# Quadratic Relations

### By: Bilal Baig

## Table of contents

__Types of Forms__

Vertex Form

Factored Form

Standard Form

__Vertex form y= a(x-h)² + k__Axis of symmetry

Optimal Value

Transformations

X-intercepts or zeroes

Step pattern

Graphing parabolas

__Factored Form y= a (x-r)(x-s)__

Zeroes or x intercepts

Axis of symmetry

Optimal value

__Standard form y= ax²+ bx + c__

Zeroes

Axis of symmetry

Optimal Value

Completing the square to turn to vertex form

Common factoring

Factoring to turn to factored form

Factoring to complex trinomial

Perfect squares

Difference of Squares

__Word Problems__

__Reflection__

## Types of forms

- Vertex form -
**y = a(x-h)²+k** - Factored form -
**a(x-r)(x-s)** - Standard form -
**y = ax**²**+ bx + c**

## Vertex form

## Axis of symmetry

The axis of symmetry is the center point of the parabola on the x axis. In vertex form, the axis of symmetry is the opposite symbol of the h value. For example in the equation y= (x+5)² - 3 the axis of symmetry would be -5 because h is a negative so when you put two negatives together they become a positive.

## Optimal Value

The optimal value is the maximum or minimum point of the parabola on the y axis. In vertex form the optimal value is K value. In the equation y= (x-5)² - 3 the optimal value would be -3 and minimum.

## Transformations

The (-h) moves the parabola left or right. The parabola moves left if (-h) is a positive like (+5) and it moves right if it is a negative like (-5).

The (K) moves the parabola up or down. The parabola moves up if K is a positive and it moves down if K is a negative.

The (A) stretches the parabola. If the A is negative the parabola opens down, if A is a positive it opens up

## Zeroes

For example:

## Step Pattern

## Graphing parabolas

## Factored Form

## Zeroes (X-intercepts)

## Axis of Symmetry

To find the Axis of Symmetry you must solve for x by setting y=0 and solve each bracket separately then put the x intercepts together and divide by 2.

## Optimal Value

## Standard Form

## Zeroes

## Axis of Symmetry

## Completing the square

- First you must put ax + bx in brackets, then common factor the numbers and place the factor in front of the brackets.
- After that you do (b/2)² and add it to bx, but since you can't just add to an equation you also have to subtract it too.
- Next you'll notice in the bracket you have a perfect square (x² + 4x + 4) so now you square root the x² and 4 leaving you with x and 2. Then you put the x and 2 together in brackets and square it.
- After that you multiply the - 5 with the - 4.
- Then add the + 20 to +15 and you have your K value which is also the optimal value.

## Optimal Value

## Common factoring

## Factoring to Factored Form

## Factoring Simple Trinomials

## Factoring complex Trinomials

## Perfect Squares

## Difference of Squares

## Word Problem

## Motion problem

1.a) What is the max height? When does it happen?

1.b) When will the ball hit the ground?

1.c) Sketch a graph.