12 - graphing quadratic eqations

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  • 8/14/2019 12 - Graphing Quadratic Eqations

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    Chapter 12

    W

    hile the straight lines

    of linear equations

    are easy to graph,they are not the most exciting.

    Graphing quadratic equations,

    however, opens the door to the

    wonderful world of parabolas.

    Chapter 12 introduces these

    U-shaped curves and givesyou the tools to graph them.

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    In this Chapter...Introduction to Parabolas

    Write a Quadratic Equation in Vertex Form

    Graph a Parabola

    Test Your Skills

    GraphingquationsEQuadratic

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    Introduction to Parabolas

    230

    Steepness or Flatness of a ParabolaParabola Basics

    When you graph a

    quadratic equation,you will end up witha U-shaped curveknown as a parabola.

    Note: A quadraticequation is an equationwhose highest exponentin the equation is two,such as x 2.

    The standard way to write

    a quadratic equation youwant to graph is in the formy = a(x h ) 2 + k , where a , hand k are numbers. Writinga quadratic equation in thisform provides information tohelp you graph the parabola.

    Note: To write a quadraticequation in the formy = a (x h) 2 + k , see page 232.

    The number represented

    by a indicates thesteepness or flatness ofa parabola. The largerthe number, whetherthe number is positiveor negative, the steeperthe parabola.

    The number represented

    by a also indicates if aparabola opens upwardor downward. When a isa positive number, theparabola opens upward.When a is a negativenumber, the parabolaopens downward.

    Remember being a kid and throwing that baseball

    as high as you could? The ball never landed inthe same spot from which you tossed it because,

    whether you noticed it or not, the ball always

    followed a curve up and then fell back down on a

    similar curve. Those curves formed a symmetrical,

    U-shaped curve called a parabola.

    When you graph a quadratic equationan

    equation in which the highest exponent is

    two, such as x2

    the graph of the equation isa parabola. Parabolas open either downward

    or upward.

    The lowest point in an upward-opening parabola

    or the highest point in a downward-openingparabola is called the vertex.

    While quadratic equations can be expressed

    in the form y= ax2 + bx+ c, for the purposes ofgraphing, parabolas are generally rewritten as

    y= a(x h)2 + k. This is called the vertex form

    of an equation. In this form, the variable a

    determines how steep a parabola will be and

    whether it will open upward or downward.The variables h and k represent the position

    of the vertex (h,k) in the coordinate plane.

    1

    -1

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    2

    1

    3

    4

    5

    -1-2-3-4-5 2 3 4 5

    y= 2(x 1)2 1

    1

    -1

    -2

    -3

    -4

    -5

    2

    1

    3

    4

    5

    -1-2-3-4-5 2 3 4 5

    y= (x 1)2 11

    2

    y= (x 1)2 11

    2

    y = 2(x 1)2 1

    y= 2(x 1)2

    1

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    Graphing Quadratic EquationsChapter 12

    231

    Vertex of a Parabola

    PracticeTip

    1

    -1

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    -4

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    2

    1

    3

    4

    5

    -1-2-3-4-5 2 3 4 5

    Determine the vertex of eachof the following parabolas andindicate if each parabola opens

    upward or downward. You cancheck your answers on page 264.

    1) y = (x 1) 2

    2) y = (x )2 5

    3) y = 3(x + 1) 2 + 2

    4) y = 4x 2 +

    5) y = x 2 + 2

    6) y = (x 2)2 3

    Can the graph of a quadraticequation with just one termbe a parabola?

    Yes. The graphs of eventhe simplest quadraticequations, which bydefinition must contain thex2 variable, form a parabola.For example, a graph ofthe equation y= x2 formsa parabola and can bewritten as y= 1(x 0)2 + 0in the vertex form. In thisequation, you will noticethat a = 1, so the parabolaopens upward, and becauseh and k both equal 0, thevertex is at the origin (0,0)in the coordinate plane.

    1

    -1

    -2

    -3

    -4

    -5

    2

    1

    3

    4

    5

    -1-2-3-4-5 2 3 4 5

    y= 2(x + 3)2 2

    y= 2(x 1)2 1

    y= 2(x 3)2+ 2

    1

    -1

    -2

    -3

    -4

    -5

    2

    1

    3

    4

    5

    -1-2-3-4-5 2 3 4 5

    y= 2(x + 3)2 1 y= 2(x 2)

    2 1

    y= 2(x + 3)2+ 2 y= 2(x 2)

    2+ 2

    x-axis

    y-axis

    The numbers

    represented by hand k , when writtenas (h,k ) , indicatethe location of thevertex of a parabola,which is the lowestor highest point of aparabola.

    The number represented by h

    indicates how far left or rightalong the x-axis the vertex ofa parabola is located from theorigin. The origin is the locationwhere the x-axis and y-axisintersect.

    Note: When a minus sign () appearsafter x in the equation, the parabolamoves to the right. When a plus sign (+)appears after x in the equation, the

    parabola moves to the left.

    The number

    represented by kindicates how far upor down along they-axis the vertex ofa parabola is locatedfrom the origin.

    Note: Positive knumbers move theparabola up. Negativek numbers move the

    parabola down.

    Note: If a minus sign () appearsafter

    xin the equation,

    hin

    (h,k) is a positive number. Ifa plus sign (+) appears after xin the equation, h in (h,k ) is anegative number.

    For example, in theequation y = 2(x + 3) 2 2 ,the vertex of the parabola islocated at (3 , 2) .

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    Write a QuadraticEquation in Vertex Form

    232

    Standard Form Example

    y = ax 2 + bx + c y = 2x 2 4x + 3

    Vertex Form Example

    y = a (x h )2 + k y = 2(x 1) 2 + 1

    Quadratic equations

    are often writtenin the standard formy = ax 2 + bx + c ,where a , b and c arenumbers and one sideof the equation is setto equal y .

    Note: A quadratic equationis an equation whosehighest exponent in the

    equation is two, such as x2

    .

    When you want to graph

    a quadratic equation, youwill want to write theequation in the vertexform y = a (x h) 2 + k ,where a , h and k arenumbers. Writing aquadratic equation in thisform provides informationto help you graph theequation.

    To change a quadratic

    equation from the standardform to the vertex form, youfirst need to identify thenumbers a , b and c in thestandard form of the equation.

    Note: If the x variable or cnumber does not appear inthe equation, assume the numberis 0 . For example, the equation

    y = 2x 2 4x is the same as

    y = 2x2

    4x + 0 .

    To determine the

    value of a in thevertex form of theequation, use thevalue of a from thestandard form ofthe equation, sincethe numbers will bethe same.

    In this example, aequals 2 .

    1 2

    Graphing a parabola is simple if you first change

    the quadratic equation from standard form(y= ax2 + bx+ c) to vertex form (y= a(x - h)2 + k).

    The vertex form of a quadratic equation gives you

    the direction that the parabola opens, as well as

    the coordinates of the vertexthe highest point

    of a parabola that opens downward or lowest

    point of a parabola that opens upward.To convert a quadratic equation to vertex form,

    you must determine the values of h and kthe

    value of a is the same in both equations. To

    determine the value of h, you use a simpleformula (h = ) that uses the standard form

    values of a and b. To determine the value of k,

    go back to the standard form of the equation.

    Replace the variable ywith the variable k and

    replace the xvariables with the value of h. You

    can then solve for the variable k in the equation.When you have all three valuesa, h and kyou can write the equation out in vertex form.

    b2a

    Standard Form Example

    y = ax 2 + bx + c y = 2x 2 4x + 3

    a = 2, b = 4, c = 3

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    Graphing Quadratic EquationsChapter 12

    233

    PracticeTip

    Write the following quadraticequations in vertex form. You cancheck your answers on page 264.

    1) y = x 2 2x + 3

    2) y = x 2 + 4x 1

    3) y = 2x 2 4x 3

    4) y = x 2 6x + 7

    5) y = x 2 + 3

    6) y = x 2 + 10x + 15

    How do I change an equation from vertex formto the standard form of a quadratic equation?

    If you multiply and then simplify the terms of

    an equation in vertex form, you will arrive atthe standard form of the quadratic equation.This is a great way to check your answer afterconverting an equation from standard form tovertex form. For information on multiplyingpolynomials, see page 154.

    3 4 5To determine thevalue of

    hin the

    vertex form of theequation, use thenumbers a and b fromthe standard form ofthe equation in theformula h = .

    In this example, hequals 1 .

    In the vertex formof the equation,replace the numbersyou determined fora , h and k .

    You have finishedchanging the quadraticequation from thestandard form to thevertex form.

    To determine the valueof

    k, use the standard

    form of the equation andreplace the variable y withthe variable k and replacethe x variables with thevalue of h you determinedin step 3. Then solve for kin the equation.

    In this example, k equals 1 .

    b2a

    h =

    = = = 14

    4(4)

    2 x 2

    b2a

    k = 2h 2 4h + 3

    = 2(1)2 4(1) + 3

    = 2 4 + 3

    = 1

    y = a (x h )2 + k

    y = 2(x 1)2 + 1

    y = 2(x 1) 2 + 1

    y = 2(x 1)(x 1) + 1

    y = 2(x2

    2x + 1) + 1y = 2x 2 4x + 2 + 1

    y = 2x 2 4x + 3

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    Graph the equationy = 2(x 1)2 3.

    y = a (x h )2 + k

    y = 2(x 1)2 3

    Location of parabolas vertex:(1,3)

    To determine the location ofthe vertex of the parabola,which is the lowest or highestpoint of the parabola, look atthe h and k numbers in theequation.

    Write the h and k numbers asan ordered pair in the form(h ,k ) . An ordered pair is twonumbers, written as (x ,y) ,which gives the location of a

    point in the coordinate plane.

    Note: If a minus sign ( )appears after x in theequation, the value of hin (h ,k ) is a positivenumber. If a plus sign (+)appears after x in theequation, the value of hin (h ,k ) is a negativenumber.

    Plot the point forthe parabolasvertex in thecoordinate plane.

    Note: To plot pointsin a coordinateplane, see page 82.

    To determine if the parabola willopen upward or downward, lookat the number represented by a .When the value of a is a positivenumber, the parabola will openupward. When the value of a isa negative number, the parabolawill open downward.

    In the equationy = 2(x 1) 2 3 ,the parabola will open upwardsince the value of a , or 2 , is a

    positive number.

    3 41

    2

    When you graph a quadratic equation in the form

    ofy= ax2 + bx+ c in the coordinate plane, you willalways end up with a U-shaped curve known as a

    parabola. For information on the coordinate

    plane, see page 80.

    When you want to graph a quadratic equation, it

    is useful to have the equation written in vertex

    form (y= a(x h)2 + k). The vertex form tells you

    whether the parabola opens upward or downwardand provides you with the coordinates of the

    parabola's vertexthe highest or lowest point of

    the parabola. In an equation in vertex form, if

    the value of a is positive, the parabola opensupward, while if the value of a is negative, the

    parabola opens downward. The values h and k

    represent the coordinates (h,k) of the parabolas

    vertex.

    After you plot the vertex, plotting one point oneither side of the vertex is sufficient to draw the

    rest of the curve. If you want to make sure youdid not make a mistake, you can plot more

    points.

    Graph a Parabola

    234

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    -1-2-3-4-5 2 3 4 5

    x-axis

    y-axis

    (1,3)

    y= 2(x 1)2 3

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    PracticeTip

    Graphing Quadratic EquationsChapter 12

    Graph the following parabolas. You

    can check your graphs on page 265.1) y = (x 1) 2 + 3

    2) y = (x + 2) 2 4

    3) y = x 2 1

    4) y = x 2 + 3

    5) y = (x 2) 2 1

    6) y = 2(x + 3) 2

    Can graphing a parabola help me solve

    a quadratic equation?Yes. Look at where a parabola crossesthe x-axis. The values forxat thosepoints will give you the solutions tothe quadratic equation. If the parabolacrosses the x-axis at two points, theequation has two solutions. If theparabola touches the x-axis at just onepoint, the equation has one solution.If the parabola does not cross the x-axis,the equation has no real solutions.

    Choose a randomnumber for the xvariable. For example,let x equal 0 .

    Place the number youselected into theequation to determinethe value of the yvariable. Then solvefory in the equation.

    Plot the two points ina coordinate plane.

    Connect the points todraw a smooth curve.Draw an arrow at eachend of the curve toshow that the parabolaextends forever.

    The parabola shows allthe possible solutionsto the equation.

    Write the x andy valuetogether as an orderedpair in the form (x ,y ) .

    Repeat steps 5 to 7to determine anotherordered pair. The orderedpair should be locatedon the other side of theparabolas vertex so youhave one point on eitherside of the parabolas

    vertex.

    5

    6

    7

    235

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    108

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    -1-2-3-4-5 2 3 4 5

    x-axis

    y-axis

    (0,1) (2,1)

    y= 2(x 1)2 3

    Letx = 0

    y = 2(x 1) 2 3

    y = 2(0 1)2

    3y = 2 3

    y = 1

    Ordered pair(x,y) = (0,1)

    Letx = 2

    y = 2(x 1) 2 3

    y = 2(2 1)2 3

    y = 2 3

    y = 1

    Ordered pair(x,y) = (2,1)

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    236

    Determine the vertex of the following parabolasand whether the parabolas open upward or

    downward.

    Question 1.

    Graphing Quadratic Equations

    a) y = (x 1)2

    + 2

    b) y = (x + 6)2

    3

    c) y = 4(x + 4)2

    +

    d) y = (x )

    2

    e) y = 200(x 50)2

    75

    Write the following quadratic equations in vertexform. Determine the vertex of each parabola andwhether the parabola opens upward or downward.

    Question 2.

    a) y = x2

    2x + 1

    b) y = x2

    + 6x + 5

    c) y = x2

    2x + 2

    d) y = 4x2

    + 4x + 3

    e) y = 25x2

    + 20x + 16

    Test Your Skills

    12

    12

    13

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    237

    Graph the following parabolas.Question 3.

    a) y = (x + 1)

    2

    2

    b) y = (x 3)2

    + 1

    c) y = 2x2

    8x + 5

    d) y = x2

    + 4x

    e) y = 2x2

    12x 14

    Graphing Quadratic EquationsChapter 12

    You can check your answerson pages 280-281.