square and as a total becoming (81x²). Then the (9x) multiplies with the (-8y) which equals to becomes (-72xy). Now, multiply the (8y) with the (9x) which equal to becomes (72xy). Lastly (8y) multiplies with (-8y) equalling to (-64y²)


Overall the standard form would look like this:

81x² - 64y²

Factoring a difference of squares

How To Graph Factored Form

This is another one of my videos. Hope this helps every one :)

https://www.educreations.com/lesson/view/how-to-graph-using-factored-form/27571040/?ref=link&s=iZ5QaU


Or click on the embedded link below.

Standard Form

How to Factor Complex Trinomials #1

This is a link to another one of my videos. Click on the link below. Hope you enjoy!

https://www.educreations.com/lesson/view/how-to-factor-complex-trinomials-1/27574555/?ref=link&s=sHj7Pt


Or click on the embedded link below.

How to Factor Complex Trinomials #2

This is another one of my videos that i made for Factoring Complex Trinomials

https://www.educreations.com/lesson/view/how-to-factor-complex-trinomials/27575797/?ref=link&s=aAHitk


Or click on the embedded link below.

Factoring By Grouping

This is my own video on how to factor with grouping.

https://www.educreations.com/lesson/view/grouping-final/27576038/?s=EqkFMm&ref=appemail


Or click on the embedded link below.

How To Use The Quadratic Formula

Quadratic formula is an easier way to find the x-intercepts of an equation when its in standard form. How to use the Quadratic Formula.
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Discriminant

In quadratics the Discriminant is used to determine whether the parabola has 2, 1, or 0 x-intercepts. The Discriminant is simply the formula with the number underneath the square root in the quadratic formula. The Discriminant can be calculated through the following: b²-4ac. Simply plug in the a,b, and c values. If the value is more than 0 then there are 2 x-intercepts. If the value is less than 0 then there are no x-intercepts. If the value is equal to 0 then there is one x-intercept which is on the x-axis.

Common Factoring

Common Factoring is when you take the common factor in the equation.

Example 1: 2x² + 8

Common factor between 2x² + 8x is:

2x² - (2) (x) (x)

8x - (2) (4) (x)

The common factors in this equation is: (2) (x)

So you put the 2x in front and then put a bracket and then divide 2x² with 2x... and 2 is left and then you divide 8x with 2.....and 4 is left


This is how the answer will look like:

2x(x+4)


This is common Factoring

Completing The Square

This is my own video on completing the squares

https://www.educreations.com/lesson/view/completing-the-square/27617938/?ref=link&s=MwtnC0


Or click on the embedded link below.

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How To Graph Vertex Form

This is another one of my videos on graphing vertex form https://www.educreations.com/lesson/view/how-to-graph-vertex-form/27619657/?ref=link&s=GNft3U


Or click on the embedded link below.

Word Problems

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Connections

The connection between quadratics 1 is the graphing it was similarly used to the part 2 of the quadratics factoring. They were similar because once you factored them you were able to get the x-intercepts, then find the AOS and then sub in the x value to find the y value and after doing this from the factoring you were able to graph it by putting the x intercepts as points and then use the AOS and the y-value making the vertex, to make the parabola. Quadratics 2 related to quadratics 3 because in quadratics 2 you would factor and once you factored you would get 2 of the x-intercepts. The same way in Quadratics 3 we used Quadratic Formula to find 2 of the x-intercepts and once x-intercepts were found we can basically isolate for the y to get the y-value to find the vertex. Altogether the quadratics unit were summed up in different forms like Standard, Vertex and Factored form.

They all used the graphing method by find the x-intercepts to find the vertex. The Standard form looked like: y=ax² + bx + c. You could use standard form to answer word problems like to find the maximum height or when the ball will hit the ground it etc. It is also used to for Quadratic Formula when as the middle value is b and the last value is c and the first value is a. Secondly Factored form looks like: y=a(x-r) (x-s). This formula is used for factoring all types like Simple and complex trinomials and etc. Lastly, Vertex form looks like: y=a(x-h)² + k. This formula is used for perfect squares and many more. But mostly these forms relate to the graphing because these forms are just showing different ways you can do to graph. Like Factored form finds two x-intercepts to graph and so does quadratic formula these forms relate to our understanding to make life easier. So whenever there is a hard question we use the easiest way to answer the question instead of the long way. For example, Quadratic Formula and Factored Form both solving for x. Depending on the form and the question you use either one applicable.

Reflection

In my opinion at the beginning of the unit i didn't like quadratics but as i started learn the different forms and factoring and using quadratic formula to find x-intercepts. I started to like Quadratics. In quadratics the most favourite part was factoring. Especially factoring by grouping. I scored really well on my test for the Factoring. The part I had a little struggle on was the word problems but as i started to put more effort on them at home . I started to understand how to do these word problems. As a also startedto have a better understanding on word problems in Optimization i liked the revenue questions. Overall, I think i did excellent throughout this unit.
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