# Quadratic

### Relation, Expression and Equations

## About Quadratics

So get ready with your pens and notebooks :D

## Table of Context

- Introducing the Parabola.
- Analyzing quadratics
- Graphing from vertex form
- Transformation.
- Finding the equation given vertex
- Graphing from factored forming
- Expanding
- Common factoring
- Factoring by grouping
- Factoring simple trinomials
- Factoring complex trinomials
- Factoring special trinomials
- Video on Factoring
- Completing the Square
- Solving by isolating x
- Discriminant
- Quadratic Formula
- Shape problems
- Revenue problems
- Motion problems
- Consecutive intergers
- Reflection
- Links between topics

## Introducing....The Parabola

Main ideas:

- Parabolas can open up or down
- The zero of a parabola is where he graph crosses the x-axis
- Zeroes can also be called x intercepts or roots
- The axis of symmetry divides the parabola into equal halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet.
- The optimal value is the value of the y co- ordinate of the vertex
- The y intercept of a parabola is where the graph crosses the y axis.

## Analyzing Quadratics.

For example:

## Graphing from vertex form

** y = a(x - h )2+ k**

In order to graph from vertex form, you need to know what each variable represent:

For example:

For example:

- The original step pattern of a parabola is over one up one, over two up four.
- The vertex decides where we start the step pattern.
- The 'a' value decides if the step pattern changes (it multiples the vertex part of the step pattern

## Graphing transformations of Parabolas

## Finding the equation given vertex

## Graphing from factored form

**y=a(x-r)(x-s)**

You need to know how each variable affects the graph before we start learning how to graph

**1. a - causes the vertical stretch or compression. If a>1 it is a vertical stretch. If -1<a>1 its is a vertical compression.**

** - if the parabola face up or down. If a is negative, the parabola face down. If a is positive, the parabola faces up.**

**2. r - first x-intercept**

**3. s - second x-intercept**

Below is a YouTube video to help you understand how to graph a parabola from factored form:

## Expanding

The example below explains how expand an equation

## Common Factoring

When you find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".'

## Factoring by grouping

## Factoring Simple Trinomials

Equation- **x2 + bx + c**

Step 1: Find multiples of x2 and c

Step 2: Form two brackets

Step 3: Insert the multiples of x2 and c in both of the brackets

Step 4: Expand the brackets to check if you got the right answer

Below is an example of simple trinomials question.

## Factoring Complex Trinomials

Equation - ax2 + bx + c

Step 1: Find multiples of ax2 and c

Step 2: Form two brackets

Step 3: Insert the multiples of ax2 and c in both of the brackets

Step 4: Expand the brackets to check if you got the right answer

Below is an example of complex trinomials

## Factoring Special Trinomials

**Perfect squares**- a value and z are squares

**(x+z)2**

Step 1: Find the multiples of a and c

Step 2: Multiply the numbers by 2 to get the middle number

Step 3: Insert the multiples of the squares in the equation above

**Different of squares** - ( x + z ) ( x - z )

Step 1: Find the multiples of x and c

Step 2: Insert both the numbers you got in the brackets one + and the other -

It is very important to know the difference between them. Below is an example of this equations

## Video on Factoring

## Completing the square

Example:

**Steps**- Remove the common factor form x2 and x term -coefficient
- Find the constant that must be added and subtracted to create a perfect square
- Group the 3 terms that form the perfect square. (move the subtracted value outside the bracket b multiplying it by the common factor first)
- Factor the perfect square and collect like terms

## Solving by isolating for x

Below is a step by step example that shows you how to answer questions like this.

## Discriminant

**discriminant**is the name given to the expression that appears under the square root (radical) sign in the quadratic formula.

**Discriminant - b2 - 4ac**

It quickly tells you the number of real roots, or in other words, the number of *x*-intercepts, associated with a quadratic equation.

D>0 - 2

D<0- 0

D=0- 1

Below is an example of how to find a discriminant

## Quadratic formula

A new formula is introduced which is:

Below is an example of a quadratic formula question

## Shape problems

The questions mostly asked is to find the area, perimeter and missing sides.

Below is an example about a right triangle

## Revenue Problems

Below is an example of a revenue problem

## Motion Problems

Example:

## Consecutive Integers

Below is an example of a consecutive integers

## REFLECTION

## Links between topics in Quadratics

## Factoring connects to Graphing

- Factoring helps you understand equations such as complex trinomials and helps you know what each variables does while graphing
- It also helps you know what the x-intercepts are and where the vertex is by calculating

## Standard form connects to Vertex form

- Both topics are linked back together through the topic of Completing the square
- You convert standard form to vertex form in order to get the vetex

## Discriminant connects to Quadratic Formula

- The discriminant equation goes back to half of the quadratic equation

## Simple trinomials connects to Complex trinomials

- Simple and Complex have the same equation except for how the a-value is equal to 1 in simple but in complex, the a-value equals to more than 1