# Intro to Quadratic Relationships

### BY: NUFAEL NASIR

## Welcome to the world of Quadratics! My name is Nick and I will guide you through this magical journey!

## Why do we use quadratics?

## Overview of the Topics to be Covered

__What is a Parabola?__

__Types of quadratic equations__

- Vertex form

- Standard form

- Factored form

__Vertex form__

- Axis of symmetry

- Optimal Value

- Transformations

- Translations
- Stretched/Compressed
- Reflection

- Step pattern

__Standard form__

- Zeroes

- Axis of symmetry

- Optimal value

- Completing the squares to convert to vertex form

- Factoring to turn to factored form

- Common
- Simple trinomial
- Complex trinomial
- Perfect squares
- Difference of squares

__Factored Form__

- Zeros or X-Intercepts
- Axis of Symmetry
- Optimal Value

__Word Problems__

-Sam's Room word problem

__Reflection on the Unit__

- Assessment reflections

## What is a Parabola?

A parabola has a few terms that make up the parabola, these include:

**The vertex**

This is the highest point of the parabola, it is made up of the axis of symmetry and the optimal value.

**x-intercepts**

This one is straight forward; it is the points of the parabola that are on the x-axis, where they "intercept". They are also called zeroes.

**Axis of symmetry**

This is the very middle of the parabola, it can be found by adding the x-intercepts and dividing them by 2. It is an x value.

**Optimal Value**

This is the highest point on the y-axis, it makes up the vertex with the axis of symmetry.

## Types of Quadratic Equations

-STANDARD FORM

-FACTORED FORM

## Vertex form

## This is a quadratic equation in vertex form:

## Axis of Symmetry

## Optimal Value

## transformations

## Translations

y=a(x-h)² -k is what allows the parabola to move left or right. If the h is +3, then you move left by 3 units, if the h is -7, then you move right by 7 units.

## Stretched/Compressed

y=a(x-h)² +k. If the "a" value is positive that means it's being vertically stretched by the amount of "a". If the "a" value is negative that means the parabola is being vertically compressed by the amount of "a". Compressing or stretching would make the parabola wider or thinner.

## Reflection

## X-intercepts/Zeroes

For example let's use this equation:

y= 4(x-4)² - 20

0=4(x-4)² -20

20/4=4(x-4)²/4

5=(x-4)²

√ 5=(x-4)²

±2.24=x-4

±2.24+4=x

The x-intercepts would be

x=6.24

and

x=1.76

## step pattern

## Standard form

## zeroes

Example:

To explain how zeroes are found using the quadratic formula, I will be using this equation: x² + 3x - 4

Firstly, we must find the a, b, and c parts of the equation, they are:

a = 1

b = 3

c = -4

Now you can simply substitute those into the quadratic equation:

x= -3+/- √3²-4(1)(-4)/2(1)

x= -3+/- √9+16/2

x= -3+√25/2

x= 1

x= -3-√25/2

x= -4

Therefore, the zeroes of the equation are x=1 and x=-4.

Next I will teach you how to find the axis of symmetry!

## axis of symmetry

= (1+(-4))/2

=-3/2

=-1.5

This is how the equation would look when graphed on Desmos. A link to Desmos is below.

## Optimal Value

x² + 3x - 4

-1.5² + 3(-1.5) - 4

2.25 - 4.5 - 4

= -6.25

Therefore, the optimal value of the equation would be -6.25. The vertex if the equation would be (-1.5,-6.25).

## Completing the squares (factoring to vertex form)

Below is a video by Mr. Anusic teaching how to use the 'completing the squares' technique to change a standard form equation into vertex form:

## Factoring standard form to factored form

- Common factoring
- Simple trinomial
- Complex trinomial
- Perfect squares
- Difference of squares

Below is a great video by Mr. Anusic that covers the different forms of factoring.

## Factored form

## Zeros or X-Intercepts

## Axis of Symmetry

(x=(r+s)/2). For the parabola above, the axis of symmetry would be:

(AOS=(3+1)/2)

AOS=4/2

AOS=2

Therefore the axis of symmetry would be 2.

## Optimal Value

y=(2-1)(2-3) +3

y=1(-1) +3

y=-1 +3

y=2

Therefore the optimal value in this equation would be 2!