Quadratics

Grade 10

What is Quadratics

Quadratics comes from the word " Quad" which means square.

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

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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

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Example 1: Quadratic Relations

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Example 2

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Example 2: Quadratic Relations

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First and Second Differences

In order to calculate first differences you need to subtract the second y value from the first y value. You do this with each pair of y values. If the first differences are constant that means the pattern is linear. If the second differences are constant that means the pattern is quadratic.
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Vertex Form

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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

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Transformations of Parabolas

* Remember the vertex: y = a (x-h)²+h


The basic parabola has the formula y=x² (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


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Graphing from vertex form ( step pattern)

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General rule : 1a, 3a, 5a

3.2 Graphing from Vertex Form

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

The general expression: y = a (x - r) (x - s)


What do you learn from this equation:

  • "a" - direction of opening
- stretch or compression

- 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)

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

* 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

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


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

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


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

To make a graph, plot all the points: X- intercepts, Axis of symmetry, optimal value and vertex. The vertex is the axis of symmetry and the optimal value.


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

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Graphing in vertex form

Standard form

Factoring

We factor expressions to turn them into factored form. There are multiple ways to do this.


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

* Whenever you factor an equation and you and you are unsure about your answer, you can always check by expanding and simplifying. By expanding it simply means multiply both you binomials, then simplify to get you original equation.


Example 1: ( j + 2 ) ( j - 5 )


  1. Multiply the fist binomial with the second binomial
* Multiply j with j and -5 ( j x j ) and ( j x -5 )


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

* Use the expression FOIL to help you remember the process


  • First
  • Outside
  • Inside
  • Last


= 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)



  1. First multiply the binomials together
= -3 ( x - 4) ( x + 5 )


= -3 (x² + 5x - 4x - 20)


2. Simplify the equation by adding the like terms

= -3 (x² + x - 20)


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

= -3 (x² + x - 20)

= -3x² -3x + 60

Word Problem

A kid is practicing to throw a football. He throws the first shot, the estimated flight path of the ball is shown by the relation: 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 (d² - 7d -5d -35) * Multiply the a value with the trinomial

h = -4d² + 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

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


Example 1: 9x + 6


  1. First you need to figure out the GCM (Also known as the Greatest Common Factor)

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

  1. GCM: 7x²
  2. (2x - 1)
  3. 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

    * Polynomials that have 3 terms and the "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 ax² + 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: 3x² + 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

    *The complex trinomial has "a" value with a 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

    The area of a rectangle is 24. The length is 2 more than the width. Write the expression in vertex form.


    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

    2w² + 2w - 24 = 0 *Move the 24 over and make the right side equal zero

    Perfect squares

    *The perfect squares is a polynomial that can be factored into two identical binomials.


    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

    *This is when there is a subtraction sign between two squared terms. When you factor them out using perfect squares you will end up with two binomials.


    Formula: a²- b² = (a+b)(a-b)


    Example 1: 16m²-4r²


    1. Find the perfect squares for both terms. (4m²)(2r²)
    2. Put it into two binomials (a+b) (a-b)
    (4m²+2r²) (4m²-2r²)


    Example 2: 12k²+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

    Factoring the Difference of Two Squares - Ex 1

    Quadratic Formula

    *We use the 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.

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    Completing the square

    Standard Form: 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= 2x² + 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 {(x² + 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

    * If you do not want to do all this factoring. If you want to find the easy way out. This formula will be your best friend. Also if you need to find the vertex and there are no x- intercepts you can use this formula to help you


    -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)

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    Connections can be made

    This unit is one whole puzzle, everything connects together. From the vertex form, to standard form , to the factored form.


    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.


    This unit has been a crazy unit. There were a lot of new parts we learned and a lot of things we refreshed on. This unit has been really long and tiring but i have learned a lot in this unit. The main thing you needed to do in this unit was pay attention. If you started to slack off you will get a lot of load on your shoulders, you needed to keep up or else you were in trouble. From parabolas to the quadratic formula everything connects. You will be shocked on how everything you learned in this unit connect together. This unit is like a puzzle, you have to use your mind. But the good thing about this unit was, you could check your answer, this way you would know if you are right or wrong. This unit has been a fun unit, it was fun to learn and TEACH.
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