Quadratics

By: Mohit Devassar

Introduction

Welcome to my website this website should help you better understand the concepts of quadratics. Most people struggle with quadratics this website is made to help those and to break down the different things of quadratics.

Table of content

Parabolas

- Step Pattern

- First differences

- Second differences

- Vertex Form

- Optimal value

- X-intercepts/zeros

- Transformations

Factoring

- Trinomial Factoring

- Binomial factoring

- Algebra Tiles

- Factoring by grouping

- Factor form

- Perfect squares

- Square differences

- Standard form to factored form

Solving

- Standard form to vertex form

- Quadratic formula

- What is the quadratic formula

- Discriminant

- Solving from vertex form

- Solving from standard form

- Word problem

- Reflection

Parabolas vertex form

Step Pattern

The Vertex is the point you start off at and it is at the point of (0,0). Then you follow the step pattern which is 1 right 1 up which would be one point and the next point would be 2 right and 2 up. After that you repeat the same steps on the left side of the vertex and you have your basic Y=X(squared) parabola and why Y=X(squared) basically means your X value is to the power of 2 of your Y value. EX: is your Y is 4 than your X value would be 16.

Graphing parabola in vertex form

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First Differences

You find the first differences by subtracting the Y values and if the first differences are the same then it is linear.
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Second Differences

If the first differences are not the same then you would start to find the second differences and if the second differences are the same then it is quadratic.
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Vertex Form equation

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Vertex Form

From vertex form you easily find your vertex by looking at your h and k value where h represents the x value and k represents the y value once you graph that point, that will be your vertex.

Axis of symmetry

What is the axis of symmetry: The axis of symmetry splits the parabola into 2 equal parts and crosses the vertex. To find the axis of symmetry you would have to look at the H value in the equation and it tells us where the vertex would be going left or right. If it is negative then you would go to the right on the x - axis and if it is positive then you would go to the left side on the x - axis.

Optimal Value

The optimal value is the absolute last value in the equation or in other words the it is the K value. The K value tells us where the vertex is going to be vertically and when you combine the H and K value you get your vertex point of the parabola. If the K value is positive then the point will be above the X - axis and if the K value is negative then the point will be below the X - axis.

How to graph with Vertex form

You would use the H and K value to find the vertex of the parabola then use the normal step pattern to find your other points but if your A in the equation has a value then you would multiply your step pattern by that number and get your points from there.

X - intercepts/zeros

To find your zeros you would need to solve for it or isolate it. You set Y to zero then solve for X


Example:

y = 2(x+2)^2-8


0 = 2(x+2)^2-8

8 = 2(x+2)^2

8/2 = (x+2)^2

4 = (x+2)^2

+-2 = x+2

-2 = x+2

-2-2 = x

-4 = x

Transformations

y=a(x-h)^2+k is the original/general equation for quadratics.

The A value controls if the parabola open up or down.

The H value controls the horizontal movement.

The K value controls the vertical movement.


All of these values must be used when using transformations.


If the A value is greater then 1 then the parabola is being stretched and if it is smaller then 1 then the parabola is being compressed and if it is 0 then nothing changes.

If the K value is increased then it would move up by that many numbers and if it is decreased then it would move down by that many numbers.

If the H value is increased then the it would move left by that many numbers but if it is decreased then it move right by that many numbers

FACTORING

Algebra Tiles

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Factoring Binomials

2x+4

= 2(x+2)


I got this by finding the greatest common factor (GCF) and putting it to the side and dividing everything by the GCF which got me what was left in the bracket. You can check if your answer is right by distributing the 2 back in the bracket. It should equal back to the original equation.


Check:

2(x+2)

= 2x+4

Factoring trinomials (complex) (simple)

Complex:

2x+4x+16x

= 2x(2+8)


Simple:

6x+15

=3(2x+5)


The process is the exact same as factoring Binomials but there is just three numbers that is why it makes it trinomials. A simple equation does not have a a value. (x+4x+16x) but you would factor it the same way.

Factoring by grouping (simple) (complex)

Complex:

2x(x+3)+5(x+3)

= (2x+5) (x+3)


Simple:

x(x+2)+4(x+2)

= (x+4) (x+2)


All you have to do is find something common like (x+3) and (x+3) and make them represented by one binomial instead of two and then just group your remaining numbers in this case it was (2x+5) and then make it into a factored form equation by putting them beside each other like in the example.

Factoring standard form

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Factored Form y=a(x+a)(x+b)

Factored form gets its name because if you were to factor a standard form equation the end result would look like y=a(x-a)(x-b) that is why this is called factored form

Factoring Perfect squares

A perfect square is any trinomial that follows a^2+2ab+b. Another way the perfect square equation can look is ax^2+b^2. If this is the form, make sure all coefficients are square rooted. An example to this would be turning 9x^2+81 into 3x^2+9^2.

Differences of Squares a2-b2

This is similar to perfect squares but a little different. Difference means subtract so there is not +2ab in the equation and it is just a2-b2. To solve this you need to put in the values of your 2 squares in the equation and just subtract them to get your difference of the squares. (There also can be binomials and trinomials)

Standard Form to Factored Form

This can get pretty difficult to understand but is possible. Just remember you have to get from y=ax^2+bx+c to y=a(x-a)(x-b). This very easy to do but still challenging. This is basically like trial and error you have to find numbers that multiply to your c and a value and add up to the middle number which is bx and once you have that, you have your factored form

Factored form to Standard form

All you have to do is use the distributive property and then collect like terms which will get you your standard form equation.
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Solving

Standard form to vertex form

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Quadratic formula

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What is the quadratic formula

The quadratic formula works all the time and it is used to find the two X-values, and then to find the vertex of the parabola.

Discriminant

The Discriminant is the number inside the square root of the equation.

1) If the Discriminant is negative then there is no solution.

2) If the discriminant is 0 then there is 1 solution.

3) If the Discriminant is positive then there are 2 solutions.

Solving from vertex form

From vertex form you already have your vertex and all you need is your 2 X-values and to do that, all you have to do is isolate your X and once you have your X just solve for Y.

Solving from standard form

For this you basically gotta complete the square, you would use the quadratic formula and sub in the values that are in the equation and find your 2 zeros/X-values and then you would add them up and divide by 2 and that will you give you your X-value for the vertex and to find the Y value you just sub in the the X value you got into the vertex and solve for Y

Word problem (example)

You want to build a fence but one side is covered by a building and you have 500m of fencing.


a) what are the dimensions of the sides?


b) what is the maximum area?

Word problem (solution)

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*Quality may be bad of the video(s)*

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Reflection

This unit was really hard and confusing for me in the beginning but once i got use to it and started understanding the concepts of quadratics, i knew what i was doing when it came to Parabola's, factoring and solving.

Test reslults

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