# Quadratic Relations

### By: Shayan Ahmed

## Table of Contents

__What is Quadratics?__

- Introduction
- Parts of a Parabola

__Vertex Form__

- How to Graph

__Factored Form__

- How to Graph
- How to find the AOS
- How to find the optimal value
- Finding the x-intercepts

__Using Polynomials__

- Multiplying Binomials
- Standard Form ---> Factored Form

__All Types of Factoring__

- Common Factoring
- Factoring by Grouping
- Simple Trinomial
- Complex Trinomial
__SPECIAL PRODUCTS__

- Difference of squares
- Perfect squares

__Standard Form__

__How to Complete the Square__

- Standard form ---> Vertex form

__Quadratic Formula__

- Applying the quadratic formula
- Discriminant

__WORD PROBLEMS__

- Revenue
- Height & Distance
- Area

__Connections__

__Reflection and Assessment__

__External (Helpful) Links__

## Introduction

A quadratic relation has a second difference. If you have a linear relation (grade 9), you realize that is a first. The second difference of quadratic relations are the same. For a better understanding, refer to the diagram below

## Key Terms

In a quadratic relation, there as several key terms as follows

Vertex - The maximum or minimum point on the graph. It is the point where the graph changes direction

How to label ---> (x , y)

Minimum /Maximum Value - The highest or lowest point on the y- coordinate

How to label ---> y= ___ (min or max)

Axis of symmetry - A line which equally divides the parabola vertically

How to label ---> x= ___

y- intercept - Where the parabola intersects with the y- axis

How to label ---> y=___

x- intercept - Where the parabola intersects with the x- axis

How to label ---> x=___

Zeroes - same as the x-intercept

Opening - the direction in which the parabola opens. If the two arms are pointing up, it is opening up and vice versa.

## Vertex Form: y=a(x-h)^2=k

The basic equation for a parabola is y=x^2. Although in vertex form there is a, h, and k. Let me show you what their jobs are.

__Step Pattern:__The basic step pattern is as follows, you take the value of y, and square it. So if y=1, x=1 or if y=3, x=9 etc.

__a:__Is the stretch factor in the equation. When "a" is changed, the step pattern of the equation changes. If "a" value changes, the step pattern is altered as well. If "a" is greater than 1, the line gets stretched. If "a" is less than 1, the line becomes compressed. Lastly, if "a" is a negative number, the graph is reflected on the x- axis.

__h:__Is the opposite reciprocal of the x-coordinate in the vertex. It moves the parabola left or right on the x-axis. If "h" is positive, it moves that many units to the left. If its negative, it moves that many units to the right.

__k:__Is the y-coordinate of the vertex. "k" has the ability to move the parabola up or down on the y-axis. For example, if the value of "k" is 6, it would move 6 up. If the value is -6 it would move 6 down.

## Factored Form

**How to graph:**

Factored form is simple but it requires several steps to locate key

points which are then used to graph.

__Direction of Opening:__Is determined by "a", if "a" is positive, parabola will be opening up. If "a" is negative, it will open down.__Zeroes:__In the equation, "r" and "s" are the zeroes/ x-intercepts. "r" and "s" are the opposite reciprocal of the zeroes. If the equation was, y= (x+4)(x+2), the zeroes would be -4, and -2.__Axis of symmetry:__The axis of symmtery is the x-coordinate of the vertex, which brings you one step closer to completing your parabola. Using the given zeroes, the axis of symmetry is [(r+s)/2]. Using the example above, the axis of symmetry is -3.__Optimal Value:__The optimal value is the y-coordinate of the vertex and the last step of factored form. In order to get the optimal value, you need to sub in the AOS. For example, y= (x+4)(x+2), the AOS is -3. Which makes it y= (-3+4)(-3+2), once solved, you get: min, y=-1.

**Finding the x-intercepts:**

The basic equation for factored form is y = ( x - n) ( x - n ) , where n is equal to any number. Much like the h value from the vertex form equation, n is always opposite in the factored form.

Look at the following, if an equation is y = ( x + 3 ) ( x - 5 ) , then the zeroes are located at (-3, 0) and (5, 0)

## All Kinds of Factoring

## Common Factoring

## Simple Trinomial Factory

## Complex Trinomial Factoring

## Using Polynomials

**polynomial**is an expression consisting of variables (or indeterminates) and coefficients" - Dictionary. To make it simple, it's a expression which has multiple terms. The naming of polynomials comes down to the number of terms, eg. two: binomial. A common or possible binomial in quadratics can be, x^2+4x+20

Multiplying Binomials, two term expressions, is going from factored form to standard form, if you watch the following video it explains how it is done.

## Multiplying Binomials (expanding)

## Completing the Square

The following steps show how to go from standard to vertex form.

- Is the standard form.
- Common factor out any term from the x^2 and the x.
- Using the "x" term, in this case "t", divide it by 2 and then square it. Since you are not allowed to change the equation, you have to add and subtract the number.
- Multiply out the subtracted value by the common factor and write it outside the bracket.
- Factor the perfect square (the number in the brackets) and rewrite it.
- The vertex form

## Applying the quadratic formula

This is a good way to find the vertex and graph from standard form. If the discriminant is positive, you can easily graph.

## Revenue Problem

## Height&Distance Problem

## Area Problem

## Connections

__All three forms__

All the quadratic forms interact, they each are good for certain tasks. Factored form is a almost instantaneous way to get the zeroes. Vertex form quickly shows you the vertex and is easy to find. Standard form is good for quadratic equation and getting roots as well. They all can be graphed so that is one more common thing. The expression is a good way to draw a rough graph which helps to image a question before solving. To conclude, all of the forms are useful in their own way and are a advantage if used properly

## Thinking Test (three pages)

## Reflection

I did fairly well on some tests but I felt I could of done a lot better if I understood certain concepts better. The thinking test was probably my best test or one of the best, I was well prepared and made on minor, silly mistake. Otherwise I was satisfied.