# Quadratic Relationship

### by: Malika Ramesh

## Key features of Quadratic Relations

**Vertex-**the max or min point on your graph, which is where the parabola will change its direction.

**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.

**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.

## Finding First and Second Differences

## Transformations of Quadratics

**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)**

## Graphing Vertex Form using transformation

**IMPORTANT!!**

**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 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 x intercept to find the x intercept begin with setting y=0. after doing so, proceed with bringing all number to the left side. So for this example the only number you can bring to the left side right now is 75. After bringing the 75 to the left side, divide the right side by the a value to eliminate it, so in this case divide the ride side by 3. Whatever you do to the right side must happen to the left so divide the left side by 3 as well. Now to eliminate the ^2 you must perform the opposite operation of squaring, which is square rooting. Square root both sides and for the left side square if plus and negative. Then proceed to bring in the last number to the left side, leaving x on the right side. Then everything else is pretty simple. See pic on right for further information. | ## Finding y intercept begin with setting x=0. Then using BEDMAS begin solving the equation until you have 1 number left equaling y. |

## Finding x intercept

**x intercept**begin with setting

**y=0.**after doing so, proceed with bringing all number to the left side. So for this example the only number you can bring to the left side right now is 75. After bringing the 75 to the left side, divide the right side by the a value to eliminate it, so in this case divide the ride side by 3. Whatever you do to the right side must happen to the left so divide the left side by 3 as well. Now to eliminate the

**^2**you must perform the opposite operation of squaring, which is square rooting. Square root both sides and for the left side square if plus and negative. Then proceed to bring in the last number to the left side, leaving x on the right side. Then everything else is pretty simple. See pic on right for further information.

## Finding y intercept

__BEDMAS__begin solving the equation until you have 1 number left equaling

**y**.

## 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**.

## 2nd word problem (Profit)

## Factored Form

## Multiplying Binomials (expanding)

**FOIL**

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

**FOIL **stands for:

**F**irst

**O**utside

**I**nside

**L**ast

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!)

step 1: Find GCF (Greatest Common Factor)

step 2: Write solution with brackets

example:

(8x+6) GCF= 2

**2(4x+3)**

## Factor Simple Trinomial

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

***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***

## Factoring Special Quadratics

**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)

**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?

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

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

## The Quadratic Formula

__-b+-b^2-4ac__(square root everything besides -b+-)

2a