# Quadratic Relations Website

### Everything you need to know about Grade 10 Quadratics!

## Made by Kamran Samra. Grade 10. Novemeber 2014

## What is Quadratics anyway? Give an example

For example, Quadratics can be used to graph the flight path of an object, for example, in football when you throw the football it makes a U shape and we can use quadratics to ensure that the ball will go in the direction and location you want it to go in. You can also use quadratics when it comes to business, you can use them in business meetings with charts when you want to make the highest amount of money possible and the highest point (the vertex) will be that mark. Quadratics are everywhere!

## Table of Contents

**Introduction to Quadratics**

- What is a parabola?
**(Basic Terminology)** - Second Differences

**Vertex Form y=a(x-h)²+k**

- Effects of Vertex form
- Axis of symmetry
- Optimal Value
**(y=k)** - Transformations
- Step Pattern
- X-intercepts or zeros
**(Subbing in y=0 and solving)** - Graphing Vertex Form

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

- Zeroes or x-intercepts
**(r and s)** - Axis of symmetry
**x=(r+s)/2** - Optimal Value
**(Subbing in)**

**Standard Form y=ax²+bx+c**

- Zeroes
**(Quadratic Formula)** - Axis of Symmetry
**(-b/2a)** - Optimal Value
**(Subbing in)** - Completing the square to turn to vertex form
- Factoring to turn to factored form

**Factoring**

- Common Factoring
- Simple Trinomial
- Complex Trinomial
- Perfect Squares
- Difference of Squares
- Factoring by Grouping

## Welcome to the world of Quadratics

## What is a Parabola?

**graphed equations**a

**parabola**. All Parabolas have a:

**Vertex**- where the optimal value and the axis of symmetry meet. (The highest and lowest part of the line)

**Axis of symmetry**- the half line of the curve (vertical line that splits the parabola in half)

**Optimal Value-** the highest or lowest point of the parabola (determined by the vertex)

**Y-intercept**- the point when the curve crosses the y-axis ( all numbers above optimal value)

**X-intercepts/Roots/Zeros- **the points where the curves cross the x-axis (all real numbers)

## Second Differences

**first differences**are the same the relation is

**linear**. If the

**second differences**are equal are the same, the relation is

**quadratic**.

## Now that we have a basic overview of Quadratics, Lets Begin with Vertex Form

## Effects of Vertex Form. What is it?

**vertex**and the one of the x-intercepts of an equation. We can sub it into the equation

**y=a(x-h)²+k**where the vertex is represented as

**(h,k)**to solve for

**a**.

## Transformations. Each variable represents a transformation

The **(-h) **value moves the vertex of the parabola **left **or **right **

When (-h) is negative move right and when its positive move left.

The **(k)** value moves the **vertex **of the parabola **up** or **down **

When **(k)** is **negative **move **downwards **and when its **positive **move **upwards**.

The** (a)** stretches the parabola and if the

If **(a) **is **positive **the parabola opens **upwards**, if **negative **it opens **downwards**.

Vertex= **(h,k)**

## Step Pattern

**(a)**value. determines the orientation and shape of the parabola

**(the stretch)**.

## x-Intercepts (y=0)

To find the** x-intercepts** in **Vertex Form**, we must sub **y=0.**

Example: **y=4(x-3)****²-1**

y=0.

0 = 4(x² - 6x + 9)-1

0 = 4x² - 24x + 35

0 = (2x-5)(2x-7)

x = 5/2 and x = 7/2

**x-intercepts (2.5,0) (3.5,0)**

## Graphing Vertex Form y=a(x-h)²+k

**Step 1:**Find the

**Vertex (h,k)**

**Step 2:** Find the **y-intercept **(Let x=0 and solve for y)

**Step 3**: Find the **x-intercepts **(Let y=0 and solve for x)

**Step 4:** Graph the parabola using all 3 points (**vertex, x-int, y-int**)

Please see the video attached for clarification.

## Factored Form y=a(x-r)(x-s)

## Zeroes or x-intercepts (r and s)

A **zero **of a parabola is another name for the** x- intercepts.**

in order to find the **x- intercepts** you must set** y=0**

**Example: y=-(x-2)(x+4) **Each bracket represent**s a x-intercept.**

**0=(x-2) **Move the **-2 **to the **opposite side of the equal sign** to **isolate **x and change it to **+2**

**x=+2 (2,0)**

**0=(x+4) **Move the **+4** to the **opposite side of the equal sign** to **isolate **x and change it to **-4**

**x=-4 (-4,0)**

The zeroes **(x-intercepts) **are **(2,0) (r) **and **(-4,0) (s) **in the equation **y=-(x-2)(x+4)**

## Axis of Symmetry x=(r+s)/2

To find the axis of symmetry, we must get our two zeros from before **(2 and -4)**, **add **them and then **divide **them by 2. Using the following formula **x=(r+s)/2**

**-4+2=-2 **Add both the zeroes together.

**-2/2=1 **Divide the number by two.

**Therefore the axis of symmetry is -1.**

The image to the right shows what an Axis of Symmetry is (blue line)

## You can now get your Optimal Value and x-intercepts and plot them on the graph to create your parabola

## Please refer to the attached video for clarification on how to graph using Factored Form

## Standard Form y=ax²+bx+c

## Axis of Symmetry (-b/2a)

Example: 5x²+6x+1. **b=6 a=5**

**-(6)/2(5)**

**-6/10**

**=-0.6**

Therefore the axis of symmetry **(-0.6,0)**

## Zeroes (Quadratic Formula) x = -b ± √b² - 4ac / 2a

Identify the coefficients **(a=5 b=6 c=1) **

Now **sub **the values into the **formula **

**x=-(6)± √(6)²-4(5)(1) / 2(5)**

x=-6± √(36)-(20) /10

-6± √(16) /10

(-6±4) /10

**-6+4**/10=-0.2** (x-intercept (-0.2,0)**

Or

**-6-4**/10=-1 **x-intercept (-1,0)**

Two x-intercepts are are** (-0.2,0) **and **(-1,0)**

**NOTE: **You **cannot **take the square root of a **negative** number. (No x-intercepts)

## Discriminates (D)

**inside**of the

**square root**in the equation ( -b ±

**√b² - 4ac**/ 2a)

It shows how many **solutions (x-intercepts)** the equation has

**Note: **

If the Discriminate is **negative**, there are **no solutions** to the equations

If the Discriminate is **positive**, there are **2 solutions** to the equation

If the Discriminate is **zero**, there is **1 solution** to the equation.

## Completing the Square to Turn into Vertex Form

**(ax²**+bx+c) to a(x-h)²+k

**Example: y=2x**²+8x+5

** First**,

**Group like terms**like (x² and x terms) together by putting brackets around them.

y=**(**2x²+8x**)**+5

** Second**, Common factor

**(ONLY THE CONSTANT TERMS)**

y=**2**(x²=4x)+5

** Third**, Complete the square

**inside the bracket**

y=**2**(**(x**²**+4x+4) -4**

__)__+5

** (4/2)****²**** **** ^ ^ Note: **You need to **subtract** 4 so you **don't make any changes** to the equation!

__Fourth__, Write the trinomials a binomial squared . We must also expand into the brackets.

y=2** (x+2)**² **-8+5**

__Vertex Form:__ **y=2(x+2)****²-3 **__The Vertex__** (h,k) **is** -(2,-3)**

**Please use the video below for reference**

## Factoring

## Common Factoring

**Example: 8x+6**

First, we have to find the GCF (Greatest Common Factor) which is **2**

Divide **8** and **6** by **2. **

Your new answer should be

**=2(4x+3) **

**Please refer to the example and video posted below**

## Simple Trinomials x²+bx+c

**ax²+bx+c**(when

**a=1**) we have to find

**2** numbers that **add** up to the **b value**

**2** numbers that **multiply** together to give the **c value**

**Example: x²-7x-18**

**=(x-_)(x+_) **

So what 2 numbers that are **added** together equal **-18 **and **added** together equal **-7**

__-9__x__2__=-18

__-9__**+**__2__**=-7**

Therefore the answer is **(x-9)(x+2)**

**For any further questions please refer to the video below**

## Complex Trinomial x²+bx+c

There are two ways to factor Complex Trinomials, the first way is:

__Decomposition__

**Example: 5x²+12x+4**

The **first** and **last** **value** multiply together **(5x4=20)** We have to find what two numbers **multiply **together that **add** to 20 and add together to get **12 (10 and 2)**

Now we replace** 10x **and** 2x **into** 12x value **and our trinomial now looks like

**(5x²+10x)+(2x+4)**

Now we have to find the **GCF **for the first and second brackets and divide them. **First Pair=5x Second Pair=2**

5x**(x+2)**+2**(x+2)**

Therefore the answer is** (5x+2) (x+2)**

__Trial and Error__

Here, we have to guess and check and keep substituting numbers until we get the correct answer.

Example:** 4x²+12x+9**

First we start off with two brackets, we know that we can use (2x+__)(2x+__) that will get us 4x². Now we have to substitute numbers in that will get us 4x²+12x+9.

(2x+3)(2x+3) Expanded: 4x²+6x+6x+9 (Collect like terms and we get **4x²+12x+9**)

**For any further questions please refer to the video I prepared. **

## Perfect Squares and Difference of Squares

**Perfect of Squares include 1,4,9,16,25,36,49,64,81,100 etc.**

First, the **first** number and **third **number can be square rooted

**Example**: **4x²+12x+9**

**√4x²=2x**

**√9 =3**

**(2x+3)²**

__Check your answer__

**(2x)(3)(2)=12x**

**Difference of squares**

** Two numbers in the expression where both numbers can be square rooted and the second number has to be negative. **

## Sample Assesments

## Reflection

## Word Problems

## Motion Problems

## Area Problems

**Area of 204cm**²

**. The length**of the rectangle is

**5cm more**than

**the width.**Find the dimensions of the rectangle.

Calculators are sold to students for **20 dollars** each.** ****300 **students are willing to buy them at that price. For every **5 dollar** increase in price, there are **30** fewer students willing to buy the calculator. What selling price will produce the **maximum ****revenue** and what will the **maximum revenue **be?