# Quadratic Relations

### Final Submission || Harman Virk

## Table of Contents

**Introduction:**

- The Parabola

**Graphing Vertex Form:**

- First and Second Differences
- Identifying Transformations of a Parabola
- Word Problem

**Factored Form:**

- Video of Factoring
- Different Types of Factoring and Examples
- Factoring to determine x-intercepts
- Word Problem

**Standard Form:**

- Completing the Square
- Quadratic Formula
- The Discriminant
- Word Problem

**Conclusion:**

- Assessment
- Reflection

## The Parabola

**Everything you ever wanted to know about parabolas...**

- Parabolas can open up or down
- The zero of a parabola is where the graph crosses the x-axis
- "Zeros" can also be called "x-intercepts" or "roots"
- The axis of symmetry divides the parabola into two equal halves
- The vertex of a parabola is the point where the axis of symmetry and the parabola meet. It is the point where the parabola is at its maximum or minimum value.
- 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

## Graphing Vertex Form

## First & Second Differences

## Identifying Transformations of a Parabola

**The 'a' value**

If **a > 0**, the parabola opens up.

If **a < 0**, the parabola opens down.

If **-1 < a < 1**, the parabola is vertically compressed.

If **a > 1 or a < -1 **the parabola is vertically stretched.

**The 'k' value**

If **k > 0**, the vertex moves up by k units.

If **k < 0**, the vertex moves down by k units.

**The 'h' value**

If **h > 0**, the vertex moves to the right by h units.

If **h < 0**, the vertex moves to the left by h units.

## Word problem

## Factored form

## Types of Factoring

## Different Types of Factoring & Examples

- The greatest common factor (GCF) is the largest integer that divides evenly into each of a given set of numbers.
- When factoring polynomial expressions we must find the greatest common factor.
- We look for the greatest common numerical factor, and for the variable with the highest degree of the variable common to each term.
- We are looking for a whole number factor

- A simple trinomial is a quadratic where 'a' = 1
- Given a quadratic in standard form, you can factor to get factored form

- 'a' does not equal 1
- Always look at the common factor first when factoring trinomial
- Not all quadratic expressions can be factored.

- Always look for a common factor first when factoring a trinomial
- We have two different terms

## Factoring to determine x-intercepts

**1) Replacing y with 0 and factoring the quadratic **

**2) Setting each bracket = 0 and solving for x**

*x*^2 + 12

*x*+27

0 = (x + 9) (x + 3)

x + 9 = 0 --> x = -9

x + 3 = 0 --> x = -3

**The x-intercepts are -3 and -9 **

## Word Problem

## Standard Form

## Completing the Square

y = ax^2 + bx + c ----> y = a (x - h)^2 + k

## Quadratic Formula

## The discriminant

**D > 0**, there will be two solutions.

When **D < 0**, there will be zero solutions.

When **D = 0**, there will be one solution.