Quadratic Relations

By: Manmeet Chatwal

Summary of website

Throughout this website, I will summarize the entire quadratics unit! I will teach you all the topics throughout the entire quadratics unit using a variety of different videos, math questions (knowledge, application and communication based), variety of different media (text, photos and videos).

Table Of Contents

TABLE OF CONTENTS

Understanding a Parabola

-What is a parabola?

-Analyzing a parabola (second differences)

Types of equations

Vertex Form

- Understanding the step pattern

- Describing Transformation

- Graphing vertex form x

- Isolating for a variable

-Finding optimal value

-Finding axis of symmetry.

Factored form

What Is A Parabola

A parabola represents a quadratic equation graphed. A parabola takes shape of a aligned U shape. A parabola can be open in the direction of up or down depending on the equation.
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A parabola can be compared to a guy shooting a basketball, the arch of a basketball in air represents the symmetrical shape a parabola forms. Parabola's have many different properties which are essential to learn for doing well in this unit!

Parabola use in real life

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Labeled parabola

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Properties of a parabola

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FIND OUT IF AN EQUATION IS QUADRATIC

when using a table of values, you can then tell if the equation is quadratic or not. A similarity all quadratic equations have is that the y values have the same second differences , and the first differences are unequal; compared to linear equations which have the same first differences.

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VERTEX FORM EXAMPLE EQUATION

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FINDING AXIS OF SYMMETRY IN VERTEX FORM

To find the axis of symmetry (y=a(x-h)+k) in a vertex form equation you need to look at the h value as explained briefly before. If the (h) value is positive you will simply need to change the sign, and what the value then becomes is the axis of symmetry.


For example:

y= 2(x+3)+4

----a----h----k

In the equation the value of h is +3, so when the sign is flipped the axis of symmetry is -3.

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THE STEP PATTERN

The step pattern is used to create the u-like shape of the parabola; the step pattern is as follows


Over 1 up 1

Over 2 up 4

Over 3 up 9


Note: *When using the step pattern we just square the number we are going over by to find the number we go up by

Note: *When there is a whole positive/negative: number, fraction, decimal in place of variable a value you just simply multiply the a value to the squared part of the step pattern

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BREAKING DOWN AND UNDERSTANDING THE EQUATION & TRANSFORMATIONS

After you plot your vertex using the transformations explained you can use the step pattern to plot your vertex form equation!


Term Representations

-The Y variable represents the Y-intercept

-The A variable represents whether the parabola opens up or down (positive a value results to parabola opening down, negative a value results to the parabola opening down), it also represents if the parabola will have a compression or stretch

- The X variable represents the x intercept

- The H variable represents where the X point is on the graph (right or left) (axis of symmetry) (If h is positive 6 the parabola will be horizontally translated 6 units to left)

- The ^2 creates the arch shape of the parabola

- The X variable is in charge of whether the parabola will be vertically shifted up or down (according to positive or negative sign) (positive K value will shift parabola up and negative K value will shift parabola down)

UNDERSTANDING TRANSFORMATIONS

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Y=a(x-h)+k

With all variables at 0 the parabola would be at the origin at 0,0


When a equation is in vertex form each letter variable is in charge of some type of transformation.


- The (-h) represents whether the parabola moves left or right. *RECORD (When the h value is positive the parabola moves left and when the h value is negative the parabola moves right). This is called a horizontal translation.


- The (a) value stretches or compresses the parabola. If the (a) value is greater than one the parabola will stretch by the factor of the worth of the variable; This is what is called a vertical stretch. If the (a) value is less than one the parabola will compress by the factor of the worth of the variable; this is called a vertical compression.


Also, the H value and K value make up the vertex


- The (k) variable represents whether the parabola moves up or down. *RECORD (When the k value is positive the parabola moves up, when the k value is negative the parabola moves down). This is called a vertical translation.

FINDING OPTIMAL VALUE OF A STANDARD FORM EQUATION.

The optimal value in a standard form equation is entirely dependent on the equations k value. If the k value is positive it will go above the x axis. If the k value is negative it will go below the x axis. *NOTE: Do not be confused, we do not flip the signs in order to find the optimal value.


For example:

y= 2(x+3)+4

----a----h----k

Therefore the k value is positive 4, meaning the optimal value of the parabola will be +4 over the x-axis.

FINDING X INTERCEPTS/ZEROS in vertex form

We isolate for the x variables to find the values of the x-intercepts in a vertex form equation. The next following steps will teach you to find the x-intercepts of a vertex form equation!
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put in an example for finding x intercepts here

GRAPHING VERTEX FORM

Graphing from Vertex Form

FACTORED FORM

In factored form we will learn how to factor a variety of equations into factored form equations; throughout the unit this part is the most tough but most important topic of the unit. It is very important that you learn this part properly and you completely understand it!


common factoring: is when you find the factors or two or more numbers , and they both get some factors that are the same, which means they are common


expanding: expanding means we remove the brackets and what ever is inside them by multiplying, which is called the parentheses


grouping: means to group terms with common factors before factoring


simple trinomial: a trinomial is a polynomial with 3 terms


complex trinomial


special factoring

FINDING ZEROES IN FACTORED FORM

The zeroes of a parabola are called the parabola's x- intercepts, and to find the x- intercepts you want to use zero for the y-value, In other words you have to be compelled to take the co-efficient's within, and change the sign which would be your zero.

FINDING AXIS OF SYMETTRY IN FACTORED FORM

To find the axis of symmetry you need to use the zeroes we just learned to find and find the average of them both. To find the average you will add both a and b values and divide them by two.


For example: X int @(8,0), (-6,0)

(a+b)/2

(8+-6)/2

2/2

also the number one would be the Axis of symmetry because after you average the x intercepts 1 is the outcome meaning it is also the axis of symmetry.

FINDING THE OPTIMAL VALUE FOR EQUATIONS IN FACTORED FORM

To find the optimal value in factored form, you have to sub the axis of symmetry as the x value in the original equation, which the allows you to solve for y (max/min).

STANDARD FORM= ax^2 + bx + c

THE QUADRATIC FORMULA


The quadratic formula allows us to solve for the zeros/x-intercepts and also allowing us to find the axis of symmetry by utilizing standard form equations. Following this further, if the equation was to be in a different form, it can simply just be expanded into standard form anytime, and then use the quadratic formula to solve for it. Below there is a picture of a quadratic equation, which we will also learn to apply.
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SONG ABOUT THE QUADRATIC FORMULA

Quadratic Formula Song Rockford Christian

FINDING X INTERCEPTS OF STANDARD FORM EQUATIONS USING THE QUADRATIC FORMULA

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Quadratic equation : Using the Quadratic formula to find the x-intercepts
The video explains what I have taught from the lesson in the pictures above. This video is great for people that learn best through listening.

FINDING THE AXIS OF SYMMETRY OF A STANDARD FORM EQUATION

In order to find the axis of symmetry of a standard form equation we take both x intercepts (found from the quadratic formula) and add them together and divide by 2, which is also known as finding the average. But we can also use part of the quadratic formula to find the axis of symmetry (-b/2a). If correctly do both of these methods you will get the same axis of symmetry.



example: let's find the axis of symmetry for the equation we found x-intercepts for above (4x^2-5x+1).


Method 1 (adding both x intercepts and divide by 2)


Axis of symmetry: (1+0.25)/2

=0.625


Method 2- Using -b/2a (variables were identified the in previous method above)


Axis of symmetry: -b/2a

=-(-5)/2(4)

=5/8

=0.625


Therefore we used both methods to get the axis of symmetry of a standard form equation and we managed to get the same answer from both methods proving that our answer is correct.

FINDING THE OPTIMAL VALUE OF A STANDARD FORM EQUATION

The axis of symmetry we just found above, we know apply it as a variable value. To do that, you simple just sub in the values and solve, in order to find for a optimal point.


With the example used above for explaining how to calculate x intercepts and axis of symmetry, we will also use the same example to explain how to find the optimal value.


Equation being subbed into: (4x^2-5x+1=0, X=0.625 (Axis of symmetry)


4(0.63)^2-5(0.63)+1

= -3.75


(I rounded 0.625 to 0.63 in example below just incase you never knew or got confused)


Therefore you have just figured out your optimal value using few simple and easy steps.

DISCRIMINANTS

So now that we know the quadratic formula, we can now concentrate on finding discriminant's. Discriminant's just inform us on the information we did not know about the standard form equation.
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As you can see the b^2-4ac term within the top part of the equation where it is being square rooted. The value that is being square rooted, before being divided by 2ac is the discriminant. The discriminant tell us the number of solutions it has within the equation. The value of the square term being squared rooted is:

  • negative, then there is no solution
  • if it is equal to 0 there is only 1 solution
  • if it is greater than 1 there is 2 solutions.

Finding the discriminant of a quadratic equation

COMPLETING THE SQUARE

Completing the Square - Solving Quadratic Equations
this video shows above talks about completing the square and shows us examples on how to do it too.

A CHART TO HELP YOU UNDERSTAND BETTER

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COMMON FACTORING

3.7 Common Factoring
the video above talks about common factoring and also tell us how to common factor

FACTORING SIMPLE TRINOMIALS

Factoring Simple Trinomials
this video above talks about factoring simple trinomials and shows us great examples in order for us too understand

FACTORING COMPLEX TRINOMIALS

3.9 Complex Trinomial Factoring
this video above talks about factoring complex trinomials and shows us great examples in order for us too understand

DIFFERENT OCCURRENCES OF FACTORING (DIFFERENCE OF SQUARES AND PERFECT SQUARES)

Factoring Perfect Square Trinomials and the Difference of Two Squares

EXPANDING FACTORED FORM EQUATIONS TO STANDARD FORM

Factored form of a quadratic equation makes life much easier when trying to find x intercepts/zeros; but sometimes you might have to expand a factored form equation to maybe simplify a equation. Expanding factored form equations includes the distributive property of numbers. When you are done then you can expand a equation from (x+4) (x+5) into =x^2+9x+20 which is also know as expanding the equation into standard form. Below i have included a picture from online on how to expand factored form equations into standard form equations.
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both methods will provide you with the same answer, if not then please go back and correct your mistakes
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My application part of the test was exceptional i got 9/10, which was is really good for me and the application part is the second picture above. but the knowledge part was not so good. i got 20/ 27. I over thought some of the questions which i found the correct answer the first time i tried the question. Secondly, I also made silly mistakes on the test which cause me to get a lower mark. In the first picture was basically the main part of knowledge , and the second picture was the application part

revenue word problem video made by me

in this video i show you how i solve my revenue word problem using simple steps, i have included a video and pictures of the problem and solution after my video
Video 1
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fencing word problem video mad by me

this is a video of me solving a fencing problem, i have showed step by step how i have solved the problem, i have also included pictures of my problem and solution after the video
Video2
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MY REFLECTION ON THIS UNIT

Overall, I found this unit very fun, exciting as i got to learn so many new things i didn't know. But, at times I found it hard at times because i didn't understand few things. Because this unit was a very long unit it was important to do your homework up to date, some of the I found difficult to understand was factoring, and this was due to my lack of completed homework. But thanks to Ms. Molyneux, a great teacher, learning was a lot easier and understanding the unit was easier too! but some of my suggestions are to complete your homework at all times, and ask questions about what you don't understand, as Ms. Molyneux is always more than happy to answer them and help any student whenever they need help! The Quadratics unit was finally a unit where we were able to connect in real life; making it so interesting. I found that we can apply concepts learned in the unit in real life for example: revenue word problems how they relate to real life business a lot. Even though I feel as though i could have done better in this unit, no body is perfect; therefore improvements can be made! One thing I encourage everyone to do is complete your homework, it gives you a better idea of what you are learning and an opportunity to ask any questions which you found difficult. Also in my quadratics test 1, as shown above, i did okay overall. Although I did very well during application I struggled in knowledge, though I am not proud of this section, we learn from our mistakes! I set a goal to review the whole quadratics unit and review factoring, to make sure i know everything before any big test or assignment

CONCLUSION

In conclusion you got to learn many different things during this quadratics unit. I will always encourage you to keep up to date by completing your homework and paying attention and asking questions in class if you do not understand one concept or another. I hope the concepts/terms I showed you through a variety of media use (pictures, text, videos) helped you understand this unit! I would like to thank you for your time!

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