Conics - Parabola

Conics - Parabola

Lessons

parabola: a curve formed from all the points that are equidistant from the focus and the directrix.
vertex: midway between the focus and the directrix
focus: a point inside the parabola
directrix: a line outside the parabola and perpendicular to the axis of symmetry

conics formula for parabola:
p=14ap = \frac{1}{{4a}} p: distance between the vertex and the focus / directrix.
a: coefficient of the squared term
  • 1.
    vertical parabola VS. horizontal parabola
    Sketch the following vertical parabolas:
    i) y=x2y = {x^2}
    ii) y=2x2y = 2{x^2}
    iii) y=2(x+3)2+1y = 2{\left( {x + 3} \right)^2} + 1

  • 2.
    Sketch the following horizontal parabolas:
    i) x=y2x = {y^2}
    ii) x=12y2x = \frac{1}{2}{y^2}
    iii) x=12(y1)23x = \frac{1}{2}{\left( {y - 1} \right)^2} - 3

  • 3.
    converting quadratic functions to vertex form by “completing the square”
    Convert each quadratic function from general form to vertex form by completing the square.
    a)
    y=2x212x+10y = 2{x^2} - 12x + 10

    b)
    y210y4x+13=0{y^2} - 10y - 4x + 13 = 0


  • 4.
    finding the focus and directrix using the formula: p=14ap = \frac{1}{{4a}}
    For each quadratic function, state the:
    i) vertex
    ii) axis of symmetry
    iii) focus
    iv) directrix
    a)
    y=18(x6)2+3y = \frac{1}{8}{\left( {x - 6} \right)^2} + 3

    b)
    12(x+1)=(y+4)2 - 12\left( {x + 1} \right) = {\left( {y + 4} \right)^2}

    c)
    y210y4x+13=0{y^2} - 10y - 4x + 13 = 0


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