# Complex algebra known as QUaDRATiC

### By Thrisha balakrishnan

## What is quadratics?

## Quadratics RELATIONS is not just about numbers it's also about remebering special words Later in we will get in deep with quadratics relations ! | ## Quadratics EXPRESSIONS is learned throught formulas and diagrams so its not that hard So later we will also learn deep into quadratics expressions! | ## Quadratic EQUATION is changing different form s into a another such as from standard form to vertex form and etc... or using formulas such as the quadratic formula So therefore u will be learning more depth into each step of converting from one eqaution into another and the quadratic formula |

## Quadratics RELATIONS is not just about numbers it's also about remebering special words

## Quadratics EXPRESSIONS is learned throught formulas and diagrams so its not that hard

## 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 .

## TABLE OF CONTENTS

Quadratics relation part 1

- Parts of a parabola

- X-intercepts/zeroes/roots
- Axis of symmetry
- Optimal value(maximum or minumim)
- vertex
- openings (up or down)

- Quadratics relationships/relations
- Graphing in vertex form
- Transformation of quadratics
- Word problems :

- Finding an equation in vertex form when given a vertex and point

## PARTS of PARABOLA

**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 .

## The zeroes/x-intercepts/rootsThe equation:y=-2(x-2)(x+3) This graph has 2 zeroes/x-intercepts/roots | ## The zeroes/x-intercepts/rootsThe equation :y=2(x-2) This graph has 1 zeroes/x-intercepts/roots | ## The zeroes/x-intercepts/roots The equation :y=2(x-2)+2 So this graph has no intercepts since its above the x-axis line |

## Analyzing quadratics relation / relationship

## Transformation of quadratics

- 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).

## A affects the vertical stretch if its greater than 1 and vertical compression if less than 1 Compares to the base quadratic. | ## H affects the horizontal translation /shift Compares to the base quadratic. | ## K affects the vertical translation /shift Compares to the base quadratic. |

## Graphing from vertex form

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

- 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 .
- 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
- 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.
- Then connect all the dots with a curve and draw the arrows on at the end of the curve

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

## Now we're going to learn about the second part to quadratics which is the expression part.

## Table of contents

Quadratics expressions part 2

- Expanding binomials
- Multiplying polynomials ,common factoring
- Factoring by grouping
- Factoring simple trinomials
- Factoring complex trinomials
- Special factoring cases(perfect squares trinomials and differences of square)
- 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 :

## Multiplying polynomials ,common factoring

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

## This is one way to transform standard to factored form

## Factoring by grouping

## Factoring simple trinomials

## Factoring complex trinomials

## Factoring special cases

## Factoring difference of squsares

*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:

*x*2 – 4 = (*x* – 2)(*x* + 2)

## Graphing in factored form

## Table of contents

Quadratics eqaution part 3

- Completing the square
- Solving from vertex form
- Quadratic formula
- Discriminant
- Graphing in standard form

## Completing the square

## Solving from vertex form

## Quadratic formula /eqaution

## Final topic the discriminant

## Graphing in standard form

## Different types of word problems

- Geometry (area/perimeter/shape)
- Motion
- Revenue #1
- Revenue #2
- Number

## Geometry( area/perimeter/shape)

## Motion problems

## Revenue #1

## Revenue #2

## Number

## 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