Complex algebra known as QUaDRATiC

By Thrisha balakrishnan

What is quadratics?

Quadratics is simplified into three categories to help us students to understand quadratics better ,so therefore the categories are quadratics relations ,quadratics expressions and equations ,so it's categorised by just knowing the basic to solving quadratics relation unit and then goes on to using our tiny little brain for the quadratics expressions and equations.
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To start off ,quadratics relations is the first thing I'm going to teach because without a base we can't get into to the harder parts such as the quadratics expression and equation .

So quadratics relation is just as what you may have learned last year in grade 9 or in your previous years , so if you have remembered informations about the unit where it starts talking about the y=mx+b , y- intercepts ,x-intercepts and etc... It's the same topics with a little more details and it goes more depth into to each topics ,so here we go

TABLE OF CONTENTS

Topics we will touch up on :

Quadratics relation part 1

  • Parts of a parabola
  1. X-intercepts/zeroes/roots
  2. Axis of symmetry
  3. Optimal value(maximum or minumim)
  4. vertex
  5. openings (up or down)
  • Quadratics relationships/relations
  • Graphing in vertex form
  • Transformation of quadratics
  • Word problems :
  1. Finding an equation in vertex form when given a vertex and point

PARTS of PARABOLA

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A parabola is a curve formed by the intersection of a cone with a plane parallel to a straight line in its surface : a curve formed by a point moving so that its distance from a fixed point is equal to its distance from a fixed line

You're probably confused looking at the picture and looking at the words so here I will explain to you

  • The ZEROES can be called the x- intercept or roots ,so when the parabola is drawn it crosses the x-axis that is representing the x-intercept and also there could be 0 so it doesn't crosses the line and it could be below or above the x-axis , 1 when its facing upward or downwards it crosses only one spot in the x-axis or 2 when it touches two spots on the x-axis
  • The parabola can be opened /facing upwards or downwards depending on the equation( shown as a smile or frown)
  • The axis of symmetry is a invisible line in the picture that divides the parabola into two equal halves
  • The optimal value is the value of the co-ordinate of the vertex which could be expressed as the highest or lowest point (maximum and minimum )
  • The vertex of the parabola is the point where the axis of symmetry and the parabola meet .It is the point where the parabola is at its maximum or minimum value .
  • The y- intercept of a parabola is where the graph crosses the y -axis .
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This parabola is facing upwards since the a value was a positive number
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This parabola is facing downwards since its a value is a negative number
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Analyzing quadratics relation / relationship

We use table of values to help us identify the the quadratic realtionship by its second differences like we learned last year in grade 9 when we used the same chart to identify a linear relationship so if the fist difference was equal it would be a linear realtionship but for quadratic we will have to do an extra step if the fist difference was uneqal by finding the second difference to state if the quadratic realtionship is constant but if it's not a a constant relationship in linear nor quadratic it's neither .
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Transformation of quadratics

What you need to know about vertex from equations y=a(x-1)2+k


  • A is is the one controls on many parts such as if it has a negative sign then opens down it gives us a frown ,if it's doesnt have any symbols it's positive so therefore it will open upwards giving a nice happy smile ,then if it's negative A it reflects the x-axis ,A is greater then 1 its a VERTICAL stretch and if the A is less than 1 ( fraction ) it's a VERTICAL compression
  • H is the one that moves on the the x-axis I'm mathematical term the horizontal shift where it moves right or left on the x-axis . Right if it has negative symbol in front of the number and left if it is a positive number .
  • K is the ne that moves up and down on the y-axis ,we call it the vertical shift to be more consisted this the tranformation that controls the dot from going up and down so if it has a negAtive sign it goes down and if it has a positive it goes up .
  • The vertex (H,K)
  • Another importAnt information is on a graph always ,always compare to y=x2 because it's the base quadratic .
  • After finding the vertex you ,that not all you also have to find other points and u do that with something we call "the step pattern ".

Transformations on a graph (from y=x^2 to vertex form equation)

Transformations on a graph (from y=x^2 to vertex form equation)

There are 3 main things you should know while, identifying the transformations if an equation/parabola; and the way to know it , is by remembering a simple acronym

"S-R-T" .

The " S " stands for stretch/compression, you would explain how much of a vertical/horizontal stretch or compression the parabola has.

The " R " stands for reflections, so here you would state whether the parabola is upwards facing, or downwards facing, and whether or no is has a reflection.

The " T " stands for translation, which means how far has the parabola moved horizontally (left or right) and vertically (up or down).

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Graphing from vertex form

Graphing form vertex form

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

Graphing from vertex form is a way of graphing the vertex form equation y=a(x-h)+k on a graph ,so the steps to graph any equation is by following these steps:

  1. Take out the vertex , how we do that is by taking out the h and k value but when taking out the h value we need remember an important rule that is to change the original sign. To its opposite so for example if we had y=(x-6), it would be positive 6 instead or if we had y=(x+3), would be negative 3 .
  2. Then , you would take the (h,k) plot it and use the Transformation rule that is written above which is the H value is the horizontal shift left to right depending on its symbol and the K value is the vertical shift moving up or down depending on its symbol
  3. So then we can't just hand in the graph with only one point we need at least 4 ,but the quesstion comes how do we get the other 3 points ,it's by using "the step pattern " . The step pattern is basically if there's no a value ,is for the first Pint from the vertex you move over 1 and up 1 then on the other sire u do the same over 1 up1 .than for your 3 point you would go over 2 and up 4. We got this formula by base quadratic ( y=x^2) in the table of value chart ,but if we had a a value you would just multiply the step pattern with watever the a value is.
  4. Then connect all the dots with a curve and draw the arrows on at the end of the curve

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Finding equations of the parabola with the given vertex . Y=a(x-h)+k

The vertex of a parabola is @ (-2,8).one of the x-intercepts (-2,0). What is the equation of the parabola?
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Now we're going to learn about the second part to quadratics which is the expression part.

This part of quadratics it uses information from the relation and makes into expression which will help this and the last part of quadratics( the quadratics equation ) .

Table of contents

Topics we will touch up on

Quadratics expressions part 2

  1. Expanding binomials
  2. Multiplying polynomials ,common factoring
  3. Factoring by grouping
  4. Factoring simple trinomials
  5. Factoring complex trinomials
  6. Special factoring cases(perfect squares trinomials and differences of square)
  7. Graphing in factored form

Factored form


The form of an algebraic expression in which no part of the expression can be made simpler by pulling out a common factor.

ex. The factored form of the expression x3+x2-2 is x(x+2)(x-1).


Expanding binomials is really simple when you follow the following steps :

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Multiplying polynomials ,common factoring

What this means is taking a given equation x^2+bx+c in this format and making into a factored form so something like this (a+b)(a-b). Two important rules you 'll need to know is that x^2+bx+c, 2 number multiplied equals c and that two same number added equals

Bx ( note: two negative # equals positive #,1 negative and 1 positive equals a negative #

This is one way to transform standard to factored form

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Another example could also be bigger numbers where you 're able to factor out a common number like for example
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Factoring by grouping

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Factoring simple trinomials

Basically simple trinomials is the same as factoring this form x^2+bx+c into a factored form (a+b)(a+b),you might wonder why am I telling you this part again but I need because this is part of a another lesson so what special about this is that it always has a 1 as its a value .no munger greater or less than 1

Factoring complex trinomials

This the factoring that is more depth then factoring simple trinomials which might get a little confusing so pay attention :
3.9 Complex Trinomial Factoring

Factoring special cases

There 2 special cases you will need to pay attention to but they're really easy to figure it out like the first one is just perfect squares so when using tiles in a chart u should be able to see a perfect square ,it's just a factore. Multiplied by 2 times so for example :(x+2)(x+2) would also be (x+2)^2
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Factoring difference of squsares

And now we're going to leans the second special case which is "differences of square", in this special case what we do is simple . So remember from your previous years of learning that "difference" means "subtraction". So a difference of squares is something that looks like x^– 4. That's because 4 = 2^,so you really have x^– 2^, a difference of squares. To factor this, do your parentheses, same as usual:

    x^– 4 = (x )(x )

    You need factors of –4 that add up to zero, so use –2 and +2:

      x2 – 4 = (x – 2)(x + 2)

Graphing in factored form

Graphing in factored form
3.5 Graphing from Factored Form

Table of contents

Topics we will touch up on

Quadratics eqaution part 3

  1. Completing the square
  2. Solving from vertex form
  3. Quadratic formula
  4. Discriminant
  5. Graphing in standard form

Completing the square

So until this point we have learned how to convert one from into another like for example from standard to factored form or factored to standard . Now we 're going to learn how to convert standard from into vertex form
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Solving from vertex form

Before this we have learned how to convert standard from to vertex form but we haven't learned how to solve an equation to find the x-intercepts and y-intercepts and etc... So here we go
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Quadratic formula /eqaution

This is another easier form to solve standard form by using a formula but only when the equation is not factorable,here is more information on it
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Final topic the discriminant

This is a form which you need to know if your going to use the quadratic formula ,you will know when I tell you now
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Graphing in standard form

Graphing in standard form

Different types of word problems

  1. Geometry (area/perimeter/shape)
  2. Motion
  3. Revenue #1
  4. Revenue #2
  5. Number

Geometry( area/perimeter/shape)

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Motion problems

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3.12 Motion problems

Revenue #1

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Revenue #2

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Number

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Connection between topics of quadratics

Standard form and vertex form - These two forms are capable are transporting one form from another form because they both have similar functions that helps it connect . Such as the a ,the b and c

Equations and graphing - all three equations we have learned in this unit are able to graph in different ways by their own equations by using the same rules such as by using the vertex ,the 2 x-intercepts and y-intercepts and the axis of symmetry

The Quadratic Formula and Solving in Vertex Form - both these type of solving vertex form, help the vertex form find the 2 x-intercepts /zeros/roots

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Reflection about a piece of assessment and overalls about this unit

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My assessment was almost all well done except for my first test I did well on every part such as the knowledge and application but I didn't do well on the communication but I know what we're the mistakes done and I have learned it .So if I was a little more careful with the wordings I could of passed that part too .Overall about this unit is that this unit was different from units I have done in my previous years because this was separated into 3 parts which had a lot of details to remember and it had a lot of difficult and easy parts to remember but one thing I learned was that if you did the homework and practised it several times it's not hard because i had experienced it when I thought the quadratic formula was so hard but I did several practises and watched different videos on it and now it's my favourite part .another part I had difficult was on optimization problems ,it's not like the quadratic formula case where I learned and I liked it but it's still not my best and favourite part.i also learned that several topics such as the graphing was my best but for others it was the opposite but I think maybe it's my best because I like graphing and I took a lot of time practising it .so now this is the end of my websites to tell you the truth this was my favourite unit in this one year of learning math in high school .