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Unit III Quadratic Equations - · PDF file Unit III Quadratic Equations 1 Section 7.1 – Solving Quadratic Equations by Graphing Investigating Solutions to Quadratic Equations Example:

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  • Unit III Quadratic Equations 1

    Section 7.1 – Solving Quadratic Equations by Graphing

    Investigating Solutions to Quadratic Equations

    Example:

    A missile fired from ground level is modeled by the quadratic function

    h(t) = –20t2 + 120t, where h(t) represents height in meters and t is

    the time in seconds.

    (a) Indicate the positions along the quadratic trajectory above, where the rocket

    attains a height of 0?

    (b) For the function above, h(t) = –20t2 + 120t which variable is replaced

    by the height 0 for those positions?

    (c) Write the resulting quadratic when the variable mentioned above is

    replaced by 0.

    t

    h(t)

    Goal:

     Solving Quadratic Equations by Graphing

  • Unit III Quadratic Equations 2

    (d) Use graphing software (https://www.desmos.com/) to determine the times

    below where the rocket attains a height of 0.

    Remember:

    Zeros of a Quadratic Function

    What is a Quadratic Equation?

     A second degree polynomial equation.

     Standard Form of a quadratic Equation ax2 + bx + c = 0.

    t

    h(t)

    x

    y

    Zeros of a Quadratic Function

    ●The zero(s) of a quadratic function represent

    the position(s) where the height is _________

    ●The zero(s) of a quadratic function

    are also referred to as the _________________

    Solutions to Quadratic Equation

    ●Solutions to the quadratic equation

    –20t2 + 120t = 0 are the ________________

    of the quadratic function

    ●times where height is 0 are:

    t1 = _____ and t2 = ______

    h(t) = –20t2 + 120t

    https://www.desmos.com/

  • Unit III Quadratic Equations 3

    (e) Use the graph below to determine the times where the rocket attained a

    height of 160 m.

    t1 = _____ t2 = _____

    (f) Use the quadratic function h(t) = –20t2 + 120t to write the quadratic

    equation that represents the height at 160 m.

    t1 2 3 4 5 6 7 8

    h(t)

    20 40 60 80

    100 120 140 160 180 200

    h(t) = –2t2 + 120t

    How can we algebraically attain the time(s) to solve a quadratic

    equation ax2 + bx + c = 0 such as: ______________________ ?

  • Unit III Quadratic Equations 4

    Section 7.2 – Solving Quadratic Equations by Factoring

    (I) Solving Quadratic Equations by Factoring

    Example:

    A missile fired from ground level is modeled by the quadratic function

    h(t) = –20t2 + 120t, where h(t) represents height in meters and t is

    the time in seconds.

    t

    h(t)

    20 40 60 80

    100 120 140 160 180 200

    At a height of 160 m the quadratic function h(t) = –20t2 + 120t

    develops into a quadratic equation:

    160 = –20t2 + 120t

    We can determine the two times where the missile attains a

    height of 160 m by FACTORING the quadratic equation

    160 = –20t2 + 120t

    and using the zero product property to isolate and solve for t.

    Goal:

     Solving Quadratic Equations by Factoring

  • Unit III Quadratic Equations 5

    Solve for t by factoring:

    160 = –20t2 + 120t

    (II) Review of Factoring

    (A) Factoring by removing a common factor.

    Example: Solve by factoring.

    (i) –20x2 + 120x = 0 (ii) 12x2 = –8x

    Express in the form ax2 + bx + c = 0

    Remove the greatest common factor.

    Factor the remaining trinomial.

    To isolate t, apply the Zero Product Property

    If the product of two real numbers is zero ( a • b = 0)

    then one or both must be zero.

    In other words: a = 0 and b = 0

  • Unit III Quadratic Equations 6

    (II) Review of Factoring

    (B) Solving quadratic equations by the difference of two squares.

    Perfect Square Numbers or Factors to remember:

    22 = 4

    32 = 9

    42 = 16

    52 = 25

    62 = 36

    72 = 49

    82 = 64

    92 = 81

    102 = 100

    112 = 121

    122 = 144

    132 = 169

    142 = 196

    152 = 225

    Example: Solve by factoring.

    (i) 81x2 = 49 (ii) 121 – 4x2 = 0

    Perfect Square Numbers or Factors

    REMEMBER: The difference of two squares factors by pattern

    a2 – b2 = (a – b)(a + b)

  • Unit III Quadratic Equations 7

    (II) Review of Factoring

    (C) Solving quadratic equations of the form x2 + bx + c = 0.

    Example: Solve by factoring.

    (i) x2 + 3x = 18 (ii) –2p2 – 20p + 48 = 0

    (D) Solving quadratic equations of the form ax2 + bx + c = 0.

    Example: Solve by factoring.

    (i) 6x2 – 11x = 10 (ii) 8x2 = 22x – 15

    P.411 #1 – #4

  • Unit III Quadratic Equations 8

    Section 7.2 – Solving Quadratic Equations by Factoring

    (Day 2)

    (I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

    ●The roots of a quadratic equation represent the TWO

    x values that are solutions to ax2 + bx + c = 0.

    Example: Determine the ROOTS for:

    (a) x(3x + 7) = 6

    (b) –3.2x2 + 6.4x = –25.6 (c) − 1

    3 𝑥2 = −𝑥 + 6

    •Express each quadratic equation in

    standard form ax2 + bx + c = 0 and factor.

    •Set each factor equal to zero and solve

    each linear equation.

    Goal:

     Solving Quadratic Equations by Factoring

  • Unit III Quadratic Equations 9

    (VI) Applications of Quadratic Equation ax2 + bx + c = 0

    Example:

    The path of a missile shot into the air from a ship is modeled by the

    quadratic function y= –4.9x2 + 39.2x + 44.1 where y represents the

    height in meters and x is time in seconds.

    Algebraically determine:

    (a) How long will it take for the missile to hit the ocean?

    (b) How long will it take for the missile to attain a height of 122.5 metres?

    x

    y Quadratic Path is modeled by:

    y = –4.9x2 + 39.2x + 44.1

    http://www.google.ca/imgres?q=war+ships&hl=en&safe=active&biw=1024&bih=568&gbv=2&tbm=isch&tbnid=hH0m3J9Vc7hPTM:&imgrefurl=http://captainsblog-ryley.blogspot.com/&docid=QbaRKJtLzQ6H2M&imgurl=http://2.bp.blogspot.com/-sqghzeBwYrA/TZAzwrLdrfI/AAAAAAAAABI/ExOJd3W6A3U/s1600/Battleship_Bismarck_Firing_A_Salvo.jpg&w=1024&h=768&ei=QiG0Tq2KMsr40gGM4Nm0BA&zoom=1&iact=rc&dur=0&sig=116108977142721622264&page=21&tbnh=162&tbnw=209&start=170&ndsp=8&ved=1t:429,r:7,s:170&tx=88&ty=82

  • Unit III Quadratic Equations 10

    Height

    time

    10

    10

    20

    20

    30

    30

    40

    40

    50

    50

    Example:

    An osprey dives toward the water to catch a salmon. Its height above the water, in

    metres, t seconds after it begins its dive, is approximated by 2( ) 5 30 45h t t t   .

    Algebraically determine the time it takes for the osprey to reach a return height of

    20 m.

    Example:

    A travel agency has 16 people signed up for a trip. The revenue for the trip is

    modeled by the function

    R(x) = –100x2 + 800x + 38400

    where x represents the number of additional people to sign up. How many

    additional people must sign up for the revenue to reach $40000?

    P.411 – 413 #6, #8, #13, #14

    http://www.google.ca/imgres?q=osprey&start=133&um=1&hl=en&safe=active&sa=N&biw=1024&bih=498&tbm=isch&tbnid=kt3t2QzoP7RNWM:&imgrefurl=http://www.stories-for-children.ca/fishing.php&docid=91eulttajWkO8M&imgurl=http://www.stories-for-children.ca/resources/osprey_fishing.jpg?timestamp=1334791817921&w=629&h=419&ei=PhOPULXfHsSG0QGOrYDgDg&zoom=1&iact=hc&vpx=706&vpy=184&dur=250&hovh=183&hovw=275&tx=208&ty=155&sig=107937703016156733068&page=11&tbnh=121&tbnw=182&ndsp=14&ved=1t:429,r:13,s:133,i:176 http://www.google.ca/imgres?q=osprey&start=120&um=1&hl=en&safe=active&sa=N&biw=1024&bih=498&tbm=isch&tbnid=Dho7Hj4voCWqDM:&imgrefurl=http://www.theballroomblog.com/2012/04/transcript-parrot-vs-raptor/osprey-in-flight-tom-munson/&docid=wRiNrTte5feTHM&imgurl=http://www.theballroomblog.com/wp-content/uploads/2012/04/osprey-in-flight-tom-munson.jpg&w=800&h=596&ei=PhOPULXfHsSG0QGOrYDgDg&zoom=1&iact=hc&vpx=340&vpy=70&dur=953&hovh=194&hovw=260&tx=166&ty=104&sig=107937703016156733068&page=10&tbnh=143&tbnw=257&ndsp=13&ved=1t:429,r:10,s:120,i:124

  • Unit III Quadratic Equations 11

    Section 7.2 – Developing Quadratic Functions/Equations

    (Day 3)

    (I) Review of zeros and roots

    Determining the zeros of a quadra