# Quadratics

### Grade 10

## What is Quadratics

## Table of contents

**Parabola**

- What is a Parabola?

- Where can we find it?

- Key features of a parabola

- Tips

- Examples

- First and second differences

**Vertex Form**

- Parts of the vertex form equation

- Examples

- Transformations of parabolas

- Step Pattern

- How to find a equation given the vertex

- Word Problem

**Factored Form**

- Parts of the standard form equations

- x- intercepts/zeroes (r and s)

- Axis of Symmetry (x=h)

- Optimal value

- Graphing

**Standard Form**

- Expand and Simplify

- Common factoring

- Simple Trinomial

- Complex Trinomial

- Word Problem

- Perfect Squares

- Difference of Squares

- Quadratic Formula

**-Connections**

**- Reflection**

## Lets Start From The Begining

## What is a Parabola ?

**A parabola is a graph of a quadratic relation.****Parabolas can open down or up ( negative or positive)****You can find them everywhere, just look around**

## Key Features of Quadratic Realations

## Tips

- Positive: The parabola is positive when it opens down
- Negative: The parabola is negative when it opens up
- Maximum: The parabola is maximum when it is negative because that is the highest it can get
- Minimum: The parabola is minimum when it is positive because that is the lowest it can get

## Example : 1

## Example 1: Quadratic Relations

## Example 2

## Example 2: Quadratic Relations

## First and Second Differences

## Vertex Form

## Tips

- If the equation is "- a " the direction of opening will be negative ( the parabola will open up)
- If the equation is "a" the direction of opening will be positive (the parabola will open down)

## Example of vertex form

## Transformations of Parabolas

*** Remember the vertex: y = a (x-h)²+h**

The basic parabola has the formula **y=x****²** (a=1 ,h=0 ,k=0)

**How does the value of "a" determine the orientation and shape of the parabola?**

**Orientation: **

- If "a" is greater than zero, the parabola
**opens up**

- If "a" is less than zero, the parabola
**opens down**(reflection)

**Shape:**

- If "a" is greater than -1
**or**if "a" is less than 1, the parabola is**compressed** - If "a" is greater than 1
**or**if "a" is less than -1, the parabola is**stretched**

**How does the value of "k" determine the vertical position of the parabola?**

- If "k" is greater than zero, the vertex
**moves up**by k units - If "k" is less than zero, the vertex
**moves down**by k units

**How does the value of "h" determine the horizontal position of the parabola?**

- If "h" is greater than zero, the vertex moves
**to the right**h units - If "h" is less than zero, the vertex moves
**to the left**h units

## Graphing from vertex form ( step pattern)

## General rule : 1a, 3a, 5a

## Example: 1y = x² + 5Step pattern:
1 x 1 = 1 x 3 = 1 x 5 = | ## Example: 2y = 2 (x + 7) - 3Step pattern:
1 x 2 = 3 x 2 = 5 x 2 = |

## How to find a equation given the vertex?

*** In order to find the equation using the vertex you will need**

__two points__from the graph**Example: **Sub in the points and solve

Vertex:** (3,-1)** (h, k)

Points: **(1, 7)** (x,y)

**y= a (x-h) ² + k** **(sub in the vertex and points)**

**7= a (1 - 3) **² - 1

**7 = a (-2)**² - 1

7 = a (4) - 1

7 = 4a - 1

7+1 = 4a

8 = 4a

**8/4 = 4a/4**

**2 = a**

To form the equation sub in the vertex and "a"

**Vertex: (3, -1)**

**"a": 2**

y = a (x - h)² +k

__Equation:__ __y= 2 (x - 3 )² -1__

## Word Problem

Joey kicks a soccer ball into the air. Its height in meters after "t" seconds is shown by

h= -5 (t - 2)² + 4

a) What is the height of the soccer ball when it was kicked?

* Find the "h" value

t =0

h= -5 (t - 2)² + 4

h= -5 (0 - 2)² + 4

h= -5 + 4 +4

**h=3 m**

**Therefore soccer ball was 3 m off the ground**

b) What was the maximum height of the soccer ball?

* Find the vertex

**(2,4)**

**The maximum height of the ball was 4 m after 2 seconds**

c) How high was the ball after 6s?

* Sub in 4 into the "t" value

**h= -5 (t - 2)² + 4**

**h= -5 (6 - 2)² + 4**

**h= -5 + 16 + 4**

**h= 15 m**

**Therefore after 6s the ball reached the height of 15m**

## Factored Form

## Second Quadratic function

__y = a (x - r) (x - s)__What do you learn from this equation:

- "a" - direction of opening

- step pattern

- The "r" and "s" equal to the zeros/ x-intercepts
- The Axis of symmetry (x=h) = r + s / 2
- Optimal value = sub "x" into the original equation

## Step 1: Zeroes/x-intercepts (r and s)

*** Make "y" into zero**

0 = 4 (x + 5) (x + 7)

*** set the x's to zero**

x + 5 = 0

**x = - 5**

x + 7 = 0

**x = - 7**

**The x- intercepts equal (x = - 5), (x = - 7)**

## Step 2: Axis of symmetry X=h

X - intercepts : (x = - 5) (x = -7)

**Axis of symmetry: **

*** add "r" and "s" together and then divide by two**

x = r + s/ 2

x = - 5 + (-7) /2

x = -12/2

**x = - 6**

## Step 3: Optimal value

X intercepts: (x = -5) (x = -7)

Axis of symmetry: x = -6

*** sub in the axis of symmetry into the original equation to get optimal value (y value)**

y = 4 (x + 5) (x + 7)

y = 4 (- 6 + 5) (- 6 + 7)

y = 4 (- 1)(1)

y = - 4(1)

**y = - 4 **

## Final step: Graphing

Original expression: **Y = 4 (x + 5) (x + 7)**

X - intercepts: **x = - 5**,** x = -7**

Axis of symmetry: **x = - 6**

Optimal value: **y = - 4**

Vertex: (Axis of symmetry, optimal value) **(- 6,- 4)**

*** **One way to check if your graph is right, look at the original equation.

- If your original equation is negative [For example: y = - 4 (x - 3) (x - 4) ], your graph will open downwards

- If your original equation is positive [For example: y = 4 (x + 5) (x + 7) ], your graph will open upwards

## Standard form

## Factoring

**1. Common Factoring:** This is when you divide the whole equation by its common factor

**2. Simple Trinomial: **This is when you have 3 terms in you equation, but you "a" value has an co-efficient of 1. You can factor the equation to get 2 binomials

**3. Complex Trinomial:** This is when you factor out a 3 term equation, but unlike the simple trinomial the "a" has a co- efficient other than 1. You can factor the equation to get 2 binomials

**4. Perfect Squares: **This is when a quadratic equation can be factored into two identical binomials

**5. Difference of Squares: **This is when there is a subtraction sign between two squared terms. The equation can be factored to get 2 binomials.

## Expand and Simplify

**. By expanding it simply means multiply both you binomials, then simplify to get you original equation.**

__expanding and simplifying____Example 1:__ ( j + 2 ) ( j - 5 )

- Multiply the fist binomial with the second binomial

* Multiply 2 with j and -5 ( 2 x j ) and ( 2 x -5 )

* Use the expression FOIL to help you remember the process

**F**irst**O**utside**I**nside**L**ast

**= j² - 5j + 2j - 10**

2. Simplify the equation

* Add the like terms together

**= j² - 5j + 2j - ****10**

**= j² - 3j - 10**

__Example 2: __ -3 (x - 4 ) (x + 5)

- First multiply the binomials together

**= -3 ( x - 4) ( x + 5 )**

**= -3 (x****² + 5x - 4x - 20)**

** **2. Simplify the equation by adding the like terms

**= -3 (x****² + x - 20)**

3.** **Multiply the equation by the a term (In this example -3)

**= **-3 (x**² + x - 20)**

**= -3x**

__² -3x + 60__## Word Problem

**h= -4 (d-5) (d-7).**Where

*d*is the distance represented in metres and

*h*is the height represented in metres.

a) Expand ans simplify the relation

**h = -4 (d-5) (d-7)** * Multiply the two binomials together

**h = -4 (d**² - 7d -5d -35) * Multiply the a value with the trinomial

**h = -4d**² + 48d - 140

b) Using the simplified relation in part a. Determine the height of the shot if *d* equals 2

h = -4d² + 48d - 140 *Sub in the 2 in the d spots

h = -4(2)² + 48(2) - 140

h = -16 + 96 - 140

h = - 60

## Common factoring

**opposite of expanding**. In common factoring you are adding brackets instead of taking them away

__Example 1__: 9x + 6

- First you need to figure out the
**GCM**(Also known as the**G**reatest**C**ommon**F**actor)

**9x + 6**

**GCM = 3**

2. Secondly, write the factored from in brackets

**9x + 6**

**9x/ 3 + 6/3 **

**3 (3x + 2) **

*** ALWAYS REMEMBER TO PUT THE GCM INFRONT OF THE BRACKETS**

__Example__ __2:__ **14x³ - 7x²**

**GCM: 7x****²****(2x - 1)****Final answer:**7x² (2x - 1)

** **

*To check your answer expand and simplify

7x² (2x - 1)

= (7x² x 2x) ,(7x² x -1)

**= **14x³ - 7x²

## Simple Factoring

*"a" value has a co-efficient*, it is called a simple trinomial. When you factor out this simple polynomial you will end up with 2 binomials of the form:

**( x + r )** and **( x + s )**

* A simple trinomial is in the form of **ax****² + bx +c (Standard Form)**

**2 numbers that add up to give b****2 numbers that multiply to give c**

***BOTH THE FACTORS NEED TO BE THE SAME**

__Example__ __1__ : x² -8x +12

**Step 1: **Find factors that multiply to give 12 and two numbers that add up to give -8

___ x ___ = 12

___ + ___ = -8

(-6) x (-2) = 12

(-6) x (-2) = -8

**Step 2: **Put the factors into **(x+r)** and **(x+s)**

(x+r) (x+s)

**(x-6) **and** (x-2)**

__Example__ __2__: 3x**² + 15x + 18**

*** **Before factoring** ALWAYS **check if the equation can be **COMMON FACTORED**

3x² + 15x +18

3x²/3 + 15x/3 + 18/3

**= 3(x + 5x + 6)**

3 x 2 = **6**

3 + 2 =** 5**

**3 (x+3) (x+2)**

## Complex Trinomial

**co-efficient other than 1.**

The complex trinomial involves **multiple steps** to get it factored into 2 binomials

__Binomial common factoring__

**6x (m-4) + 3 (m-4)**

*Hint a binomial can be used as the common factor

(That means **m - 4 **will go infront of **6x + 3**)

**= (m-4) (6x+3)**

__Factor by grouping__

**jl + kl + jm + km**

*When you can not common factor all the terms you can group all the terms together

**( jl + kl ) ( jm+km )**

**= l ( j+k ) + m ( j+k )**

*The binomials **always** should be the same when you group all the terms together

**= (j+k) (l+m)**

***NOW YOU WILL USE ALL THESE STEPS TO FACTOR THE COMPLEX TRINOMIAL**

__THE__ __DECOMPOSITION__ __METHOD__:

__Example__ **1****:**

**3x² + 8x +5**

**1. Product: (a) x (c)**

= 3x5

**= 15**

3 x 5** = 15**

3 + 3** = 8**

* The product is the number that **replaces the "c" value. **

**2. Sub in the factors into the middle of the equation**

**= **3x² **+ 3x + 5x **+5

**3. Factor by grouping**

=** (**3x² + 3x**) **+ **(**5x +5**)**

=** 3x (x+1) + 5 (x+1)**

**4. Put it all together**

= **(x+1) (3x+5)**

## Word Problem

Let "2w + 2" represent the length

Let "W" represent the width

**(L)(W) = 24** *the formula for area is length times width, so just plug the numbers in

**(2w+2)(w) =24** *Multiply the binomials together

**(2w² + 2w) = 24 **

**2w****² + 2w - 24 = 0** *Move the 24 over and make the right side equal zero

## Perfect squares

Formula:

**a² + 2ab +b² = (a+b)²**

**a² - 2ab + b² = (a-b)²**

__Example__ __1__: y² - 8y + 16

* find the factors that **multiply to give you "c"** (16) and the factors that **add up to give you "b"** (-8). But make sure both the factors are the same.

___ x ___ = 16

___ + ___ = -8

(-4) x (-4) = 16

(-4) + (-4) = -8

* Form the factors into binomials

**(y-4)(y-4) * Since both of these binomials are the same you can form it into one**

**= (y-4)²**

## Difference of squares

Formula: **a²- b² = (a+b)(a-b)**

__Example__ __1__: 16m²-4r²

- Find the perfect squares for both terms. (
**4m****²**)(**2r****²**) - Put it into two binomials
**(a+b) (a-b)**

**(4m²+2r²) (4m²-2r²)**

** Example 2: 12k**²+27c²

*Always check if you can **common factor**

12k²+27c²

= 12k²**/3** + 27c²**/3**

*Find the difference of squares

= 3(4k² - 9c²)

The difference of sqaures **(2k²) - (3c²)**

*Put it all together **(a+b) (a-b)**

**3 ( 2k+3c) (2k-3c)**

***Don't forget to put the GCM. The GCM only goes infront of the first bracket**

## Quadratic Formula

**QUADRATIC FORMULA**when there isn't any other possible way to factor or when the numbers don't work out.

**Exact Roots: **The exact roots are the ones where the square root sign is still there.

**Approximate Roots: ** The approximate roots are the ones that are fully solved and there is no square root sign.

## Completing the square

**y = ax² + bx + c**

Vertex From: **y = a (x-h)² + k**

**The confusing question is HOW DO YOU GET FROM STANDARD TO VERTEX FORM**

**Example 1: Convert y= 2x**² + 8x + 5

Step 1: Group the x² and x terms together

y = (2x² + 8x) + 5

Step 2: Common factor only the constant terms

**y = 2 (x² + 4x) + 5 **

**Step 3: **Complete the square inside the bracket

y = 2 {(**x****² + 4x + 4**)-4} + 5 * The bold numbers is a perfect square trinomial **4 = (4/2)²**

**y = 2 (x + 2 ²)-4) + 5 ***Expand and simplify

**y = 2 (x + 2)² - 8 +5 **

**Vertex Form: y= 2 (x + 2) - 3**

## A little gift

**-b/ 2a ***That is all you have to remember

__Example__ __1__: y= x² + 12x + 32

* Find the a,b and c

**a = 1, b = 12, c = 32**

* Sub into the formula

**-b/2a**

**-(12) / 2(1)**

*Then solve to get the **x** value (**h value**) of the vertex

**-(12) / 2(1)**

**x = -6**

* To find the y value (k value) sub in x to the original expression

y= x² + 12x + 32

y= (-6)² + 12(-6) + 32

**y= -36 - 72 + 36**

**y= -72**

**Vertex (-6 , -72)**

## Connections can be made

When you have the vertex form you can factor it to get the x-intercepts. Than you add the zeroes and divide them by two, to get the A.O.S. Sub in the A.O.S into the original equation to get the optimal value. Then all you need to do is graph your parabola.

Whenever you factor you will be left with two binomials. Once you have the binomials you can get the two x-intercepts you can graph your parabola by finding the A.O.S and the optimal value. This is the main part in the whole unit.

## The END

## Reflection

After this quiz, I started to do more practice questions. After doing some more practice question I understood the vertex form better. It took some time because there were a lot of new things to learn, but i understood it better after doing some practice questions. Next time If I don't understand anything I will ask my teacher, try some more questions, ask my friends and try my best.

__TEACH__.