Quadratic Star BY: AKASH K.

Learn everything to know about Quadratics!

What is a Quadratic Relation?

A quadratic relation on a graph would typically make a nice curve. The word quadratic, has the root word 'quad' which represents a square. The square is put with the variable (like x^2). This can come helpful when calculating or drawing out the curve of a ball or thrown object. It can also be used in the military and business fields when calculating maximum profit or when launching a missile. THESE ALL USE QUADRATICS!
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Real World Examples of Parabolas

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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

When you chart data of a quadratic relation, you would notice the first differences will not be constant, but the second differences will always be constant. This is different when being compared to a linear relation, because a linear relation has all of its first differences constant.
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This chart displays a relation of y=x^2. This is due to the fact that the first differences are different while the second differences are constant. Since all the second differences are all 2 then this is a quadratic relationship.

Graphed Quadratics- Parabolas

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


Content


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 "

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Transformations

There are certain transformations or 'movements' made by the parabola with the changing of values in an equation.


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

a(x-r)(x-s)

Standard Form

ax²+bx+c

Vertex Form

a(x-h)²+k

Vertex Form

a(x-h)²+k

Key Components

The vertex form helps to show the location of the vertex easily and also the vertical stretch/compression.


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

You can use different types of given information to generate a vertex form equation. For example, I can use the vertex and a point the parabola has to create an equation.
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Graphing From Vertex Form

Chart Base Parabola

Have a table of values and use it to plot the points onto a graph. Know that every base parabola has a vertex at the origin (0,0).
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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.

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Then you would use the vertical stretch/compression factor to figure out the step pattern. The vertical stretch/compression factor can also help to determine the direction of opening for the parabola.

Factored Form

Factored form, the product of a constant and two linear terms
y=a(x-r)(x-s)

Key Componenets

The Factored Form method helps to identify the 'zeros' (aka the x intercepts). In order to find the zeros, you must set y=0. The 'q' and the 'p' are the zeros.

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

The use of expanding with factored forms are to take a factored equation to a standard equation. For example, (x+2)(x+3), is a factored form expression. You can expand that to get, x^2+5x+6, which is a standard form expression. Below is a step by step sheet to expanding expressions.
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Common Factoring

Factoring only occurs when there is a common factor that can be divided from all the terms. It is recommended to find the 'Greatest Common Factor' and not just any factor. When you factor something out it goes out front of the brackets and the expression gets rearranged. Bellow is a step by step sheet to common factoring.
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Common Factoring

Common Factoring Binomials

Common Factoring Binomials is just like factoring out single terms. One you find a common binomial in the terms you can factor it out of the expression. A step-by-step example is shown below.
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Common Factoring Binomials

Factoring By Grouping

You can only do this if you have expressions with an even number of terms and pairs with common factors. This can come useful when you have a large expression with no overall common factor. A step by step example is shown below.
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Factoring By Groupong

Factoring Simple Trinomials

Form: x² + bx + c


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

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factoring trinomials

Factoring Complex Trinomials

Form: ax² + bx+ c



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:

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Perfect Squares

Form: a² + 2ab +b²

Factored: (a + b)²


Are Trinomials that have its leading coefficient and its the last term to be square rooted. When you factor perfect squares you get a term that gets multiplied by itself (Perfect Square).
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Factoring Perfect Square Trinomials - Ex1

Difference of Squares

Form: a² - b²

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

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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

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The Optimal Value

This is the highest or lowest y value of the graph


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)

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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

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Standard Form

ax²+bx+c

Solving Using The Quadratic Formula

The Quadratic Formula helps by using information from a Standard Form equation to find the zeros (x-intercepts)


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

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Using the Quadratic Formula

Discriminant

The discriminant is the, b²-4ac, part of the quadratic formula


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

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Factoring Standard Form (Standard Form To Factored Form)

This process aids in converting Standard Form equations to Factored Form equations


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

Completing The Square (Standard Form To Vertex Form)

This process helps to convert standard form equations into vertex form equations


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:

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Graphing Standard Form

In order to graph, you need three pieces of information:


  1. The vertex
  2. 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

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Types of Math Problems

Motion Problems

This includes problems about ball, throws, rocket launching, etc.


Some key features to look for:


The X-Axis


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

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Optimization (fencing problems)

Below I have given an example of an optimization problem where you are trying to figure put the dimension of the rectangle using the information given. I have showed two different methods which you can use while still getting the same correct answer.
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Geometric problems

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Economic Problems

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Number Problems

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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

At first, I was very comfortable with quadratics and its basic fundamentals of quadratics. I understood how to factor and expand easily. I took every day's time to study, practice, and do the workbook questions for the topic. I spent hard hours understanding and teaching myself the questionable topics. When I fully understood this topic I took my time to help others and answer questions they had.
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After I got the results back for this quiz I was very satisfied and proud of the mark i felt that I earned.


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".

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After this test, I felt very sorrowed and depressed with the fact that I have turned this way. I have made a promise to my self to try to be better and put more effort and pride into my work. The later topics, quadratics 3, I have taken a lot of time to understand and apply myself when I solve equations and solve difficult word problems and then do good on my website. In the future when learning trigonometry I will ask more questions and apply my self better in practicing and in difficult challenges to come out on top.