# Quadratics Realtionships

### Grade 10 Academic Math

## Summary

## Introduction

*y*=

*ax^*2 +

*bx*+

*c*or

*y*=

*a*(

*x*-

*h*)^2 +

*k or x^2+bx or y=(x+a) (x+b)*. During this time we will graph quadratic equations; as well as learn how to find out if an equation is quadratic. When graphed quadratic equations form a parabola shape. Furthermore in the unit we visit factoring; in which we will mainly focus on how to factor simple/complex binomials and tri-nomials. We will factor equations such as a

*x^*2 +

*bx*+

*c*= 0 to find two binomial expressions, and use zeroes for graphing. Later on we will learn completing the square method to graph equations in

*y*=

*ax*2 +

*bx*+

*c*form.

*Finally we will learn a formula for equations in form of*

*ax*2 +

*bx*+

*c*= 0; which is called the quadratic equation.

## Table of contents

__Understanding a Parabola__

-What is a parabola?

-Analyzing a parabola (second differences)

__Types of equations__

__Vertex Form__

- Understanding the step pattern

- Describing Transformation

- Graphing vertex form x

- Isolating for a variable

-Finding optimal value

-Finding axis of symmetry.

__Factored form__

- Factors and zeroes

- Finding axis of symettry

- Finding optimal value

- Factoring using algebra tiles

- Expanding factored form equations to standard form

- Factoring simple trinomials

- Factoring complex trinomials

- Other forms of factoring (differences of squares, perfect squares)

- Reviewing graphing with factored form

__Standard form__

- Using quadratic formula

- Completing the square

__Word Problems__

## What is a Parabola?

## Properties of a Parabola

## Labelled Parabola

## How to find out if an equation is quadratic

## Extra Practice on calculating second differences

## Vertex Form

## Breaking down and understanding the equation & transformations

__Term Representations__-The Y variable represents the Y-intercept

-The A variable represents whether the parabola opens up or down (positive **a** value results to parabola opening down, negative **a **value results to the parabola opening down), it also represents if the parabola will have a compression or stretch

- The X variable represents the x intercept

- The H variable represents where the X point is on the graph (right or left) (axis of symmetry) (If h is positive 6 the parabola will be horizontally translated 6 units to left)

- The ^2 creates the arch shape of the parabola

- The X variable is in charge of whether the parabola will be vertically shifted up or down (according to positive or negative sign) (positive K value will shift parabola up and negative K value will shift parabola down)

**After you plot your vertex using the transformations explained you can use the step pattern to plot your vertex form equation!**

## Understanding Transformations

**Y=a(x-h)+k**

**With all variables at 0 the parabola would be at the origin at 0,0**

- The (a) value stretches or compresses the parabola.** If the (a) value is greater than 1 the parabola will stretch** by the factor of the value of the variable; This is what is called a **vertical stretch. If the (a) value is less than 1 the parabola will compress** by the factor of the value of the variable; this is known as a** vertical compression.**

- The (-h) represents whether the parabola moves left or right. ***RECORD** (When the h value is positive the parabola moves left and when the h value is negative the parabola moves right). This is called a** horizontal translation.**

- The (k) variable represents whether the parabola moves up or down. ** *RECORD** (Whenthe k value is positive the parabola moves up, when the k value is negative the parabola moves down). This is called a **vertical translation.**

**Did you know? The H value and K value make up the vertex?**

## The Step Pattern

Over 1 up 1

Over 2 up 4

Over 3 up 9

**Note**: *When using the step pattern we just square the number we are going over by to find the number we go up by

**Note:** *When there is a whole positive/negative: number, fraction, decimal in place of variable a value you just simply multiply the a value to the squared part of the step pattern

## finding Axis of symmetry in vertex form

**(y**

**=a(x-h)+k**) in a vertex form equation you need to look at the h value as explained briefly before. If the (h) value is positive you will simply need to change the sign, and what the value then becomes is the axis of symmetry.

For example:

y= 4(x+5)+6

----**a**----**h**----**k **

** **In the equation the value of h is +5, so when the sign is flipped the **axis of ****symmetry is -5.**

## Finding optimal value of a standard form equation.

For example:

**y= 4(x+5)+6**

----a----h----k

Therefore the k value is positive 6, meaning the optimal value of the parabola will be +6 over the x-axis.

## Isolating for x in vertex form (finding x intercepts/roots)

## Graphing vertex form

## Factored form

## Finding Zeroes in factored form

## Finding axis of symettry in factored form

For example: X int @(6,0), (-4,0)

(a+b)/2

(6+-4)/2

2/2

1=Axis of symmetry Therefore after averaging the x intercepts 1 is the axis of symmetry.

## Finding the optimal value for equations in factored form

## Common Factoring

## Factoring simple trinomials

## Factoring Complex Trinomials

## Special cases of factoring (difference of squares and perfect squares)

## Using algebra tiles to factor (particularly for visual learners)

## Expanding Factored Form equations to standard form

## video about Graphing with factored form (made by myself)

## Standard form

## The Quadratic formula

## Quadratic Formula Song

## Finding x intercepts of standard form equations using the quadratic formula

## Finding the axis of symmetry of a standard form equation

__Lets do an example, let's find the optimal value for the equation we found x-intercepts for above (3x^2-4x+1).__

__Method 1 (adding both x intercepts and divide by 2)__

Axis of symmetry: (1+1/3)/2

=0.666

Method 2- Using -b/2a (variables were identified in previous part)

Axis of symmetry: -b/2a

=-(-4)/2(3)

=4/6

=0.66

Therefore we applied both methods taught to find for the x intercepts of a standard form equation and we managed to get the same answer proving that our answer is correct.

## Finding the optimal value of a standard form equation

**With the example used above for explaining how to calculate x intercepts and axis of symmetry, we will also use the same example to explain optimal value.**

__Equation____ being ____subbed____ into: __(3x^2-4x+1=0 X=0.66 (Axis of symmetry)

3(0.66)^2-4(0.66)+1

=-.3332

Therefore you have your optimal value.

## discriminants

**negative there is no solution****is equal to 0 there is only 1 solution****if greater than 1 there is 2 solutions.**

## Completing the Square

## Connections chart

## Word problems

## Revenue Questions

## Optimization Fencing problem

## unit reflection

Below is a picture of my factoring unit test: