# Quadratics

### By: Dilveer Dhaliwal

## How quadratics help us in reality

## How to determine Quadratic Realtions

- Recall that linear relations had first differences

So...

- Quadratic relations have SECOND differences

## VERTEX FORM

- The vertex form is one way that a quadratic relationship can be written as it is in the form
**y=a(x-h)² +k**and the basic form of it is**y=x²**. - An example for this is
__y= 2(x-3)² +5__ - The "x" and "y" do not have a value because they are variables for the x and y intercept(s) which will have different values according to the relation

## What each variable represents in the Vertex Form

- In the vertex form
*y=a(x-h)² +k*each variable has a part to do - The
tells us if there is a**a***stretch/expansion*or*compression*to the parabola. - The
**h**tells us the x value of the vertex and the axis of symmetry. - The
**k**tells us the y value of the vertex and the optimal value.

## Transformations

- The transformations that can occur to a parabola are vertical or horizontal translations, vertical stretches and reflections.
- As we know the
*a*represents vertical stretch and compression if the value is more than 1 than it is a*compression*and if it is less than 1 it will be a*stretch/expansion* - If
*a*has a negative value then there will be a*reflection*on the x-axis (the parabola will be flipped; downwards) The

*h*tells if the parabola moves left or right (horizontal stretch). If the*h*The

*k*informs us if the parabola moves up or down (vertical stretch) so if it is negative the graph will move downwards and if it is positive it will move upwards.

## Graphing from Vertex Form using the Step Pattern

**y=x²**this is the basic parabola.- The original step pattern is
*"over 1 up 1 , over 2 up 4"*this pattern can be used for any parabola as long as it is the basic (original) one, so it will not have an*a*value to determine the compression or stretch. - The basic parabola will always open upwarads

## Graphing from Vertex Form when not in Basic Form

- If the vertex form has the
*"a"*in front of it, it is not a basic parabola anymore, it will either have a stretch or compression and will open either up or down - The value of
*a*will decide if the step pattern will change

An example...

- First, you will graph the vertex
- Then multiply the original step pattern by the value that
*a*had been given - Then plot the points (vertex, 4 step pattern points).

- It does not matter if there is a decimal, fraction, or a negative number for
*a*, you still have to multiply that number with the*original*step pattern.

## How to find the Axis of Symmetry (AOS)

- The way it is written is
**x=h**, so it would be the h in the vertex form

- The axis of symmetry is the middle point of the zeros; the x intercept of the vertex
- So the AOS of the parabola above will be -2 because it is the midpoint of the parabola and it is the x-intercept of the vertex

## How to find the Optimal Value

- It is written as
**y=k**, so it would be the*k*in the vertex form - The optimal value is either the highest point or the lowest point on the parabola; the y-intercept of the vertex
- So the optimal value of the parabola above would be -6 because in this case it is the lowest point on the parabola and it is the y-intercept of the vertex

## X-intercepts and Zeros

- To find the x-intercept/zeros in vertex form you would have to set
*y=*0 - For instance let's say you have to find the zeros in the equation
__y=-3(x+2)____²+27__the*y*would be replaced with 0

- We do negative and positive square root because the number being squared could be either one
- As a result the x-intercepts would be 1 and -5

## FACTORED FORM

- The factored form is another way that a quadratic relation can be written as and the form is
**y=(x-r)(x-s)** - An example of this form is
__y=0.5(x+3)(x-9)__

## Zero and X-intercepts

- Zeros can be found by setting each factor equal to 0 meaning
**x-r=0**and**s-r=0** - For example in the relationship
__y=0.5(x+3)(x-9)__

- Therefore, the zeros/x-intercepts are -3 and 9 for this relation so when you graph the zeros these are the values you would plot for the x-intercepts.

## Axis of Symmetry (AOS)

- To find the axis of symmetry you have to add the two x-intercepts and divide it by 2
- Therefore the formula or method you would use is
**x= (r+s)/2** - For the example we have been using previously we would find the x-intercepts (-3 and 9) add them then divide the answer by 2
- So it would be

x=(r+s)/2

x=(-3+9)/2

x=6/2

x=3

- For that reason, the axis of symmetry would be 3

## Optimal Value

- In order to find the optimal value you would have to substitute the axis of symmetry into the equation as
*x*since it is a*x*value on the parabola - If we use the example we have been using so far we would substitute x=3 into the equation
__y=0.5(x+3)(x-9)__

- Therefore the optimal value for the example is -18

## Graphing

- Now that you know how to find the x-intercepts, axis of symmetry and optimal value you can plot 3 points

- The first x-intercept which is x=-3
- The second x-intercept which is x=9
- The vertex which is (axis of symmetry;x, optimal value;y) (3,-18)

## STANDARD FORM

- Last but not least, the standard form is also a way to write a quadratic relation and the form is
**y=ax²+bx+c** - An example of this form is
__y=3x²+15x+18__

## Quadratic Formula

- The quadratic formula is

## How to find the zeros using the Quadratic Formula

- In order to use the formula the standard form should equal to 0 since you are
*solving*for the zeros - Since we know the formula and have the question all you have to do is plug in the appropriate values just like in the example below

- The square root is both positive and negative since there would be a negative and positive square root to the number which in the end would give us two zeros
- If you want to keep the exact answer leave it the way it is in step 4 but if you want to take it down to a fraction/decimal then you can continue and add the first time and subtract the next when solving

## Discriminants

- The discriminant is the number that is being square rooted
**(b²-4ac)**this number tells us if there are any zeros/solutions and if there are, how many there are. - If the discriminant is a negative there will be no zeros because a negative number cannot be square rooted
- If the discriminant is 0 then there is only one zero because it does not matter if you add or subtract a 0 the answer will be the same
- If the discriminant is greater than 0 then it will have two zeros, just as in the example above, since the discriminant is 9 there were two zeros

## Axis of Symmetry

- The formula for this is

- To find the axis of symmetry all you have to do is fill in the formula that is shown above
- And do it just as shown in the example below

- It is as simple as that so the axis of symmetry would be x=0.7 for this parabola

## Optimal Value

- To find the optimal value (y-value;y-intercept in vertex) all you have to do is substitute the axis of symmetry into the original equation to get
*y*just as we did in factored form - An example for this is...using the axis of symmetry from above

- So they optimal value for the example we have been using would be -0.45

## Word Problem

## Completing the Square

- Completing the square helps us turn from Standard form to Vertex form (y=ax²+bx+c to y=a(x-h)²+k
- There are a few steps and they are as follows:

- Take the c out and make it k since they both carry the same value and represent the same thing on the graph
- Then put brackets around the remaining two numbers
- Then factor out the GCF from the number in the brackets; after divide b by two and square to get the number you need to add and subtract
- Then add and subtract that number from the brackets
- When taking the subtraction number out of the brackets multiply it by the
*a*value;then you will be able to make a perfect square - Make the perfect square (write x and divide b by 2 then put squared outside of the bracket) and write the vertex form

Just as in the example below

- Therefore the vertex form would be y=2(x+2)²-3

## Factoring to turn to Factored Form

## Factoring-4 terms by Grouping

- If you ever end up with four terms to factor it is very simple

- We group the first two and second two terms
- Then we factor out the GCF from each factor
- Then we can common factor just as shown above

- So basically we grouped the terms into binomials first two and second two, then we found the GCF and factored it out and finally we common factored
- This gave us (x²+3)(x+7)

## Factoring Simple Trinomials

- The way that simple trinomials are written is
*x²+bx+c* - An example for this is w²+5w+6
- There are six steps to get into the factored form and they are

- Make sure the format is
*x²+bx+c* - Find two numbers that give the product of 'c'
- Add the two number to see if they give the sum of 'b'
- Factor them in for 'bx'
- Do four terms by grouping method
- Then we do common binomials

An example for this is

- Therefore w²+5w+6 in factored form is (w+2)(w+3)

## Factoring Complex Trinomials

- Complex trinomials will include an
*a*value with the a making it complex - An example of this is 8x²+6x-5
- In order to factor this we do the following:

## Perfect Squares

- Perfect Squares make expanding easier
- There are 3 steps to do them:

- Square the first term
- Double Product
- Square second term

- But you can only do perfect squares if what you are expanding is either for example (a+b)² or (a+b)(a+b) since you are multiplying the same things that would end up being squared, or (a-b)(a-b) and (a-b)² but of course the variables and values would be different and they can be a part of a larger relation.

## Factoring Perfect Square

- You may also use the acronym

Sam

Doesn`t Pull

Strings

which stands for

Square, Double Product and Square

## Difference of Squares

- Let's say you don't have a perfect square but something like it for example
*(y-7(y+7)*the only different thing are the signs and this method is called difference of squares - When we do difference of squares there are only two steps

- Square the first term
- Square the second term

- For example

- The subtraction sign will always stay as subtraction, hence the name
*difference*of squares - This method can be applied anywhere as long as the relation you are expanding is in the format (a+b)(a-b) or (a-b)(a+b)

## Factoring Differnece of Squares

Difference of Squares factors into a product of sum and difference

## Common Factoring

- Common Factors are when we factor out the
*GCF*in an expression - There are only three steps to this

- Find GCF
- Divide expression by GCF
- The GCF number will go outside the brackets as well

Example

2x+4y

2(x+2y)

Since the GCF was 2 we divided the whole expression by 2 and then put the GCF outside of the brackets as well which factored into 2(x+2y)

OR you can make a list on all the factors of each number and find the common one(s) and find the GCF that way; as shown in the example below

## Binomial Factors

- This method is known as binomial factors because we are putting the expression into only two parts; an easier way to expand
- For example

- Basically we did three things to get the answer we got

- Since (2s+5t) are the same we can write them just once
- Then we have 3s and -7t left which can be put in another set of brackets
- Both are multiplied so when you need to expand to get back to what you started you may do that

Hence we get (2s+5t)(3s-7t) as our factored form

## REFLECTION

*a*could have. Later, once I got the hang of it, they were not as hard as I thought it was. Another difficulty I had in this whole unit was when we had to find the amount of x-intercepts a parabola had by looking at the equation. I had difficulties because I never really knew how find the x-intercepts by just looking at the vertex form. Then later on we learned that all we had to do was substitute y as 0 and solve for x and that x would give us the two x-intercepts. I also used to get confused between perfect squares and difference of squares because I would confuse perfect squares for difference of squares and vice versa. So the method I used to remember which is which is that difference of squares is a product of addition and subtraction which would eliminate that, that is not a perfect square making it difference of squares. The things I could do with ease were simple trinomials and expanding especially on the assessment below.