Quadratics

By Nisha Sabharwal

What Does Quadratics Mean:

The name Quadratic comes from "quad" meaning square, because the variable gets squared. Quadratic equations make a smooth curve.

Quadratic Examples in Real Life

Reflection:

In the beginning of this unit when we started with quadratics 1 I was struggling with what to do but when we started quadratics 2 I got a better understanding and once I understood what to do in quadratics 2, quadratics 3 was much easier. This improvement didn't however show on my tests because I was still somewhat unclear with quadratics 1 and that made me unsure on my answers because they would be related on the tests.

In this quadratic unit i learned about parabolas and many ways to solve quadratic relations. In quadratics 1 I was taught about the basics of a parabola; how to read it, what it means when a parabola is opening upwards or downwards, finding the vertex, axis of symmetry, and x and y intercepts. Then in quadratics 2 I added to my knowledge of parabolas to make quadratic relations and equations and learned quadratic equations; the vertex form, standard form, and factored form. I also learned how to rearrange all these forms to make another such as changing from standard form to factored form. Lastly in quadratics 3 I was taught how to solve these quadratic equations and also introduced to the quadratic formula. In conclusion all three quadratics were interconnected with each other and breaking them apart made it easier to learn considering it was such a long unit.

Quadratic Equations:

  1. Vertex form: y=a(x-h)2+k
  2. Standard form: y=ax2+bx+c
  3. Factored form: y=a(x-r)(x-s)

Quadratics Terms:

First Differences

  • Optimal value
  • Axis of Symmetry (A.O.S)
  • x-intercepts/zeros
  • Exact roots
  • Approximate roots
  • Discriminants
  • Parabola

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    Standard Form:

    y=ax2+bx+c


    Zeros:

    Quadratic Formula (-b+- square root of b2-4ac/2a)

    5x2-7x+2=0

    • 5 is the a term 7 is the b term and 2 is the c term
    7 +- square root of (-7)2-4(5)(2)/2(5)

    • since 7 was a negative in the equation it gets changed to a positive

    7 +- square root of 9/10

    7 + square root of 9/10

    =1

    7 - square root of 9/10

    =0.4

    the zeros are (1, 0.4)


    Axis of Symmetry:

    -b/2a

    -(-7)/2(9)

    7/10

    =0.7


    Optimal Value:

    sub in 0.7 into the equation

    y=5(0.7)2-7(0.7)+2

    y=0


    Completing the Square:

    y=x2+8x+4

    y=(x2+8x=42-42)+4

    y=(x2+8x+42)-42+4

    y=(x+4)2-12


    • (2 represents squared)

    Factoring

    Factoring

    Finding the Vertex:

    The vertex is made up of the h and k value in the equation.

    Example: y=(x+6)2+12

    Vertex: (-6,12)


    *If the h value is positive in the bracket, when plotting your vertex it turns negative. And if in the bracket it is negative, when plotting the h value turns positive (i.e. y=2(x-8)2-9, vertex= (8,-9)). The k value stays the same when plotting

    A.O.S. and Optimal Value:

    • A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
    • Optimal value=the k value of vertex equation


    Example: y=-(x+5)2+9

    • A.O.S= 5
    • Optimal Value= 9

    How to Find First and Second Difference

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    Note:

    • if first difference the same=linear relation
    • if second difference same=quadratic relation (parabola)
    • if no difference the same=neither

    Reading Parabolas:

    y=a(x-h)2+k

    *2 stands for squared



    a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).


    h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).


    k=how many units the parabola vertically moves (-#=moves down, +#=moves up).


    Vertex= (h,k) the maxima or minima of the curve(parabola)

    Big image

    Transformations

    a= if the parabolas opens up or down (negative=opens down and positive=opens up), compressed or stretched (if a decimal or fraction=compressed, if a whole #=stretched).


    h=how many units the parabola horizontally moves (-#=moves to the left, +#=moves to the right).


    k=how many units the parabola vertically moves (-#=moves down, +#=moves up).


    Vertex= (h,k) the maxima or minima of the curve(parabola)

    Finding Equations in Vertex Form

    When working with vertex form, we are usually looking to solve for a. With many problems regarding this form the h,k(vetex),x,and y(another point) values will be given, as you have to do is sub in all the values bring the numbers to one side solve for a. Once get your values write your equation with the h,k,and a values.

    Finding Zeros/x-intercepts and y-intercepts in Vertex Form:

    *when finding the y-intercept, always sub x=0


    Example: y=-4(x+2)2+16


    • y=-4(x+2)2+16
    • y=-4(0+2)2+16
    • y=-4(2)2+16
    • y=-4(4)+16
    • y=-16+16
    • y=0


    *when solving to find the x-intercepts, remember to always sub y=0

    • y=-4(x+2)2+16
    • 0-16=-4(x+2)2
    • -16/-4=(x+2)2
    • 4=(x+2)2
    • take square root on both sides
    • +2 or -2=x+2
    • 0 or -4=x


    therefore the x intercepts are 0 and -4

    Step Pattern

    The step pattern determines where to plot your points in the parabola. To determine you step pattern, you must know what a equals. Once you know your a value then, multiply your a value with 1,3, and 5 to get your step pattern.
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    A.O.S

    • A.O.S=the h value of vertex equation, it is also the point the divides the parabola in half.
    • Optimal value=the k value of vertex equation


    Example: y=-(x+5)2+9

    • A.O.S= 5
    • Optimal Value= 9

    Quadratics in Factored Form

    Factored form: y=a(x-r)(x-s)

    *r and s represent zeros (x-intercepts)


    1. find zeros
    2. find A.O.S
    3. find optimal value

    Example:

    y=-2(x+4)(x+6)

    1. Find zeros:
    • x+4=0 x+6=0
    • x+4-4=0-4 x+6-6=0-6
    • x=-4 x=-6


    2. A.O.S= -4+(-6)/2

    • = -10/2
    • = -5 (x value of vertex)
    • 3.Optimal Value (sub -5 as x in original equation)
    • y=-2(-5+4)(-5+6)
    • y=-2(-1)+(1)
    • y=2 (y/k value of vertex)


    Vertex: (-5,2)

    Vertex form: -2(x+5)2+2

    Difference of Squares

    When working with difference of squares, you are multiplying two binomials to get a binomial.

    Formula: a2-b2=(a+b)(a-b)


    when expanding (a+b)(a-b):

    • (a+b)(a-b)
    • a2-ab+ab-b2
    • a2-b2

    Word Problems

    Word Problems

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