All About Quadratics

INTRODUCTION - WHAT IS QUADRATICS

The word 'Quad' in the word 'Quadratics' is Latin for squared, which explains why quadratic equations evolve around being squared or including '^2' anywhere in the equation. A quadratic equation is based upon the standard form of 'y=ax^2+bx+c'. Where the variables a,b & c are ONLY numbers. What makes the equation a quadratic equation is the variables b,c could be 0 while a cant.

A quadratic equation can be represented through what is a 'Parabola', which is a curved line created on a graph or represented through written form.

Is it a Quadratic Relation?

Finite differences can determine whether an relation is quadratic.

First differences are Linear.

Second differences are Quadratic.

Third differences are Non-Linear.

PARABOLA

As mentioned above, a 'Parabola' is a curved line that is represented through a graph. What makes a parabola, a parabola is how it is symmetrical. A parabola is used to indicate distance. For example, one could throw a basketball towards the net and calculate using any of the three forms; factored, vertex & standard formed equations the distance from where the ball was thrown to where it landed.


A parabola could either be presented in a downward or upward situation. When the parabola is downward, it has an optimal value in which it is called 'max'. The max point is basically the highest point of the curved line. Whereas, a 'min' is the lowest point of the curved line, which is shown with a upward parabola. The difference between them is that the downward parabola has a maximum value that you can physically see and figure out since the ends of the parabola ends at the x-axis. And with a upward parabola, you cannot see or figure out the highest point due to the ends of the curved line facing upwards leaving it to be infinite.


A parabola has many different parts to it, such as the 'vertex'. The vertex of a parabola is where the curved line crosses its axis of symmetry. For example, if the equation is : y = (x+1)^2-4 , you would indicate the vertex to be (-1,-4).


Another part to a parabola is the 'axis of symmetry'. This is basically an imaginary line the goes through the vertex or the highest or lowest point of the parabola. It is the separation to the two symmetrical pieces of the parabola. An example for the equation : y = 2x^2+1, the axis of symmetry will be (0).


There are also 'x-intercepts' & 'y-intercepts' in a parabola. The x-intercept is basically the 'axis of symmetry', it is where the parabola touches the x-axis. The y-intercept is indicated by sub x=0 into the equation. For example, with the equation : y = 3x^2-4 , you would plug in 0 to replace x and solve. The answer would come out to be -4 .


There is also a 'step pattern' which finds the next point of the parabola. In an equation that does not have a compression or stretch, (makes the parabola more narrow or wide) the step pattern is 'over 1 up 1 & over 2 up 4'. An equation could be : y = (x+1)^2-4. An equation that does not follow that pattern is : y = -4(x-3)^2+10. Since there is a '-4', you would need to multiple the "up 1, up 4' part by that compression or stretch. So it would be 'over 1 up 1x-4 = up -4, over 2 up 2x-4 = up -16'.

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Transformation

A dramatic change in an object. In math, the change of an object is the way it flips, slides, and reflects. In quadratics, a transformation is used to tell how the way a parabola moves.


Types of Transformations


Vertical Stretch/Compression:

-If negative, there is a vertical reflection and the parabola opens downwards.

-If the number is smaller than -1 to 1, it is considered an compression.


Vertical Shift:

-Represented by the value of 'k'.

-The movement of the parabola being up or down.

-If it is positive, it will be above the x-axis, if it is negative, the parabola will be below x-axis.


Horizontal Shift:

-Represented by the value 'h'

-The parabola moving left or right

-If it is positive, it will be on the right side of y-axis, and if it negative, it will be on the left side of the y-axis.


Example

y= -1/2x(x-2)^2-1


-The parabola will open downwards because the value 'a' is a negative number.

-The value of the number is the vertical translation. The vertical translation its '-1'.

-The value of this number is the opposite of the sign that is in the equation. The horizontal translation is +2.

VERTEX FORM

Vertex form is one of the three types of quadratic equation forms.

It is in the form of ' y = a(x-h)^2+k '.

The 'a' represents the compression or stretch of the parabola; (how narrow or wide the parabola opens)

The 'h' represents the horizontal shift; (how much to the left or right the parabola shifts from x=0)

The 'k' represents the vertical shift; (how much upward or downward the parabola shifts from y=0).


There are two types of vertex forms.

1) Without 'a' , ex. y = (x-3)^2+4

2) With 'a' , ex. y = -2(x+6)^2-9


When there is no 'a' that means there is no stretch or compression, and the step pattern stays as 'over1 up1, over2 up4'.


When there is a compression or stretch, the step pattern changes. Instead of it being 'over1 up1, over2 up4', you will multiple the up parts of the pattern by the stretch/compression.

So for the equation y = -2(x+6)^2-9. 'over1 up1x-2 = up-2, over2 up4x-2 = up-8'


How to solve for 'a' by given coordinates :

ex. (2,6) (5,3)

> y = a(x-2)^2+6

> 3 = a(5-2)^2+6

> 3 = a(3)^2+6

> 3 = 9a+6

> 3-6 = 9a

> -3/9 = 9a/9

> -1/3 = a

>> y = -1/3 (x-2)^2+6

Graphing Vertex Formed Equations

Watch video below to see a step-by-step process.


https://www.youtube.com/watch?v=y99lNRqLjBA

Factored Form

Factored Form is another type of form of a quadratic equations.

It is in the form of ' y = a(x-r)(x-s) '.

Multiplying Binomials

Multiplying Binomials is basically expanding binomial equations.

In order to do this, you would need to distribute the numbers.

Ex. (x+5)(x-4)

You would need to multiply each number by each other, so you would multiply x x x, x x -4, 5 x x & 5 x -4.

- x x x = x^2

- x x -4 = -4x

- 5 x x = 5x

- 5 x -4 = -20

After this, you would just put the whole thing into the equation = x^2-4x+5x-20

You would then simplify it by adding both -4x and 5x. So then the whole equation would then be = x^2+x-20.


Ex. (c-4)(c+6) - (c+2)(c-3)

- c x c = c^2

- c x 6 = 6c

- -4 x c = -4c

- -4 x 6 = -24

(c^2+6c-4c-24) = first half ... (c^2+2-24) = simplified


- c x c = c^2

- c x -3 = -3c

- 2 x c = 2c

- 2 x -3 = -6

(c^2-3c+2c-6) = second half ... (c^2-1c-6)


c^2+2-24 - c^2-1c-6 = now you would just subtract the two equations


c^2+2-24 - c^2-1c-6 = 3c-18

Factoring Simple Trinomials


Factoring Simple Trinomials is converting an equation that is standard form into factored form. What makes a simple trinomial, simple is that the 'a' value will always one.

e.g. x^2+5x+6

When factoring simple trinomials, the order of where you place the multiples of the 'c' value into the brackets do not matter. The reason is because since you are multiplying them by the 1 (the square rooted x^2), it'll always come out to be the same answer.


Watch Video below to see a step-by-step process.


https://www.youtube.com/watch?v=roGIlgbdbQk

Factoring Complex Trinomials

Factoring Complex Trinomials is very similar to Factoring Simple Trinomials but instead the 'a' value in the equation is greater than one.

e.g. 2x+9x+4

Now that the 'a' value is greater than one, the way you factor the equation relies on where you would place the multiples of the 'c' value into the brackets. It also depends on the different multiples of the 'a' value.


Watch Videos below to see a step-by-step process.


https://www.youtube.com/watch?v=kKL04c4ebMY&feature=youtu.be

Special Products; Perfect Squares & Differences of Squares


Finding Special Products


(2x+y)^2 = you would first expand this to be a binomial equation.

(2x+y) (2x+y) = then you would multiply the binomial equation to become standard form.

4x^2+2xy+y^2 = since there are two different variables, the second part will take both variables and the last variable will become squared (^2).


Perfect Square


Perfect Square is when the two binomials are the same "equation"

'SDPS' (Sam Doesn't Pull Strings / Squared, Double the Product, Squared).


Ex. (2x+y)(2x+y) > this is a perfect square because they are the same. A simplified way to write this is (2x+y)^2.


Watch the Video below to see a step-by-step process.


Differences of Squares


Watch Video below for a more detailed step-by-step process.


https://www.youtube.com/watch?v=XIaWKDuDjW0

Standard Form

f(x) = ax^2+bx+c


ex: f(x) = 3x^2+8x+9

Graphing From Standard Form

Watch Video below to see a step-by-step process.


https://www.youtube.com/watch?v=znAggG1UeBE

Discriminants

Discriminants are part of the quadratic equation that allows you to determine the amount of solutions there are in the equation.


The discriminant is the number inside the square root 'b^2-4ac'.


3 Different Solutions.


No Solutions

To figure out if there are no solutions in the equation, you would need to figure out if the discriminant is less than 0 'D=<0'. When the number is less than 0, you cannot square root it.


Ex. 0=3x^2+4x+5

As you can see, you cannot square root the discriminant, therefore this has no solutions.


1 Solution

When there is only 1 solution, the discriminant of the equation has to be 0 'D=0'.


Ex. 0=2x^2+4x+2

2 Solutions

An equation that has 2 solutions will need to have the discriminant to be greater than 0 'D=>0'. It will have 2 solutions because once you square root the number, it will give you two different answers; one positive & one negative.


Ex. 0=3x^2+4x-5

As you can see, there are two different answers; x=0.79 & x= -2.12.

Quadratic Formula

The formula is used when trying to find the x-intercepts in standard form.


Watch Video Below to see step-by-step process.


https://www.youtube.com/watch?v=iulx0z1lz8M

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Completing the Square

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Axis of Symmetry

The axis of symmetry is the x value in the vertex and the middle of the parabola. The axis of symmetry is important because without it, a parabola will not be complete.


How To Find Axis of Symmetry


If your equation is y=x^2+12+20.


y=x^2+12+20

(x+10)(x+2)

Once factored, next you have to zero the factors.

x=-10

x=-2

Once that is done, you simply add the two together and divide them by 2.

-10+-2=-12

-12/2=-6

therefore, x=-6 is your axis of symmetry.

Optimal Value

Optimal Value is the y value of the vertex and the minimum/maximum value in a parabola. It determine what the stretch or compression is of a parabola.


Equation: y=a(x-h)sqrt+k


Example:


Axis of Symmetry = x = -1

y= ^2-1x-2

y=(-1)^2(-1)-(2)

y=1+1-2

y=0

Graphing from Factored Form

Watch this video for a step-by-step process.


https://www.youtube.com/watch?v=PhoZOUwqWVY

Word Problems

Revenue Problems


Look at the side image for an example.


Area Problems


Watch the video for an example of an area problems.


https://www.youtube.com/watch?t=32&v=IGGnn9oa4QY

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Connection

Standard Form to Vertex Form:


When you are turning standard form into vertex form remember to use completing the squares.


y= -2x^2+12x+11

y= -2 (x^2 -6x+9-9) +11

y= -2 (x^2-6x+9)+18+11

y= -2 (x-3)^2+29


Vertex Form to Factored Form:


When you are turning vertex form into factored form remember to use expanding and simplifying.


y=(x+6)^2-7

y=x^2 +36x+12x-7

y=x^2+12x+29


Standard Form to Factored Form:


When you are turning standard form into factored form remember to use factoring.


x^2+8x+7 =0

(x+1)(x+7)=0

x=-1 x=-7