# QUADRATICS

### math by Jasmeet Nijjar

## What does Quadratic mean?

## TABLE OF CONTEXT

vertex form:

- first and second differences
- parts of a parabola/how to read parabolas
- Transformations
- how to make a parabola +step pattern
- finding zeros

Factored form:(what you need to find out to graph factored form)

- finding x intercept
- finding axis of symmetry
- finding optimal value

- Quadratic formula
- Discriminant
- Axis of symmetry
- Optimal value
- Completing a square
- Factoring to turn into factored form (common factoring, factor by grouping, factor simple trinomials, factor complex trinomial, difference of squares, perfect square)

## First and Second Differences

## How to read Parabolas

Axis Of Symmetry/Vertex: This is the point that divides the parabola in two and also where it meets. (add the two x int and divide by 2)

X intercept: where the parabola hits the x axis, is also called zeros (plug y as 0)

Y intercept: When the parabola hits the y axis (plug x as 0)

Optimal value: The highest or lowest point (max or min)

Direction of opening: when the graph is smiling its opening upward and when its frowning its opening down

## Transformations:

## The original parabola is y=x² every graph is added onto the original

y=a(x-h)² +k

a- tells if the parabola stretches up or gets wider

h- tells you if the parabola moved left or right

k- tells you if the parabola moves up or down

ex.

narrow: y=2x²

wider: y=o.5x²

move up: y=x² +3 *for left and right your probably wondering why

move down: y=x²-3 moving to the right actually and move to the left

move right: y=(x-1)² you actually move to the right. This is because

move left: y=(x+1)² you have to use the opposite sign from the bracket

flip: y=-2x (x-1)² is actually x= 1

y=a(x-h)²+k

y= -4 (x-3)² + 4

- narrowed by 4 (a)
- flipped (-a)
- move to the right (opposite of everything in bracket)
- up 4 (k)

## How to make a parabola in the video below

- find vertex; y=a(x-h)²+k opposite of h and k are your vertex
- use step pattern: original- over 1 up 1, over 2 up 4 now you do ax2 and ax4
- AND YOUR PARABOLA IS DONE

in vertex form opp h and k are your vertex on the graph

ex y=a(x-4)²+6

vertex is (4,6)

## How to find zeros/x intercepts

## factored form

y=a(x-r)(x-s)

This is called factored form because you need to factor to get x intercept.

## step 1 for graphing

y=a(x-r)(x-s)

1. separate x-r and x-s and put =0

x-r=0 x-s=0

2. now bring your r and s to the other side and x=r x=s switch your signs

x=r x=s

ex.

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

x-6=0 x-4=0

x=6 x=4

## step 2 for graphing

you do r plus s and divide the answer by 2, which should give the half of the two roots.

y=a(x-r)(x-s)

r+s/2

ex

(x-6)(x-4) *opposite of whats in bracket

6+4=10/2

aos=5

## step 3 for graphing

*we already got our roots, and our axis of symmetry

to find optimal value you have to plug the x in the factored form equation with the axis of symmetry you found earlier.

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

y=(5-6)(5-4)

y=(-1)(1)

y= -1

now all we have to do is plot the 4 things you got on the graph and use the step pattern from the video how to make a parabola.

## Standard form

y=ax²+bx+c

- A formula you learn that can help you find the x intercept for ANY equation is called quadratic formula.

which looks like...

ex

y=2x²+9x+10

- 2 is a
- 9 is b
- 12 is c

## discriminant

## How to find axis of symmetry using as similar fromula

using the other equation it will be:

-9/2x2=AOS

-2.25=AOS

## OPTIMAL VALUE

y=2x²+9x+10

y=2(-2.25)²+9(-2.25)+10

y=2(5.0625)-20.25+10

y=10.125-20.25+10

y=10.125-10.25

y=0.125

## now you got your x intercept, aos and optimal value all you have to do is graph it

## completing the square and turning it into vertex form

- factor whats common and bring it out from the first two terms and kick the last term out. put the two terms in brackets.
- make it a perfect square trinominal by getting the second term, dividing that by 2 and power it by 2
- the answer you got for step #2, you write it twice; first add second subtract
- keep the 3 numbers in the bracket and kick the last one out, but first you have to multiple it with the common number we found in step #1
- now you solve and get your answer

## Factoring to turn into factored form

- common factoring
- Factoring by grouping
- simple trinomial
- complex trinomial
- perfect square
- difference of square

## Common factoring

## Factoring by grouping

## fatoring Simple Trinomial

## Factoring complex trinomial

## DIfference of squares

a²-b²

square root a² = a

square root b² = b

and write your answer in brackets, the first one use a '+' and the second one use a '-' sign.

(a+b)(a-b)

TO check your answer you have to use distributive property;

axa=a²

ax-b=-ab

bxa=ba

bx-b=-b²

-ab and ba cancel out leaving you with a²-b²

## perfect square

a²+2ab+b²

square root a²=a

square root b²=b

write in a bracket and put '²' sign outside of it

(a+b)²

Check by multiplying inner and outer

expand (a+b)²

(a+b) (a+b)

axa=a²

bxb=b²

and for the middle do 2xaxb=2ab

## If you found my videos on factoring confusing to understand watch this

## LINKING

- All three forms can be graphed
- you can ind all of there Axis of symmetry, optimal value, y and x intercepts, and vertex
- in vertex form you can find the vertex by using opposite of h and original k
- Use step pattern to graph all forms
- Discriminate will tell you in advance how many roots there will be
- x+x/2=aos and -b/2a= aos
- can use quadratic formula to solve any type of x, even complex trinomials
- all forms can be created into eachother
- distributive property works on all factoring to turn into factored form

## How to solve a quadratic problem using quadratic formula

b) same thing as a vertex is (3,19) so i looked at y axis (height) and wrote 19

c) punted is like something being kicked started above the ground so what we are finiding is how much in the air was it before kicked

i used the original equation h=-2(t-3)²+19 and substituted t as 0 and you should get your answer

vertex (10,20)

c) i subsitute 1 as d in the original equation and solved it

first you make a chart on her revenue, which means her income and price change

a)

20-price change times 15+ 2 times the price change

b)

you have to find the x intercepts, (20-p)(15+2p) is already in factored form so i easily found it.

then you will have to find the axis of symmetry from there which is 6.25 and plug it in (15+2p) and solve

after you get your answer solve the whole equation

the question is askign for the largest fencing, and finding aos gives you the maximum value

the total area is 500 which is 2x and 2yfor perimeter you add 500=2x+2y

we can make the equation more simple

250=x+y

to find y we will bring x over = 250-x=y

plug that in for y times x (lxw for area)

its in factored form so you can find the x intercepts

x=250 x=0

find the axis of symmetry by addign the zeros and dividing it by 2 which is 125

now plug that in as x in the A=(250-x)(x) equation

and solve to get your answer

The quadratics unit has been a bit challenging compared to the other units because I do not work well with graphs, I just freak out. At the beginning I understood it throughout part 1 but then,i wasn't here for like 2 classes in part 2 which got me a bit confused and the part 3 came, which was added onto part 2 and became complicated. I am still slowly trying to understand this unit well. I also struggled with the website because on my terrible memory, after i do a test i forget everything i learned which is a terrible habit. on my tests i tend to do better in application than in multiple choice because those questions just confuse me. There are some parts i enjoy doing like complex trimomials, common, grouping, simple trinomials ect, because I've done them so much that i can't forget then and they start to get easier. I am very happy that quadratics is split into three sections because if it wasn't for that I would be having alot of trouble trying to remember everything and do MUCH poorly on my test. Just like everyone else I go through struggles, sometimes its in mat but I with help from peers and teachers, quadratics isn't as bad as i thought it would have been.