### by: Malika Ramesh

There are 3 different types of equations:

1. factored form a(x-r)(x-s)

2.standard form ax^2+bx-c

3.vertex form a(x-h)^2+k

## Key features of Quadratic Relations

To understand Quadratic relations you're going to need to understand the terminology being used in the equations. These terms will help you locate important points on your parabola and make life ALOT easier.

• Vertex- the max or min point on your graph, which is where the parabola will change its direction.
How would you label you're vertex? good question, you would label the vertex as (h,k)

• Maximum/Minimum value- the maximum or minimum value on your graph is decided by the direction your parabola is placed in. If your parabola looks like a U then the vertex will determine the minimum value, if your parabola looks like a upside down U then the vertex will determine the maximum value.
the max/min will be represented by the k value.

• Axis of Symmetry- the axis of symmetry (A.O.S) is exactly half of the parabola. the A.O.S will run through the middle of the vertex dividing the parabola into 2 halves.

the A.O.S is represented by the h value.

• Optimal Value- the optimal value is the y value of the vertex.
How to find Vertex, Optimal Value, A.O.S

## Step 1

An easy way to see whether an equation is linear or quadratic is finding the first and second differences. To determine whether your equation is linear or quadratic you have to put your x and y values in a chart.

## Step 2

Use the y values to find the first difference. Use the value underneath the top one and subtract in order to get your first difference.

## step 3

if the first differences are the same then its linear and you do not need to find the second difference, but if your first differences are different then read step 4.

## step 4

if your reading this step then your first differences probably didn't match up and you're going to have to find the second difference. to do so all you have to do repeat step 2 with the first differences you got in the previous step.

## Step 5

if you get a pattern in step 4 then its quadratic. Keep in mind that if at this point you don't get 1st or 2nd differences then there it is neither.

if you want to transform a parabola, an easy way to do so is with using y=a(x+h)^2+k which is vertex form.

• the a in the formula determines if the the parabola will stretch or be compressed. If the a value is positive, then your parabola will open upwards. If the a value is negative then your parabola will open downwards.

• the h value in the formula represents the horizontal translation. it will decide whether the parabola will move towards the right or the left . If the h value is positive, it will move left. If the h value is negative, then it will move towards the right.

• the k value will represent the vertical translation. The vertical translation is whether the parabola will move up on the y-axis or down. If the k value is positive then your parabola will move up, if the k value is a negative number then your parabola will move downwards.

• (h,k) represents the vertex of the parabola. Make sure you change the h value to the opposite sign. For example:
y=3(2-6)^2+9
(h,k) = (+6,+9)
Picture by: Namrita Chouhan

## Graphing Vertex Form using transformation

How to: Graph Quadratics in Vertex Form

IMPORTANT!!

The video doesn't discuss the a value. For the a value because it is a positive number you will already know that the parabola will open up!

## Graphing with Step Pattern

step pattern is very easy, as long as you know at your a value.

step 1: 1 x a

step 2: 3 x a

step 3: 5 x a

then with the product received after doing steps 1-3, plot your point by either going to the right by 1 or to the left by 1. The products from step 1-3 will determine how many spaces up or down the points will move.

## Finding x and y intercepts in Vertex Form

finding the x and y intercepts from vertex form is fairly easy. All you have to do to find the x intercepts is set y to 0. Same thing to find y-intercept, set x to 0

## Solving vertex word problems

word problem: A football is kicked from one side of the field to the other. The height is represented by the equation h=-3(t-5)+5.9. While solving keep in mind the vertex (5,5.9)

In the vertex the h value represents the time and the k value represents the height.

a) what is the maximum height of the football?

the maximum height of the would be the k value because the k value represents the vertical placement of our parabola. Therefore our maximum height would be 5.9m of the football being kicked.

b)how long did it take the football to get there?

the maximum time would be the h value because the h value represents the horizontal placement on our parabola and normally time is on the x-axis therefore the h value will represent the max time. Max time is 5 seconds.

## Multiplying Binomials (expanding)

An easy way to multiply binomials is using FOIL

you must be asking what is FOIL and how can it help?

FOIL stands for:

First

Outside

Inside

Last

there are also special cases:

Perfect Squares

(a+b)^2= a^2+2ab+b^2

Difference of Squares

(a+b)(a-b)=a^2-b^2

## Common Factoring (OPPOSITE OF EXPANDING!)

If you are given 2 binomials try seeing them as one, this will help immensely.

step 1: Find GCF (Greatest Common Factor)

step 2: Write solution with brackets

example:

(8x+6) GCF= 2

2(4x+3)

## Factor Simple Trinomial

polynomials like x^2+bx+c can be written as the result of 2 binomials of the form (x+r) (x+s). In order to correctly factor simple trinomials your standard form (x^2+bx+c) must change to look like (x+r)(x+s)

for example:

x^2+7x+12 would equal (x+3)(x+4) after being factored.

when you want to factor a polynomial of the form x^2+bx+c (when a=1), you want to find:

1) 2 numbers that add to give b

2) 2 numbers that multiply to give c

## Factoring Complex Trinomials

complex trinomials have a coefficient that another number other than 1 in front of the x^2 term.

*Before you can start factoring complex trinomials you must know how to*:

- Binomial common factoring

-factor by grouping

Example #1: Binomial common factoring

remember: Binomial can be a common factor

Factor: 3x(2-2) + 2(2-2)

= (2-2) (3x+2)

Example #2: Factor by grouping

remember: there is never a common factor in all terms, but you can group terms that have common factors

factor:

df+ef+dg+eg

(df+ef_dg+eg)

=f(d+e)+g(d+e) *make sure the 2 brackets are the same*

=(d+e)(f+g) *factor out binomials*

Perfect squares are when a number is multiplied by itself, for example:

1,4,9,16,25,36,49,64,81,100,121,144 etc these numbers are all perfect squares

Factoring a difference of Squares:

factor: 9x^2-1b *apply (a+b)(a-b)=a^2-b^2*

a=squareroot of 9x^2= 3x

b = squareroot of 16=4

## Graphing Quadratics in Factored Form y=a(x-r)(x-s)

standard form: y=ax^2+bx+c to factored form: y=a(x-r)(x-s)

to solve a equation that involves anytype of factoring you must make one side equal to 0

x^2-3x-28=0

-7x4=-28

-7+4=-3

(x-7)(x+4)=0 *inorder to solve for x you must set each bracket equals to 0*

^ ^

x-7=0 x+4=0

x=7 x=-4

^^^^^^^^^^^

roots or zeros

## How do you graph using the x-intercepts? How do you find the vertex?

finding the zeros- as we previously learned finding the zeroes will provide 2 x values, those 2 values will give you your 2 x-intercepts.

A.O.S- ( x value of vertex)

A.O.S= -3+9

2

=6/2

x =3

A.O.S=( y value of vertex)

y=0.5(3+3)(3-9)

y=0.5x6x-6

y=-18

after doing these 2 steps you can confirm your vertex as (3,-18)

## Maximum and Minimum values

how do you go from y=ax^2+bx+c(standard form) to y=a(x-h)^2+k (vertex form)?

step 1: group the x^2 and the x term together

step 2: complete the square inside the bracket given in the equation

step 3: write the trinomal as a binomial squared

example

y=x^2+8x+5

y=(x^2+8x+16-16)+5

y=(x+4)^2-11