# Quadratics

### By Nisha Sabharwal

## What Does Quadratics Mean:

__Quadratic__comes from "quad" meaning

*square*, because the variable gets squared. Quadratic equations make a smooth curve.

## Quadratic Examples in Real Life

## Reflection:

In the beginning of this unit when we started with quadratics 1 I was struggling with what to do but when we started quadratics 2 I got a better understanding and once I understood what to do in quadratics 2, quadratics 3 was much easier. This improvement didn't however show on my tests because I was still somewhat unclear with quadratics 1 and that made me unsure on my answers because they would be related on the tests.

In this quadratic unit i learned about parabolas and many ways to solve quadratic relations. In quadratics 1 I was taught about the basics of a parabola; how to read it, what it means when a parabola is opening upwards or downwards, finding the vertex, axis of symmetry, and x and y intercepts. Then in quadratics 2 I added to my knowledge of parabolas to make quadratic relations and equations and learned quadratic equations; the vertex form, standard form, and factored form. I also learned how to rearrange all these forms to make another such as changing from standard form to factored form. Lastly in quadratics 3 I was taught how to solve these quadratic equations and also introduced to the quadratic formula. In conclusion all three quadratics were interconnected with each other and breaking them apart made it easier to learn considering it was such a long unit.## Quadratic Equations:

- Vertex form: y=a(x-h)2+k
- Standard form: y=ax2+bx+c
- Factored form: y=a(x-r)(x-s)

## Quadratics Terms:

First Differences

## Parabola

## Standard Form:

Zeros:

Quadratic Formula (-b+- square root of b2-4ac/2a)

5x2-7x+2=0

- 5 is the a term 7 is the b term and 2 is the c term

- since 7 was a negative in the equation it gets changed to a positive

7 +- square root of 9/10

7 + square root of 9/10

=1

7 - square root of 9/10

=0.4

the zeros are (1, 0.4)

Axis of Symmetry:

-b/2a

-(-7)/2(9)

7/10

=0.7

Optimal Value:

sub in 0.7 into the equation

y=5(0.7)2-7(0.7)+2

y=0

Completing the Square:

y=x2+8x+4

y=(x2+8x=42-42)+4

y=(x2+8x+42)-42+4

y=(x+4)2-12

**(2 represents squared)**

## Factoring

## Finding the Vertex:

The vertex is made up of the h and k value in the equation.

Example: y=(x+6)2+12

Vertex: (-6,12)

*If the h value is positive in the bracket, when plotting your vertex it turns negative. And if in the bracket it is negative, when plotting the h value turns positive (i.e. y=2(x-8)2-9, vertex= (8,-9)). The k value stays the same when plotting

## A.O.S. and Optimal Value:

- A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
- Optimal value=the k value of vertex equation

Example: y=-(x+5)2+9

- A.O.S= 5
- Optimal Value= 9

## How to Find First and Second Difference

## Note:

- if first difference the same=linear relation
- if second difference same=quadratic relation (parabola)
- if no difference the same=neither

## Reading Parabolas:

y=a(x-h)2+k

***2 stands for squared**

a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).

h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).

k=how many units the parabola vertically moves (-#=moves down, +#=moves up).

Vertex= (h,k) the maxima or minima of the curve(parabola)

## Transformations

a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).

h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).

k=how many units the parabola vertically moves (-#=moves down, +#=moves up).

Vertex= (h,k) the maxima or minima of the curve(parabola)

## Finding Equations in Vertex Form

## Finding Zeros/x-intercepts and y-intercepts in Vertex Form:

Example: y=-4(x+2)2+16

- y=-4(x+2)2+16
- y=-4(0+2)2+16
- y=-4(2)2+16
- y=-4(4)+16
- y=-16+16
- y=0

*when solving to find the x-intercepts, remember to always sub y=0

- y=-4(x+2)2+16
- 0-16=-4(x+2)2
- -16/-4=(x+2)2
- 4=(x+2)2
- take square root on both sides
- +2 or -2=x+2
- 0 or -4=x

therefore the x intercepts are 0 and -4

## Step Pattern

## A.O.S

- A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
- Optimal value=the k value of vertex equation

Example: y=-(x+5)2+9

- A.O.S= 5
- Optimal Value= 9

## Quadratics in Factored Form

Factored form: y=a(x-r)(x-s)

*r and s represent zeros (x-intercepts)

- find zeros
- find A.O.S
- find optimal value

Example:

y=-2(x+4)(x+6)

- Find zeros:

- x+4=0 x+6=0
- x+4-4=0-4 x+6-6=0-6
- x=-4 x=-6

2. A.O.S= -4+(-6)/2

- = -10/2
- = -5 (x value of vertex)
- 3.Optimal Value (sub -5 as x in original equation)
- y=-2(-5+4)(-5+6)
- y=-2(-1)+(1)
- y=2 (y/k value of vertex)

Vertex: (-5,2)

Vertex form: -2(x+5)2+2

## Difference of Squares

When working with difference of squares, you are multiplying two binomials to get a binomial.

Formula: a2-b2=(a+b)(a-b)

when expanding (a+b)(a-b):

- (a+b)(a-b)
- a2-ab+ab-b2
- a2-b2