### math by Jasmeet Nijjar

Have you ever wondered how roller coasters are made? Roller coasters have math behind it. Linear relationships and parabolas are used to make them. We developed our learning from Linear relationships to parabolas to understand our world better. Quadratic comes from the word QUAD, which means square like (x²). The power to 2 is what causes quadratics to have a curve in them. Quadratics is used to measure curves or anything 'u' shaped for example, kicking a soccer ball, bridges, the moon going around the earth and more.

## 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
Standard Form:
• 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

We use tables of values to determine if the table is linear, quadratic or none. In the picture you have to see the difference in the Y column, 3-2=1, 6-3=3 ect If the difference between the Y in the first column were all the same numbers it was be linear. So we go on to the next column and find the difference between the first difference, ex 3-1=2, 5-3=2, see now the difference keeps coming as 2 which means its quadratic. If there was no common difference even in the second difference column then it wouldn't be linear or quadratic.

Just so you know a parabola is the name for Quadratic when its graphed.

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

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

Vertex form is a way to write the equation

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

1. find vertex; y=a(x-h)²+k opposite of h and k are your vertex
2. use step pattern: original- over 1 up 1, over 2 up 4 now you do ax2 and ax4
3. 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)

3.3 More Graphing from Vertex Form

## How to find zeros/x intercepts

In vertex form you are given a, h and k but not x or y. To solve for x you can plug y=0 because x intercepts have a value of 0. You should start bringing everything away from x using samdeb. At the end x should equal to one positve and negative number. solve the last part by using just a plus and the other part using just minus, this should lead to your 2 x intercepts.
x intercepts from vertex form

## factored form

Another form you learn in quadratics is called 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

Finding your x intercepts from factored form is very easy;

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

Finding axis of symmetry;

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

Finding the optimal value:

*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

The last form in quadratics is 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...

you must be wondering where a b and c came from, well

ex

y=2x²+9x+10

• 2 is a
• 9 is b
• 12 is c
Watch the video below, i will be teaching you how to use the quadratic formula

## discriminant

The discriminate is only the portion inside of the square root from the quadratic formula.

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

-b/2a= AOS

using the other equation it will be:

-9/2x2=AOS

-2.25=AOS

## OPTIMAL VALUE

to find the optimal value you have to plug your aos value as x in your original formula

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

once againg use the step patter you watched in the how to make a parabola video

## completing the square and turning it into vertex form

The video below demonstrates how to convert from standard to vertex from by completing a square.

1. factor whats common and bring it out from the first two terms and kick the last term out. put the two terms in brackets.
2. make it a perfect square trinominal by getting the second term, dividing that by 2 and power it by 2
3. the answer you got for step #2, you write it twice; first add second subtract
4. 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

Converting Standard to Vertex Form

## Factoring to turn into factored form

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

## Common factoring

Watch the video below, it should take you through the process of common factoring, but common factoring is when you find the factors of numbers and write it in factored form

## Factoring by grouping

This is basically the same thing as common factoring, but it has 4 terms, and we are get grouping but making binomials. After you group you find the common factor and write it in factored form. Watch the video below

## fatoring Simple Trinomial

factor means, all the possible ways to multiply to get that number. It is called simple because the coefficient is one, and it has three terms, which is called a trinomial. Watch the 2 videos below on factoring simple trinomial

## Factoring complex trinomial

Very similar to simple trinomial, only difference it that the coefficient ha can not be one. Watching the video below will help your understanding of complex trinomials
3.9 Complex Trinomial Factoring

## DIfference of squares

Difference of squares(dos) have 2 different symbols so problems for D.O.S are in a²-b² formula, but the end result should look like (a+b) (a-b).

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)

axa=a²

ax-b=-ab

bxa=ba

bx-b=-b²

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

## perfect square

Perfect square are the factors of themselves, they use the formula a²+2ab+b² and your answer should end up as (a+b)²

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

3.11 Factoring

• 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

More Word Problems Using Quadratic Equations - Example 1
a) the question below, in vertex form, h and k are your vertex. (3,19) is my vertex, and its also when the ball reached its highest so i looked at x axis (time) and wrote 3 sec

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

i did the same thing here just like the last question

vertex (10,20)

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

The question says bryanna sells 15 shirts for \$20, but she starts to decrease money amount by \$1 and she sells 2 more shirts on top of 15

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

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 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