Quadratics

BY: KAMILAH GAFOOR

WHAT IS A QUADRATIC?

To break it down:

-The prefix, "Quad" means a quantity of 4 (e.g quadruplets are 4 children)

-So the 4 represents a square and in quadratics, it means to be squared, as in x². This y = x² equation makes a U-shape curve on a graph that can travels upwards or downwards, this is now called the parabola.

Table of Contents

What is a Quadratic?

> Second Differences

> Parts of a Parabola

> What is Each Form & How Does it Relate?

> FOIL

> Transformations

The Vertex Form

> What is the Vertex Form?

> To Construct a Parabola With Vertex Form

> Step Pattern

> How to Find Zeros/ X-Intercepts Using Vertex Form

The Factored Form

> What is the Factored Form?

> How to Find the Vertex in Factored Form

> What is the Difference from Expanding and Factoring

> What is Factoring?

> Common Factoring

  • > Binominal Common Factoring
  • > Factoring By Grouping
  • > Simple Trinomial Factoring
  • > Complex Trinomial Factoring
  • > Special Factoring : Perfect Squares
  • > Special Factoring : Difference in Squares
  • Standard Form

    > What is Standard Form?

    > How to Get the X-Intercepts from Standard Form

    > Quadratic Formula & Discriminants

    > How to Get the Vertex by Converting Standard Form to Vertex Form

    Word Problems

    > Optimization / Area Problems

    > Revenue Problems

    > Motion Problems

    > Geometric Word Problems

    > Numbers Word Problems

    > Reflection

    INTRODUCTION

    The Difference Between Linear & Quadratic Relations (Second Differences)

    In a Linear Relation, there are an even amount of x-values in the first differences of an equation.


    But in a Quadratic Relation, there are an even amount of x-values in the second differences of an equation.

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    Parts of the Parabola

    Vertex - the point (x,y) of maximum/minimum OR the peak

    Optimal Value - the y-coordinate of the vertex

    Axis of Symmetry - the x-coordinate of the vertex

    Zero - the point(s) of where the line will meet to x-axis (the x-intercepts)

    Y-Intercept - the point(s) of where the line will meet the y-axis

    Direction of Opening - either Down(Valley)/Up(Peak) - where the parabola points up/down)

    Maximum/Minimum - the vertex can be positive/negative, positive = minimum and negative = maximum

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    Forms of Quadratics

    What Is Each Form & How Does It Relate?

    There are 3 types of expressions in Quadratics:

    1) Vertex Form

    a(x - h)² + k

    2) Factored Form

    (x -r)(x + s)

    3) Standard Form

    ax² + bx + c

    How Quadratic Forms Relate

    FOIL & Reverse FOIL Method

    FOIL METHOD & REVERSED FOIL METHOD
    NOTE: I made a mistake when doing the Reversed FOIL Method on the example, but with the one with a co-effiecent greater than one. even though I listed a pair of factors that corresponded to c but not bx, I forgot to multiply it to the 'x's. So instead of 13 it was really suppose to be 14x, which still didn't correspond to 11x.

    Tranformations

    What is a Vertical Stretch?

    - the effect of increasing the co-efficient of x²

    - it makes the parabola stretched and thinner


    e.g. 2x² OR 5x²


    What is a Vertical Compression?

    - the effect of decreasing the co-efficient of x²

    - it makes the parabola compressed and wider


    e.g. 0.5x² OR 1/2x²


    What is a Reflection?

    - the effect of a negative co-efficient of x²


    e.g. -1x² OR -0.5x²


    What is a Vertical Translation?

    - the effect of adding or subtracting a number to x²

    - this moves the parabola up or down the y-axis


    e.g. x² + k OR x² - k


    What is a Horizontal Translation?

    - the effect of adding or subtracting a number to x² but within brackets

    - this moves the parabola to the left (negative) or to the right (positive) side of the x-axis


    e.g. (x + k)² OR (x - k)²

    Vertex Form - a (x-h)² + k

    What is Vertex Form?

    CONNECTION:

    This Form can relate to Standard Form because it can convert to that form by expanding.

    This form can also relate to Factored Form because of its relationship between zeros and its relationship between the vertexes.


    In the equation, the vertex is given, and allows us to find the rest of the missing variables.


    In y = a(x-h)² + k

    The Optimal Value (y coordinate of the vertex) is k

    The Axis of Symmetry (x coordinate of the vertex) is h

    The Stretch Value is a


    E.g. y = (x + 2)² + 5


    => The number inside the bracket will be Axis of Symmetry (x-coordinate of vertex), and the number outside of the bracket will be the Optimal Value (y-coordinate of the vertex). It will be (2, 5)


    But...

    ***IMPORTANT***

    The +/- sign on the Axis of Symmetry (the x-coordinate of the vertex) will be reverse.

    [This is because it will cancel out in equation to give the y-intercept]


    Therefore, the vertex will be (-2, 5)

    To Contruct a Parabola With Vertex Form

    With the vertex, you will now construct your parabola, but you need coordinates to do so...


    After placing the vertex on the parabola, the Step Pattern will help find the missing coordinates:


    'The Step Pattern' was made to help find coordinates of a quadratic equation, using the vertex.


    (Over 1, Up 1, Over 2, Up 4)

    The Step Pattern

    This method is used to help identify the missing (x,y) values on a parabola.


    The Step Pattern is = "Over 1, Up 1. Over 2, Up 4"

    [Depending on the opening of direction, it could be up or down]


    The 'Overs' are the x-coordinates, and the 'Up/Down' are the y-coordinates.


    But this Step Pattern (Over 1, Up 1. Over 2, Up 4) only relates to y = x


    So if the quadratic equation doesn't have a co-efficient in front of x, then this is the method ("Over 1, Up 1. Over 2, Up 4")


    But if there is a co-efficient, then we must multiply that co-efficient by the "Over 1, Up 1. Over 2, Up 4" method.


    This is How:


    E.g. y = 2x² ----> 2 is the co-efficient


    "Over 1, Up 1(2). Over 2, Up 4(2)" = Over 1, Up 2, Over 2, Up 8

    *we do not multiply the x-coordinate because there are many possibilities (e.g. 1,2,3,4 etc.) , only the y-coordinate is needed*


    Therefore, for the equation, y = 2x², the Step Pattern is "Over 1, Up 2, Over 2, Up 8". And two coordinates would be (1, 2) and (2, 8).


    To continue, keep multiplying the co-efficient by the original Step Pattern.

    How to Find Zeros/ X-Intercepts Using Vertex Form

    In order to isolate for the x-intercepts (zeros) you sub in y = 0


    1) Write out the equation

    2) Sub in y=0

    3) Subtract or Add k from both sides of the equation, in order to eliminate k

    4) Square Root both sides of the equation, in order to eliminate the exponent

    5) Divide a from both sides of the equation, in order to eliminate a

    6) Take away the brackets of (x-h)

    7) Subtract or Add h from both sides of the equation, in order the eliminate h


    e.g. 2(x + 1)² - 16

    y = 2(x + 1)² - 16

    0 = 2(x + 1)² - 16

    0 + 16= 2(x + 1)²

    16/2 = 2(x + 1)²/2

    16= √(x + 1)²

    16 = (x + 1)


    NEGETIVE WAY:

    16 = (x + 1)

    4 = x - 1

    4 + 1 = x

    5 = x


    POSITIVE WAY:

    16 = (x + 1)

    4 = x + 1

    4 - 1 = x

    3 = x

    Therefore, the x intercepts of this example is x = 5 and x = 3

    With the x-intercepts and the vertex, you can easily construct the parabola!

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    Factored Form y = (x - r)(x + s)

    What is the Factored Form ?

    CONNECTION:

    To relate to the vertex form, the zeros/x-intercepts are visible in this form, and to find the vertex is uncovered. In the vertex form, the vertex is given, and the x-intercepts are unknown.


    This form can also relate to Standard Form because it is used to construct the FOIL Method, which can convert it into Standard Form

    (ax² + bx +c).


    But also when an equation is in this form, you can find the x-intercepts, which can lead to the axis of symmetry and optimal value (the vertex).


    In this equation:

    the r and s are the x-intercepts


    e.g. (x - 5)(x + 7)


    The 5 and 7 are the x-intercepts BUT the signs are reversed when in the bracket.


    This is because when finding 'x' , we sub y = 0 and change the signs to bring it to the other side of the equation. So, we change the signs (-/+) to cancel it out, thus meaning the x-intercepts' signs (-/+) are now flipped.


    Making the x-intercepts now (x = 5) and (x = -7)

    How to Find the Vertex in Factored Form

    After taking the x- intercepts, and flipping the signs (-/+), you then add the two together and divide it by 2 to find the Axis of Symmetry (x coordinate of vertex)


    eg. (x - 5)(x + 7) = y


    x = 5 x = -7


    5 + (-7) = x

    2

    -1 = x


    Then take the x value, sub it in the equation, and solve for y


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

    (-1 -5)(-1 + 7) = y

    (-6)(6) = y

    -36 = y


    Thus the vertex of the equation is (-1, -36)

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    Difference Between Expanding and Factoring

    There is a difference between these two, these terms have the opposite definition!


    Expanding: this means to reveal all operations in an equations, (e.g. exponents and brackets) so it expands in length


    BUT...


    Factoring: means to break down an equation, and make it simpler.

    What is Factoring?

    This is the opposite of expanding/distribution --> it is to make the expression simplified.


    • There are Different Types of Factoring:
    • Common Factoring
    • Binominal Common Factoring
    • Factoring By Grouping
    • Simple Trinomial Factoring
    • Complex Trinomial Factoring
    • Special Factoring : Perfect Squares
    • Special Factoring : Difference in Squares

    Common Factoring

    This form takes out a multiple that the variables/numbers share, then brings it outside the bracket so the expression is simplified.


    Eg. 2x² + 4x + 6


    All variables/numbers share a multiple, this is a COMMON Factor!


    So the expression is now--> 2(x²+ 2x+3)


    To check your answer, you must use DISTRUBUTIVE PROPERTIES to solve

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    Binominal Common Factoring

    Like Common Factoring, there was a number brought out of the brackets, now there will be two brackets that is the same, and we bring THAT bracket out to the front of the expression.


    The difference is that these contain TWO terms instead of ONE.


    *(REMINDER: Finding an expression like this is rare, there is a step before that is required to reach this stage, it is Factoring by Grouping, which will be discussed after)*


    Eg. 3x(x+5) -2(x+5)


    The brackets are the same, therefore instead of keeping both, reduce it to one by multiplying (COMMON FACTOR IT) . Then, put the remaining numbers in another bracket!


    So the expression is now --> (x+5)(3x-2)


    To check your answer, use the FOIL METHOD to solve

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    Factor By Grouping

    This expression has 4 terms. This is a combination of Common Factoring and Binominal Factoring.


    Eg, x^3 + 7x² + 3x +21


    The First Step is to break the 4 terms into 2 brackets of two terms.

    x^3 + 7x² + 3x +21 --> (x^3 + 7x²) + (3x +21)


    The Second Step is to Common Factor each bracket

    (x^3 + 7x²) + (3x +21) --> x²(x+7) + 3(x+7)


    The Third and Final Step is to Binominal Factor the expression

    x²(x+7) + 3(x+7) --> (x+7)(x²+3)


    So the expression is now --> (x+7)(x²+3)


    To check your answer, use the FOIL METHOD to solve

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    Simple Trinominal Factoring

    **This is DIFFERENT from Complex Trinomial because there is no co-efficient of x²**

    x² +4x + 3

    This expression had 3 terms and is usually in Standard Form. Using FOIL or "SPREADING THE RAINBOW" is used to make this trinomial expression (ax² +bx +c)


    Therefore, we need to convert Standard From into Factored Form (ax² +bx +c) to (a+b)(c+d)


    The relationship between the two forms is:

    1) the b +/- d in Factored Form makes a sum of bx in Standard Form

    2) the b x d in Factored Form makes a product of c in Standard Form

    3) the a & c in Factored Form makes a product of ax² in Standard Form

    *(in this type of trinomial, then a & c is just 'x')*


    The Video explains how to do the FOIL Method Reversed......

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    Complex Trinomial Factoring

    **This is DIFFERENT from Simple Trinomial Factoring because there is more than one co-efficient of x²*


    This expression had 3 terms and is usually in Standard Form. Using FOIL or "SPREADING THE RAINBOW" is used to make this trinomial expression (a² +bx +c)


    Therefore, we need to convert Standard From into Factored Form (a² +bx +c) to (a+b)(c+d)


    BUT the difference is, there could be more than one option for a & c of the Factored Form ---this is because there would be more than one set of multiples for a²---


    Eg. 2x² - 5x + 3 (ONE option)

    (2x_____)(x____)


    6x² + 20x +6 (TWO options)

    (6x____)(1x____) or (2x____)(3x____)


    The relationship between the two forms is:

    1) the b +/- d in Factored Form makes a sum of bx in Standard Form

    2) the b x d in Factored Form makes a product of c in Standard Form

    3) the a & c in Factored Form makes a product of a2 in Standard Form

    *(in this type of trinomial, then a & c must be multiples of a²)*


    The Video explains how to do the FOIL Method Reversed....

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    SPECIAL FACTORING : Perfect Squares

    This is an expression where the first and last term are the product of a squared number, and the middle term is the two squared numbers multiplied by 2.


    E.g. x²+ 8x +16 --> (x)(x) + 2(x)(4) + (4)(4)


    And when you Trinominal Factor it, it become (x+4)(x+4) --- which can also be (x+4)²


    It identify a Perfect Square, here are the steps :

    SDPS (Sam Doesn't Pull Strings) or (Squared Double Product, Squared)


    S: Square -- (x)(x) =


    DP: Double Product -- 2(x)(4) = 8x


    S: Square -- (4)(4) = 16

    Special Factoring : Difference of Squares

    In the expression, the Standard Form (a2 + bx + c) to Factored Form (a +b)(c + d) is a bit different. the b & d of the Factored Form are the reciprocals (3, -3)


    *Product of the Sum and Differences expand to give differences of squares*


    E.g.

    (a +b )(a +b) = a² + b²


    (2x -5)(2x+5) = (2x)(2x) + (-5)(5)


    Therefore, the expression is 4x² - 25

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    Standard Form

    What is the Standard Form?

    CONNECTION:

    This form can relate to Factored Form because, when using reverse FOIL Method, I can convert it into Factored Form. Also I can find the x-intercepts through Quadratic Formula.

    This Form can also relate to Vertex Form by completing the square, this way it is converted into Vertex Form.


    The Standard Form is y = ax² + bx + c

    This type of form can be converted into any form and from there, can attain the x- intercepts, axis of symmetry, optimal value and more.

    How to Get the X-Intercepts from Standard Form

    There are many ways in getting the x - intercepts in Standard Form, one way is converting it to Factored Form and using the Quadratic Formula.


    **This relates to the factored form because..**

    the a is stretch value

    the b is from the factored form (r + s)

    the c is from the factored form (r)(s)


    Therefore, the x-intercept are found as r and s.


    E.g. y = x² + 10x + 25


    Factored Form = (x + 5)(x + 5)


    The x-intercepts are (x = -5) and (x = -5)


    But factoring doesn't work all the time, especially if the equation has decimals points so there is now a formula that can be used anytime, the Quadratic Formula.


    What is the Quadratic Formula?

    The quadratic formula is a formula to find the x-intercepts when factoring does not work, and when there are decimals as variables.

    ax² +bx +c

    I

    I

    v

    -b +/- √-b² -4ac = x

    ...........2a...........


    Steps:

    1) Sub in all the variables

    2) Solve -b² and -4ac

    3) Then take the product of -b² and subtract/subtract it to the product of -4ac

    4) After, make two different equations

    -b + √-b² -4ac = x

    ...........2a...........

    AND

    -b - √-b² -4ac = x

    ...........2a...........

    5) Then square root the following

    6) After, ADD or SUBTRACT -b to the square rooted product

    7) Finally, Divide by 2a

    8) Repeat 6 & 7 again with SUBTRACTION or ADDITION of -b


    E.g. y = x² -7x + 12

    .......y = ax² + bx + c

    -b - /+√-b² -4ac = x

    ...........2a...........


    1) -(-7) -/ + √-7² -4(1)(12) = x

    .................2(1)...........


    2) 7 -/ + √49 - 48 = x

    .............2(1).........


    3) 7 -/ + √1 = x

    ........2(1)...


    4.1) 7 + √1 = x

    ..........2(1)....


    5) 7 + 1 = x

    .......2(1)....


    6) 8 = x

    ...2(1)....


    7) 8 = x

    ....2.


    8) 4 = x

    .

    OR

    .

    4.2) 7 + √1 = x

    ..........2(1)....


    5) 7 - 1 = x

    .......2(1)....


    6) 6 = x

    ...2(1)....


    7) 6 = x

    ....2.


    8) 3 = x


    Therefore, (x = 4) and (x = 3) in equation y = x² -7x + 12


    BUT when the discriminant, which is the final product before square rooting, is negative, there cannot be a solution due to error***

    How to Get the Vertex by Converting Standard Form to Vertex Form

    It also relates to vertex form because it can be converted by completing the square.


    Steps:

    1) Write out the equation in Standard Form

    2) But imaginary brackets around x² and bx (leaving out the stretch value)

    2) Then take 'b' divide it by 2, and square that quotient

    3) Now, put the new product made from Step 2, and put it in the imaginary bracket, after 'bx'

    *BUT you can never just add in a number in an equation, we must cancel it out*

    4) Now put it in the same number but with opposite signs in order to cancel it out

    5) Then, put that negative/positive number outside of the bracket and add it to 'c'

    6) After looking at the brackets, it should be a perfect square so factor it

    7) If there is a stretch value, it should remain the same.


    In the example below, the vertex is (-1, 88)

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    Con'td...

    However, there is an easier way to find out the vertex, using a shortcut.


    -b = x

    2a


    e.g. y = x² + 6x + 9

    -b = x

    2a


    -6 = x

    2(1)


    -6 = x

    (2)


    -3 = x


    Then to find the Optimal Value, sub in the Axis of Symmetry which was just recovered...


    y = x² + 6x + 9

    y = (-3)² + 6(-3) + 9

    y = 9 + (-18) + 9

    y = -9 + 9

    y = 0


    Therefore, the vertex is (-3, 0)

    Word Problems

    Optimization Word Problem

    Question:

    You have a 500-foot roll of fencing and a large field which is bordered on one side by a building. You want to contract a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?


    In this problem, I first, had to figure out the equations in order to find out the missing variables. I used my knowledge of substitution to help. Then I isolated 'x' to find my x-intercepts. Using the x-intercepts, I was able to find the Axis of Symmetry, by adding the two together and dividing it by 2. After, I used that to sub as my 'x' to find my optimal value. Finally I took my Axis of Symmetry and Optimal value and multiplied it together to find the maximum area of the playground.

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    Revenue Problem

    Question:

    A management firm had determined that 60 apartments in a complex can be rented if the monthly rent is $900 and that for each $50 increase in the rent, three tenants are lost with little chance of being replaced. What rent should be charged to maximize revenue? What's the maximum revenue?


    In this problem, I was able to make equations based on the written information. then I was able to isolate for 'x' to find my x-intercepts. After, I was able to find the Axis of Symmetry, by adding the two together and dividing it by 2. Then, I used that to sub as my 'x' to find my optimal value, which was the max. revenue.

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    Another Revenue Problem (Created)

    Revenue Problem Solution

    Motion Problem

    Question:

    The path of a football after it was kicked from the height of 0.3 m above the ground is given the equation h = -0.2d² + 2d +0.3, where h is the height , in metres, above the ground and d is the horizontal distance, in metres.


    *To do this question, you can do the quadratic formula to find the x-intercepts, and see which one would be on the positive side. This is because since this is a real life situation, negative numbers do not exist*


    dis.- height

    2m - 3.5m

    4m - 5.1m

    6m - 5.1

    8m - 3.5

    10m - 0.3m


    a) How far has the football travelled horizontally, when it landed on the floor?


    It landed n the floor at 10.1m as a distance


    b) Find the horizontal distance when the football was at the height of 1.5m above the ground.


    At the height of 1.5m, the distance was 2.8m


    c) What is the maximum height reached by the football? At what horizontal distance will it reach its height?


    0.3 + 10.1 = x 5.2m is the maximum height

    2

    5.2m = x

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    Geometric Word Problem (Rectangle)

    Question:

    The length of the rectangle is 5cm greater than its width. the area is 104cm². Find the dimensions of the rectangle.


    What I did for this question was change the equation for area and made y = 0, then used the quadratic formula to find x.

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    Numbers Word Problem

    Question:

    The product of two consecutive integers is 1482. what are these numbers?


    For this question, all I did was make the equation and isolate for 'x'.

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    Reflection

    This reflection is based upon my quiz on Quadratics 2. I chose this quiz because I did well on it but I did notice some mistakes that could have been easily fixed before. I also chose this quiz because I did poor on my Quadratic 2 Test, and I would like to reflect on how I would like to improve and how I want to fix my mistakes. In this section, I have struggled a lot with factoring. Though at times it can be easy, it is quite difficult because I am confused on which one goes with which equation and I sometimes mix it up with expanding.


    On the test I have a few mistakes in regards to factoring and expanding. For example, I did not do the expanding question correct, going back I see my mistake which was multiplying before the brackets. I had to use FOIL to make the factored form go into Standard Form, and then multiply the stretch factor after.


    Another mistake that I did was once again not expanding correctly. Now seeing this test, and my pervious test, I can now learn from my mistakes and starting emphasizing my time with practicing expanding questions.

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    All written work and watched videos were done by me!